L IBOR M ARKET M ODELS - T HEORY AND A PPLICATIONS B Y D AVID G LAVIND S KOVMAND A dissertation submitted to T HE F ACULTY OF S OCIAL S CIENCES In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics and Management U NIVERSITY OF A ARHUS D ENMARK
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L I B O R M A R K E T M O D E L S
-
T H E O R Y A N DA P P L I C A T I O N S
B Y D A V I D G L A V I N D S K O V M A N D
A dissertation submitted to
T H E F A C U L T Y O F S O C I A L S C I E N C E S
In partial fulfillment of the requirements for the degree of
Doctor of Philosophyin
Economics and Management
U N I V E R S I T Y O F A A R H U S
D E N M A R K
Table of ContentsPreface v
Summary vii
Dansk Resumé (Danish Summary) ix
Chapter I 1The Valuation of Callable CMS-spread bonds with floored coupons
Chapter II 53Fast and Accurate Option Pricing in a Jump-Diffusion Libor Market Model
Chapter III 107Alternative Specifications for the Lévy Libor Market Model: An EmpiricalInvestigation
Preface
This thesis was written in the period October 2004 to January 2008, during mystudies at the School of Economics and Management at the University of Aarhus. Inthis period I have spent an academic year at the Haas School of Business, Universityof California, Berkeley. The last 4 months have been spent at my new employer, theAarhus School of Business, University of Aarhus.
A number of people have contributed to the making of this thesis. First andforemost I thank my main advisor Professor Bent Jesper Christensen, who draftedme to the PhD program and sparked my interest in research. Secondly my co-advisorProfessor Niels Haldrup I also thank for solid guidance along the way.
A large thanks goes to my informal advisor, cooperator, and friend ProfessorPeter Løchte Jørgensen. Without our discussions and your advice on matters aca-demic, as well as real-world, I would never have gotten this far.
From September 2006 to May 2007 I visited the Haas School of Business atUniversity of California, Berkeley. My stay was extremely pleasant and I wouldlike to thank Professor Greg Duffee for inviting me and Michael Verhofen for goodcollaboration and very fruitful discussions.
I am also grateful to my new co-workers in the Finance Research Group atthe Aarhus School of Business, especially Elisa Nicolato and Thomas Kokholm, forhelping and encouraging me in the final stages of completing the thesis.
The good people at Scanrate Financial Systems I thank for providing me withadvice, data and friendship over the years. I look forward to continuing our collab-oration.At the University of Aarhus I thank my fellow PhD students for providing a friendlywork environment as well as excuses to occasionally step away from the computer.The administrative staff I thank for assistance with practical matters concerningteaching and traveling. Søren Staunsager and his staff should also be mentioned inhelping me numerous times with computer issues. Finally I thank my family andfriends for putting up with me and for general encouragement, love and support.
David Glavind Skovmand, Arhus, January 2008
v
Updated Preface
The pre-defense took place on April 1, 2008.I am grateful to the members of the assessment committee, Klaus Sandmann, ClausMunk and Peter Ove Christensen, for their careful reading of the dissertation andtheir many useful comments and suggestions. Many of the suggestions have beenincorporated in the present version of the thesis while others remain for future workon the chapters.
David Glavind Skovmand, Arhus, April 2008
vii
Introduction
This thesis consists of three self contained chapters with a common theme of pric-ing of interest rate derivatives using the Libor Market Model of Miltersen, Sand-mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997) and Jamshidian(1997). This model has become a favorite among banks and other financial institu-tions for pricing derivatives and managing risk. The original papers have spawneda new branch of academic literature that deals with the many theoretical as well aspractical issues surrounding the Libor Market model and its extensions. This thesiscan be considered an attempt at contributing to that paradigm.
Summary of Chapters
Chapter I: The Valuation of Callable Bonds with FlooredCMS-spread coupons
(With Peter Løchte Jørgensen)Published: Wilmott Magazine, December 2007In this paper we investigate a new type of structured bond that has recently beenintroduced with enormous success - primarily among private investors - in manycountries in Europe. The bonds are medium term and with fixed and very highinitial coupons. The remaining coupons are determined as a constant multipliertimes the spread between a long and a short swap interest rate. These coupons arefloored at or near zero, and the bond investment can thus be seen as a bet on thesteepening of future term structure curves. However, if the term structure becomestoo steep, the bonds may be called by the issuer. The paper studies the pricing andthe optimal call strategy of these highly exotic bonds in a stochastic interest rateframework. We implement two versions of the LIBOR Market Model as well as aGaussian two-factor short rate model. We show how to adapt the Least-SquaresMonte Carlo procedure to handle the callability of the product in a numericallyefficient manner. We also calculate lower bounds for the product as well as deltaand vega ratios.
Chapter II: Fast and Accurate Option Pricing in aJump-Diffusion Libor Market Model
Submitted to the Journal of Computational FinanceThis paper extends, improves and analyzes cap and swaption approximation formu-lae for the jump-diffusion Libor Market Model derived in Glasserman and Merener(2003).
ix
More specifically, the case where the Libor rate follows a log-normal diffusion pro-cess mixed with a compound Poisson process under the spot measure is investigated.The paper presents an extension that allows for an arbitrary parametric specificationof the log-jump size distribution, as opposed to the previously studied log-normalcase. Furthermore an improvement of the existing swaption pricing formulae in thelog-normal case is derived. Extensions of the model are also proposed, includingdisplaced jump-diffusion and stochastic volatility.The formulae presented are based on inversion of the Fourier transform which isapproximated using the method of cumulant expansion. The accuracy of the ap-proximations is tested by thorough Monte Carlo experiments and the errors arefound to be at acceptable levels.
Chapter III: Alternative Specifications for the Levy LiborMarket Model: An Empirical Investigation
This paper introduces and analyzes specifications of the Levy Market Model origi-nally proposed by Eberlein and Ozkan (2005). An investigation of the term structureof option implied moments shows that the Brownian motion and homogeneous Levyprocesses are not suitable as modelling devices and consequently a variety of moreappropriate models is proposed. Besides a diffusive component the models havejump structures with low or high frequency combined with constant or stochasticvolatility. The models are subjected to an empirical analysis using a time series ofdata for Euribor caps. The results of the estimation show that pricing performancesare improved when a high frequency jump component is incorporated. Specifically,excellent results are achieved with the 4 parameter Self-Similar Variance Gammamodel which is able to fit an entire surface of caps with an average absolute per-centage pricing error of less than 3%.
x
Dansk Resume(Danish Summary)
Denne afhandling indeholder tre selvstændige kapitler som alle omhandler prisfast-sættelse af rentederivater ved brug af Libor Market Modellen oprindeligt beskrevet iMiltersen, Sandmann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997)og Jamshidian (1997). Denne model er blevet utrolig populær blandt banker ogandre finansielle institutioner til derivatprisfastsættelse og risikostyring. De treoriginale artikler har skabt en ny gren af akademisk litteratur som omhandler demange teoretiske savel som praktiske aspekter ved Libor Market Modellen og densudvidelser. Denne afhandling kan betragtes som et forsøg pa at bidrage til detteparadigme.
xi
References
Brace, A., D. Gatarek, and M. Musiela (1997): “The Market Model ofInterest Rate Dynamics,” Mathematical Finance, 7(2), 127–154.
Eberlein, E., and F. Ozkan (2005): “The Levy Libor Model,” Finance andStochastics, 9, 327348.
Glasserman, P., and N. Merener (2003): “Cap and swaption approximationsin Libor market models with jumps,” Journal of Computational Finance, 7(1),1–36.
Jamshidian, F. (1997): “Libor and Swap Market Models and Measures,” Financeand Stochastics, 1(4), 261–291.
Miltersen, K. R., K. Sandmann, and D. Sondermann (1997): “Closed formsolutions for term structure derivatives with log-normal interest rates,” Journalof Finance, 52(2), 409–430.
xii
Chapter I
The Valuation of Callable Bonds with Floored
CMS-spread coupons ∗
David Skovmand,†
University of Aarhus and CREATES
Peter Løchte Jørgensen‡
Aarhus School of Business
May 1, 2008
Abstract
A new type of structured bond has recently been introduced with enormoussuccess - primarily among private investors - in many countries in Europe. Thebonds are medium term and with fixed and very high initial coupons. Theremaining coupons are determined as a constant multiplier times the spreadbetween a long and a short swap interest rate. These coupons are flooredat or near zero, and the bond investment can thus be seen as a bet on thesteepening of future term structure curves. However, if the term structurebecomes too steep, the bonds may be called by the issuer. The paper studiesthe pricing and the optimal call strategy of these highly exotic bonds in astochastic interest rate framework. We implement two versions of the LIBORMarket Model as well as a Gaussian two-factor short rate model. We show howto adapt the Least-Squares Monte Carlo procedure to handle the callability ofthe product in a numerically efficient manner. We also calculate lower boundsfor the product as well as delta and vega ratios.
∗Financial support from the Danish Mathematical Finance Network is gratefully acknowledged.The authors are grateful for helpful comments from Mark Joshi, Svend Jakobsen, B.J. Christensen,and Malene Shin Jensen as well as from participants at the March 2007 World Congress on Com-putational Finance in London, the conference on Quantitative Methods in Finance in Sydney,December 2005, and at a seminar at Cass Business School in London. Any remaining errors arethe authors’ responsibility.
†Current affiliation: Aarhus School of Business and the Center for Research in EconometricAnalysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus Schoolof Business, Department of Business Studies, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark,e-mail: [email protected]
‡Address: Aarhus School of Business, Department of Business Studies, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark, e-mail: [email protected]
1 Introduction
The market for complex interest rate dependent products has been growing rapidlyin the past two years. Fueled by the low interest rate environment investors havebeen seeking higher returns on their investments which has caused banks to comeup with more and more complicated products to sell.
This paper investigates one of these products called the callable Constant Matu-rity Swap (CMS)-steepener. This type of product is typically sold as a medium termbond. The bond provides the investor with a high fixed coupon (typically 6-10%)for an initial period but after this initial period the bond pays a floating coupondetermined as some multiplum of the spread between a long and short constantmaturity swap rate determined in arrears. The coupons are floored at or near zeroto avoid negative payments. The bond can also be called at par on coupon datesafter the initial period making it a Bermudan type product.
From the investor’s point of view this type of product is essentially a bet on thesteepening of the term structure. Other than in the unlikely event of issuer default,the worst case scenario for the investor is being locked into a below market couponuntil maturity of the bond. The high initial coupons are meant to compensate forthis risk. The issuer has a limited downside since he can call the bond at par if theCMS spread coupons he has to pay become too high.CMS spread related products (callable and non-callable) were extremely popularwith investors in 2005 with an estimated $50 billion sold on a Worldwide basis (seeJeffery (2006)). However, the recent flattening of the term structure of interest rateshas caused a significant drop in the secondary market value of the products. Thishas led to a wide criticism of issuers from the investor community where many feelthey have purchased something they did not understand. Indeed the largest pool ofbuyers have been unsophisticated retail investors.
Adding to the controversy banks have also been reluctant to disclose the profitsmade from CMS Steepeners. According to Sawyer (2005) there have even beensuggestions that the banks themselves have had difficulties with both pricing andmanaging risk.
In the Nordic region sales of CMS Steepeners have been particularly high. Themarket was primarily triggered by the huge success of Forstædernes Bank (Forbank)who – in cooperation with Dexia Bank – issued a DKK 2.4 Billion ($400 Million)callable CMS Steepener in the beginning of 2005. We use this particular issue as acase of study in our investigation of the products.
The academic literature has dealt with these products only on a very abstractlevel where the most notable is Piterbarg (2004b). In a sense any callable bondcan be viewed as a non-callable bond subtracted the value of the issuer’s embeddedoption to call the bond at par. Since the bond can be called only at discrete times
2
this embedded option is a Bermudan. Therefore Piterbarg (2004b) uses the well-known method of Least-Squares Monte Carlo (LSMC) (see Longstaff and Schwartz(2001), Carriere (1996), Tsitsiklis and Roy (1999) and Stentoft (2004)) to price theoption part independent from the product itself. In this paper we propose a simplerand more efficient way of pricing these bonds that is based on pricing the entirebond instead of splitting it into a non-callable and an option part. We present thetechniques in a framework that is general enough to be used for any issuer callablebond.
A well-known issue in pricing American or Bermudan options using the LSMCprocedure is finding the proper exercise strategies. This has been widely studied inthe case of simple fixed for floating Bermuda swaptions (Andersen (2000) and Sven-strup (2005)). In the case of CMS spread callables this problem is more complex,and we therefore perform a numerical investigation to evaluate the quality of differ-ent exercise strategies. The technology to perform this quality check is developedin the papers of Andersen and Broadie (2004) and Haugh and Kogan (2004). Theintuition behind these papers is basic: The price of any option is equal to the valueof the hedge portfolio, but to hedge a Bermudan option we would need an optimalstopping rule. The LSMC procedure only provides us with an estimated stoppingrule. The present value loss or duality gap from following a suboptimal stopping ruledetermines the quality of the stopping rule approximation. We adapt the procedurein Andersen and Broadie (2004) and show both theoretically and numerically howthe duality gap can be calculated in the case of issuer callable bonds.
We would not expect the pricing of these products to be independent of theunderlying assumption of the distribution of interest rates. We therefore implementthree different models of the term structure: The simple Gaussian short rate G2++model of Brigo and Mercurio (2006) as well as the log-normal (see Miltersen, Sand-mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997), and Jamshid-ian (1997)) and the constant elasticity of variance (CEV) extended version of theLIBOR market model (see Andersen and Andreasen (2000) and Hull and White(2001)). We discuss the different properties of these models as well as the specificsof their calibration.
Perhaps even more important than pricing are the sensitivities of the product– the deltas and vegas. Calculating these in the LIBOR market model is a non-trivial exercise. We show how the pathwise approach to sensitivities in Glassermanand Zhao (1999), Piterbarg (2004a), and Piterbarg (2004b) can be adapted to ourspecific case and our results are consistent with intuition.The paper is arranged as follows. First, we fix notation and describe the CMSSteepener in greater detail. Secondly, we describe the different models of the termstructure. Third, we outline the LSMC procedure as well as a theoretical derivationof the duality gap of a generic suboptimal strategy. Fourth, we show how hedge
3
ratios are calculated and outline the numerical procedure which ends the theoreticalpart of our paper. Next we deal with the calibration and numerical implementationof the different models of the term structure. Finally, we present the numericalresults and discuss the implications. We end with a conclusion.
2 Basic Setup
In this section we introduce some basic notation and briefly recapitulate some centralvaluation expressions regarding simple interest rate derivatives. These are usedlater in the paper mainly in relation to market calibration of our various dynamicmodels. The present section also describes the CMS Steepener and its valuation onan abstract level.
We assume a standard probability space (Ω,F , P) with sigma-algebra filtrationFt∞t=0 and real world probability measure P. We also make the standard assump-tions of no arbitrage and frictionless markets.
For a tenor structure T0 < . . . . . < TM and day-count fractions τk = Tk−Tk−1 wedefine the discretely compounded time t forward LIBOR rates which apply betweentimes Tk−1 and Tk as
Fk(t) = F (t, Tk−1, Tk) =P (t, Tk−1) − P (t, Tk)
τkP (t, Tk), (1)
where P (t, Tk) denotes the time t price of a zero coupon bond which matures at Tk.A call option on the forward rate giving the payoff τk(Fk(Tk−1) − K)+ at Tk is
referred to as a caplet. The time t value of the caplet can be expressed as
Cpl(t) = P (t, Tk)τkEFk
t [(Fk(Tk−1) − K)+], (2)
where Fk refers to the forward measure with corresponding numeraire P (t, Tk) (seee.g. Bjork (2004)). As is well-known, the forward rate Fk(t) must be a martingaleunder this measure.
A Tα × (Tβ − Tα) payer swap is a contract that entitles the holder to a seriesof floating LIBOR payments in exchange for paying a series of fixed payments attimes Tα+1 < . . . . . < Tβ. Tα is the starting point of the swap and (Tβ − Tα) is thetenor. The forward swap rate is defined as the level of the fixed rate that makes thepresent value of the swap equal to zero. The forward swap rate at time t ≤ Tα istraditionally written as
Sα,β(t) =P (t, Tα) − P (t, Tβ)∑β
i=α+1 τiP (t, Ti). (3)
4
An equivalent representation of the forward swap rate (see e.g. Rebonato (2002)) is
Sα,β(t) =
β∑
i=α+1
wi(t)Fi(t),
with
wi(t) =τiP (t, Ti)
∑βj=α+1 τjP (t, Tj)
. (4)
This expression shows that the forward swap rate can be seen as a weighted averageof forward LIBOR rates. Note, however, that the weights will be changing in astochastic manner over time.
It is often more convenient to express the spot swap rate in terms of the lengthof the swap tenor, l:
Sl(Ti) ≡ Si,β(Ti+l)(Ti), (5)
where β(t) = i if ti−2 < t ≤ ti−1.A Tα × (Tβ − Tα) payer swaption of strike K is a contract that gives the right
but not the obligation to enter into a Tα × (Tβ − Tα) payer swap at time Tα andwith fixed rate K. It can be shown (see e.g. Bjork (2004), p. 381) that the time tvalue of the payer swaption can be represented as
PS(t) = Cα,β(t)Eα,βt [(Sα,β(Tα) − K)+],
where Eα,βt [·] denotes expectation formed under the swaption measure Cα,β. The
associated numeraire is defined as Cα,β(t) =∑β
i=α+1 τiP (t, ti).We define a callable CMS Steepener as a bond with a principal of 1 and payments
CFi at t1 < . . . ≤ ti ≤ . . . < tN and maturity at tN . We set the day-count fractionδi = ti− ti−1. The bond can be called at par by the issuer on all payment dates afterthe lockout period ends at time tc ≥ t1. In practice a pre-advise of a few businessdays is typically required, but this will be ignored here. In general the call date isunknown and therefore a stochastic variable. We denote the time index of the calldate by η where N ≥ η ≥ c. The payments are defined as
CFi =
δiR if i ≤ cδi max [m[Sl2(ti−1) − Sl1(ti−1)], f ] if i > c
,
where R is the fixed initial coupon, and Sl2 and Sl1 are the swap rates set in arrearsas defined in (5) with l2 > l1. These rates are referred to as Constant Maturity Swap
5
(CMS) rates because as time passes the rates have a fixed time to maturity.1 m isa constant multiplier (e.g. 3) and f is a small positive constant (e.g. 5 bps) whicheffectively floors the coupon payments just above zero (for tax reasons).
Let us now consider the general valuation at time t, t < t1, of the CMS Steepenerbond.2 Assuming, for simplicity, that we know the stochastic call time η we get fromthe First Fundamental Theorem of Finance (see Bjork (2004)) that the value canbe represented as
Vt = Nt EN
t
[
N−1tη +
η∑
i=1
CFi
Nti
]
, (6)
where N is a generic martingale measure with corresponding numeraire Nt. Now,if the issuer calls the bond before expiry he avoids paying coupons but forfeits theinterest earned on the principal. Assuming that this interest rate is the LIBOR ratewe can value the issuers option as
Ct = NtEN
t
[
N∑
i=η+1
CFi − δiFi(ti−1)
Nti
]
. (7)
The entire bond value can therefore be written as
Vt = NtEN
t
[
N−1tN
+N∑
i=1
CFi
Nti
]
− Ct, (8)
where the first term on the right-hand side of course corresponds to the value ofan otherwise identical non-callable CMS Steepener bond. The LSMC algorithmcan be implemented on both (6) and (8). We shall argue below that it will becomputationally more efficient to work with the single expression (6) than to workwith the decomposition in (8) as is often done in similar contexts (see e.g. Piterbarg(2004b)).
3 Model Choice
In order to evaluate the model risk in pricing the CMS Steepener we will value thisstructured bond using three distinct models for the evolution of the term structure
1A CMS rate is not the fixed rate that sets the price of a Constant Maturity Swap equal to zero.A Constant Maturity Swap is more general expression than a standard swap in that the floatingLIBOR rates are normally replaced with a longer term fixed maturity rate for example the 20 yearswap rate. Pricing a Constant Maturity swap is a non-trivial exercise (see e.g. Hunt and Kennedy(2004)).
2The choice of valuation date before the first coupon date is made purely for notational sim-plicity. Valuation at later dates entails no further complications.
6
of interest rates. We start out with the 2-factor classical Gaussian model termedthe G2++ model and proposed by Brigo and Mercurio (2006). This model’s mainadvantage is tractability. Since the model has Gaussian state variables a closed formexpression for their transition densities can be derived. This makes the evaluationof complex floating payoffs particularly easy. But the tractability comes at a priceand the model is not able to simultaneously price a large segment of the vanillainstruments used for calibration.
An entirely different approach is the so-called market models pioneered by Mil-tersen, Sandmann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997),and Jamshidian (1997). These models differ from the classical interest rate modelsin that they model directly the evolution of discretely compounded market ratessuch as the LIBOR or the swap rate. As we will see, the main advantage of thesemodels is the ease at which they can be calibrated to price a large segment of vanillaproducts such as caps and swaptions. Many different market models exist and inthis paper will use the standard log-normal forward LIBOR market model as wellas the CEV-extension proposed by Andersen and Andreasen (2000).
3.1 The G2++ Model
In the G2++ model the instantaneous short rate under the risk-neutral measure,Qc, is given by
r(t) = x(t) + y(t) + ϕ(t), r(0) = r0, (9)
with stochastic factor processes, x(·) and y(·), described by the Ornstein-Uhlenbecksystem
dx(t) = − ax(t) dt + σ dW1(t), x(0) = 0,
dy(t) = − bx(t) dt + ξ dW2(t), y(0) = 0,
with
dW1(t) dW2(t) = ρ dt.
In this dynamic system W1(t) and W2(t) are correlated Brownian motions, a, σ, band ξ are positive constants, and −1 ≤ ρ ≤ 1. In (9), ϕ(·) is a deterministic functionused to match the initial observed term structure of interest rates. The model isaffine and the zero coupon bond price can thus be written as
P (t, T ) = A(t, T ) exp −B(a, t, T )x(t) − B(b, t, T )y(t) , (10)
7
where
A(t, T ) =PM(0, T )
PM(0, t)exp
1
2[V (t, T ) − V (0, T ) + V (0, t)]
,
B(z, t, T ) =1 − e−z(T−t)
z,
V (t, T ) =σ2
a2
[
T − t +2
ae−a(T−t) − 1
2ae−2a(T−t) − 3
2a
]
+ξ2
b2
[
T − t +2
be−b(T−t) − 1
2be−2b(T−t) − 3
2b
]
+ 2ρσξ
ab
[
T − t +e−a(T−t) − 1
a+
e−b(T−t) − 1
b− e−(a+b)(T−t) − 1
a + b
]
,
where PM(0, ·) denotes the market observed discount function. The forward LIBORand swap rates can now be generated in closed form through formula (1) and (3)-(4).A pricing formula for swaptions is described in the Appendix. For further detailswe refer to Brigo and Mercurio (2006).
3.2 LIBOR Market Models
The log-normal LIBOR market model (LMM) was derived with the intent of havinga model that was consistent with the standard market practice of using the Black-76 formula to value caplets. Consistence with the Black-76 formula requires thediscretely compounded or simple forward rate to be a log-normal martingale underthe forward measure, and short rate models are not able to achieve this. The HJM-model by Heath, Jarrow, and Morton (1992) is similar to LMM in spirit but modelsthe instantaneous forward rate instead of the simple forward rate. HJM was alsothe starting point for one derivation of the market model of Brace, Gatarek, andMusiela (1997), and indeed Hunt and Kennedy (2004) show that the market modelscan be seen as a subset of the HJM model class. A PDE approach to market modelscan be found in Miltersen, Sandmann, and Sondermann (1997) but perhaps a morenatural derivation which is also the one used by most textbooks is the change-of-numeraire technique advocated by Jamshidian (1997). We refer to these papers orthe textbooks by Brigo and Mercurio (2006) and Rebonato (2002) for an extensivetreatment.
The log-normal assumption of the forward rate distribution was known to beat odds with reality long before LIBOR Market Models were invented. Thereforeseveral extensions have been proposed some of which still nest the log-normal version.The goal of the extensions was to have a model that realistically replicates theso-called volatility skew observed in market prices of interest rate options. The
8
volatility skew refers to the decreasing curve of Black-76 volatilities as a functionof strike observed in market prices. The simplest extension is the CEV modelinvestigated in Andersen and Andreasen (2000) and Hull and White (2001). Thismodel uses a different distribution for the forward rate which allows for replicationof simple versions of the skew. In reality the skew is more like a hockey-stick shapeand if one wants to replicate this pattern exactly one would need stochastic volatilityor jumps in the LIBOR rate process. This causes problems with calibration as wellas inherent difficulties with hedging as these type of models describe incompletemarkets. We therefore stick to the CEV-extension of Andersen and Andreasen(2000) as this retains completeness.
The general version of the LIBOR market model we work with is the following:
dFk(t) = σk(t)φ(Fk(t))dZk(t), t ≤ Tk−1, ∀ k = 1, . . . , M,
where Zk(t) is a Brownian motion under the forward measure with numeraire P (t, Tk).Note that forward rates are martingales under their respective measures as theyshould be. We also assume constant correlations, i.e.
dZi(t)dZj(t) = ρi,j dt, ∀ i, j = 1, . . . , M.
The choice of φ-function determines the distribution of the forward rates. We lookat two different specifications,
φ(x) = x,
φ(x) = xp , 0 < p < 1.
The first of these yields a log-normal distribution of the forward rates and the secondimplies a non-central χ2-distribution (see Andersen and Andreasen (2000)).
The two most important products we need to price are caplets and swaptions.If we were to price a caplet the current formulation of the model would suffice sincea caplet depends on one forward rate only. Recall from (2) that the time t price ofa caplet on Fk(·) and expiring at time Tk is given as
Cpl(t) = P (t, Tk)τkEFk
t [(Fk(Tk−1) − K)+].
With φ = x this immediately leads to the well-known Black-76 pricing formula, cf.Appendix B.2. When φ(x) = xp the distribution of the forward rate is skewed tothe left and the Black-76 formula no longer applies. This skewing of the distributioncauses in-the-money caplets to be priced higher and out-of-the-money caplets to bepriced lower implying a volatility skew. A closed-form formula can also be derivedand is shown in Appendix B.2.
For swaptions no exact pricing formula exists in our reference LIBOR marketmodels and one must resort to approximations or simulation to calculate prices.
9
In such a simulation we would need realizations from the joint distribution of theforward rates comprising the underlying swap, and we therefore need the dynamicsof all relevant forward rates under a single measure. As it turns out a forward rateFk(t) is only driftless under its ”own” measure with numeraire P (t, Tk). The noarbitrage conditions determine the drift under other numeraires. The usual riskneutral measure (Qc cf. earlier) takes the continuously accrued bank account as thenumeraire, but since we are working with discrete rates it is more convenient towork under the discrete money market measure which we denote by Qd and whichhas the discrete bank account as numeraire. The discrete bank account is definedas follows:
B0 = 1,
BTk= BTk−1
[1 + τkFk(Tk−1)], 1 ≤ k < M,
Bt = P (t, Tk)BTk, t ∈ [Tk−1, Tk].
The dynamics of forward rates under this measure can be established using Ito’sLemma and Girsanovs Theorem (see Andersen and Andreasen (2000)). The resultis
dFk(t) =σk(t)φ(Fk(t))(µkdt + dZk(t)), (11)
µk =k∑
j=1
1t<Tjρk,jτjσj(t)φ(Fj(t))
1 + τjFj(t),
where we have redefined dZk(t) as a Brownian motion under Qd. We can nowsimulate from the SDE in (11) and use formula (3) to get realizations of the swaprate and value the expectation of B−1
TαCα,β(Tα)(Sα,β(Tα) − K)+ under the discrete
money market measure through simulation to get the swaption price.However, simulation is not feasible for calibration purposes. We therefore use an
approximative formula for swaption prices given in Appendix B.2. The approxima-tion is mainly founded on assuming that the swap rate follows the same distributionas the forward rates. This is of course not true because the swap rate will have anon-zero drift under any measure in the LMM. However, the discrepancy betweenthe distributions have been shown to be very small by Brace, Dun, and Barton(2001) and Hull and White (2001) for the CEV-case; making this procedure accu-rate enough for calibration purposes.
4 Pricing Methodology
Standard Monte Carlo simulation techniques cannot be used for pricing callableproducts like the CMS Steepener since these methods do not identify the stopping
10
rule (call strategy) necessary for characterizing the value of the security. One wayto estimate such a stopping rule goes via the Least-Squares Monte Carlo procedurenormally attributed to Tsitsiklis and Roy (1999), Carriere (1996), and Longstaff andSchwartz (2001). This procedure has countless extensions and applications wherePiterbarg (2004b) is the most notable in our context since he focuses exclusively onthe LIBOR Market Model. The idea of the LSMC procedure as used in Piterbarg(2004b) is described in the following.
To price any American or Bermudan product we essentially need two things ev-erywhere in the state space: The exercise value and the continuation value of theproduct. In general neither of these are readily available so they have to be esti-mated. Consider, for example, the option embedded in the callable CMS Steepenerwhose value was defined in (7). Once this option is exercised the holder ”receives”a series of payments determined by CFi − δiFi(ti−1). The value of these paymentscannot be calculated in closed form for the LIBOR Market Models or the G2++model. We must therefore either approximate or simulate the value. Piterbarg(2004b) shows in a quite general setting how this value can be approximated byregression.
However, if instead of focusing on the option to call the bond we look at thebond in its entirety, then the exercise value for the issuer is simply the principal andtherefore readily observable. Although this simple insight avoids an entire regressionstep in the LSMC procedure it is rarely used in the literature. Notable exceptions areJoshi (2006) and Amin (2003) which both use this ”trick” in different contexts. Tothe authors’ knowledge it has never been applied in the case of issuer callable bonds,and we will therefore more carefully describe the procedure in a setting appropriatefor these types of bonds.
We assume an economy with a time t vector of state variables Xt. The objectiveof our analysis is to price a bond with maturity tN and a pricipal of 1. The bond isissued at time t0 = 0, has payments CFi on payment dates T = t1, . . . , ti, . . . , tN,and is callable at par on all payment dates. The valuation of this bond at issuanceis an optimal stopping problem of type
V P0 /N0 = inf
ηEN
0 [N−1tη +
η∑
i=1
N−1ti
CFi].
The superscript on V indicates that this is the solution to what we shall later referto as the Primal problem. As before N is a martingale measure with correspondingnumeraire Nti at time ti. Note also that the index of exercise η is unknown andtherefore a stopping time in a mathematical sense. To price the product we needan estimate of the stopping time, or a stopping rule. To obtain such an exercisestrategy we proceed as follows: At each exercise time the issuer can either pay backthe principal or wait until the next exercise time. If we assume that the coupons
11
are reinvested in numeraire bonds the (accumulated) value at an exercise time istherefore
Vti = min
(
1 + Nti
i∑
k=0
N−1tk
CFk , NtiEN
ti
[
N−1ti+1
Vti+1
]
)
. (12)
A more convenient way of writing this is to define
Qti = NtiEN
ti
[
N−1ti+1
Vti+1
]
− Nti
i∑
k=0
N−1tk
CFk
= NtiEN
ti
[
N−1ti+1
CFi+1 + min(1, Qti+1)]
,
and then rewrite (12) in terms of Qti as follows,
Vti = Nti
i∑
k=0
N−1tk
CFk + min (1 , Qti) .
Vti cannot be directly evaluated since it depends on the exercise behavior of theissuer at ti+1. The LSMC procedure is essentially an algorithm to approximateVti by simulation. But instead of approximating Vti directly we approximate theQti ’s instead. The two approaches are essentially equivalent as one can alwayscalculate one from the other using the above formula. But approximating Qti meansthat we avoid approximating coupons that have already been determined. for theapproximation of Qi we choose for each time a polynomial function f(·, ·) :Rr×n+1×Rr
+ → R where n is the order of the polynomial
Qti ≈ f(βi; R(Xti)) =β(0)i + β
(1)i R1(Xti) + β
(2)i R1(Xti)
2 + . . . β(n)i R1(Xti)
n (13)
... (14)
+β(r×n−n+1)i Rr(Xti) + β
(r×n−n+2)i Rr(Xti)
2 · · · + β(r×n)i Rr(Xti)
n
(15)
Xti is a M dimensional vector of state variables and R(X) = (R1(X), . . . , IRr(X))where the Ri’s are vector functions RM
+ → R+ that each map the state variablesinto a key rate determined a priori to be important for the continuation value of thebond.The important thing here is that the rates used must be measurable with respectto the filtration Fti , as the issuers decision to call can only be based on informationknown at the call time. Choosing these rates is not an exact science, and it is veryspecific the particular problem at hand. In general one should include rates thatdetermine the level and the slope of the term structure at a given point in time.
12
Piterbarg (2004b) also advocates the use of a so-called core swap rate. That is theswap rate that sets on the day of exercise and expires on the day the entire bondmatures. For the case of CMS Steepeners we have found that using the short forwardrate, the core swap rate, and the two swap rates that determine the coupons giveexcellent results.
As the functional form for f(·; R(Xti)) we use polynomial functions which seemsto be the industry standard. Choosing f to be linear in β is a smart choice from acomputational perspective since it allows us to estimate β with standard ordinaryleast squares (OLS). This choice does not appear to be overly restrictive (see Piter-barg (2004b)) and we will therefore use the linear model throughout the paper. TheLSMC procedure goes as follows:
1. Simulate K paths of the state variables Xti(ωj)Ni=1 and calculate the nu-
meraires Nti(ωj)Ni=1 and coupons CFi(ωj)N
i=1 for each path j = 0, . . . , K,ωj ∈ Ω.
2. At the terminal date tN set QtN (ωj) = 0 ∀ j = 0, . . . , K, ωj ∈ Ω.
3. At the last exercise time tN−1 setQtN−1
(ωj) = NtN−1(ωj)NtN (ωj)
−1(1 + CFN(ωj))
and calculate the OLS estimator βN−1 =∑K
j=1 RN−1(XN−1(ωj))QtN−1(ωj)
∑Kj=1 RN−1(XN−1(ωj))′RN−1(XN−1(ωj))
.
4. For the next exercise time tN−2 setQtN−2
(ωj) = NtN−2(ωj)N
−1tN−1
(ωj)CFN−1(ωj)+min(1, f(βN−1; RN−1(XtN−1(ωj)),
and calculate the OLS estimator βN−2 in the same manner as in step 3.
5. Repeat previous step until the first the exercise time t1.
We now have an exercise strategy through the estimated parameters of the fi-functions. We can define the indicator function for exercise as
Ii(ωj) =
1 if 1 ≤ f(βi; Ri(Xti(ωj)) or ti = tN0 if 1 > f(βi; Ri(Xti(ωj)))
.
The stopping time index for each path is
η(ωj) = min(i ∈ [1, . . . , N ] : Ii(ωj) = 1).
Finally, the MC price estimate of the bond price can the be calculated as follows
V0 =1
KN0
K∑
j=1
Ntη(ωj)(ωj)
−1 +
η(ωj)∑
i=1
N−1ti
(ωj)CFi(ωj)
. (16)
13
4.1 Lower bounds
The method described in the previous section does not guarantee that the chosen fi-functions are optimal. This means that exercise might occur at sub-optimal pointsin time. If this is the case the estimate in (16) will not converge to the true valueof the bond as K is increased. Exercise is done by the issuer who seeks to minimizethe value of the bond. Therefore if exercise is done sub-optimally the value of thebond will be higher than in the optimal case. One can therefore interpret the pricein (16) as an upper bound on the true bond price. The method does not give anymeasure on how far one is from the true price and the estimate (16) could thusin principle be severely upward biased. In this section we therefore show how tocalculate a corresponding lower bound estimate to complement the upper bound.The difference between the lower and upper bound is called the duality gap and canbe seen as a measure of the quality of the estimated exercise strategy.
The problem of calculating a lower bound is similar to calculating an upperbound for American options. In the case of straight American options it is theholder of the option that exercises, and the challenge in that case is therefore toestablish an upper bound since a suboptimal strategy provides a downward biasedestimate. A method of finding proper upper bounds for American options usingsimulation has been recently developed in three seminal papers by Rogers (2002),Haugh and Kogan (2004) and Andersen and Broadie (2004) (for other methods seeGlasserman (2004)). A less mathematical approach is that of Joshi (2006) that givesa very good economic interpretation of the upper bound. In all the cited cases thetool to calculate the upper bound is the concept of duality which is well-known in,for example, the operations research literature. For more details on this subject werefer to the above cited papers and the references therein.
Below we show how a lower bound in relation to issuer callable bonds can bedeveloped using the ideas from the American options literature. The main points caneasily be lost in the mathematics and we therefore end with an economic argumentbased on the intuition provided in Joshi (2006). Recall that the problem of pricingan issuer callable bond is that of computing
V P0 /N0 = inf
η∈I
EN
0 [N−1tη +
η∑
i=0
N−1ti
CFi)], (17)
for a series of exercise times T = t1, . . . , tN, with indices I = 1, . . . , N. We haveearlier termed this the primal problem. Now, for an arbitrary finite submartingaleMtn – abbreviated Mn – we can define a so-called dual function F (n, Mn) as follows,
F (n, M) = Mn + EN
tn
[
minη∈[n,N ]∩I
(
N−1tη +
η∑
i=1
N−1ti
CFi − Mη)
)]
.
14
The dual problem consists of finding the maximum submartingale,
V D0 /N0 = sup
MF (0, M) = sup
M
(
M0 + EN
0
[
minη∈I
(N−1tη +
η∑
i=1
N−1ti
CFi − Mη)
])
.
(18)Recall from the previous section that we defined Vtn as the value of the bond newlyissued at time tn plus the coupons up until that time invested in numeraire bonds:
Vtn = min
(
1 + Ntn
n∑
i=0
N−1ti
CFi , NtnEN
tn
[
N−1tn+1
Vtn+1
]
)
. (19)
We can now show the following theorem:
Theorem 1: Given the primal and dual problems in (17) and (18), the follow-ing statements are true
1. The duality relation holds i.e V P0 = V D
0 .
2. An optimal solution M∗n for the dual problem is the discounted bond price
process Vtn/Ntn .
Proof: See Appendix A. 2
The dual problem can now be used to construct a lower bound for the bondprice. We know the optimal submartingale, but solving this process is equivalentto solving the primal problem. But if we where to use an arbitrary submartingaleinstead we get the inequality,
V0/N0 =M∗0 + EN
0
[
minη∈I
(N−1tη +
η∑
i=1
N−1ti
CFi − M∗t )
]
≥M0 + EN
0
[
minη∈I
(N−1tη +
η∑
i=1
N−1ti
CFi − Mt)
]
. (20)
Hence any submartingale will give a lower bound. Suppose we use the suboptimalstrategy estimated in the previous section and let Vti be defined as Vt in (19) when
following this estimated suboptimal strategy instead. Set M0 = V0/N0 and M1 =Vt1
Nt1
and
Mn+1 = Mn +Vtn+1
Ntn+1
− Vtn
Ntn
+ InEN
tn
[
Vtn+1
Ntn+1
−(
N−1tn +
n∑
i=1
N−1ti
CFi
)]
.
15
We first notice that this is actually a martingale. Assume that the suboptimalstrategy says ”continue” at time tn, that is In = 0. In this case we have:
EN
tn [Mn+1] =Mn + EN
tn
[
Vtn+1
Ntn+1
− Vtn
Ntn
]
=
Mn + EN
tn
[
Vtn+1
Ntn+1
− EN
tn
[
Vtn+1
Ntn+1
]]
= Mn.
This means that Mn is a martingale in the continuation region. If we assume thestrategy says exercise at tn so In = 1, then we have
EN
tn [Mn+1] =Mn + EN
tn
[
Vtn+1
Ntn+1
−(
N−1tn +
n∑
i=1
N−1ti
CFi
)]
− EN
tn
[
Vtn+1
Ntn+1
−(
N−1tn +
n∑
i=1
N−1ti
CFi
)]
= Mn.
So Mt is also a martingale in the exercise region. Martingales are also submartingalesso we can insert this in equation (20) to obtain a lower bound. Hence, a lower boundfor the bond can be written at time 0:
V L0 = V0 + EN
0
[
minη∈I
(N−1tη +
η∑
i=1
N−1ti
CFi − Mη)
]
. (21)
The first term is the upper bound calculated from the LSMC algorithm described inthe previous section. At the first exercise time (or at maturity if the bond is nevercalled) the exercise value is always equal to M . This means that the second termwill always be less than or equal to zero.
V L0 can be interpreted as the price the buyer of the bond will be willing to pay,
whereas the price in (16) can be seen as the sellers price. The argument goes asfollows. The seller of the bond decides when he wants to call the bond. This may ormay not be at an optimal time. The buyer wants to be hedged even if the bond iscalled optimally, but without knowledge of the optimal strategy, exact replication isnot possible. But the buyer can form a subreplicating strategy based on selling thebond himself and following a suboptimal exercise strategy. The initial value of thehedge will be V L
0 . When following this strategy 4 situations can occur at a givenpoint in time:
1. The buyer and seller exercise. In this case their prices will agree and the hedgewill be perfect.
16
2. The buyer and seller do not exercise. Again the prices will agree.
3. The seller exercises and the buyer does not. If the time is optimal the buyerwill take a loss on his hedging position which is financed by selling numerairebonds.
4. The buyer exercises and the seller does not. The buyer can resell the productwith 1 less exercise date to continue the hedge. If the time was not optimalthen losses from reselling are financed by selling numeraire bonds.
In all 4 cases the buyer’s price will be equal to or below that of the seller’s. All thesmall losses are discounted back to present time. The losses are equal to the secondterm in (21) which is called the duality gap. We define this term as follows:
∆0 = V L0 − V0 = EN
0
[
minη∈I
(N−1tη +
η∑
i=1
N−1ti
CFi − Mη)
]
. (22)
This term is calculated through simulation. The algorithm is as follows
1. Estimate an exercise strategy using the technique from the previous section.
2. Simulate K paths of the state variables Xti(ωj)Ni=1 and calculate the nu-
meraires Nti(ωj)Ni=1 and coupons CFi(ωj)N
i=1 for each path j = 0, . . . , K,ωj ∈ Ω.
3. For the first path, ω1, do the following: Start from t1 and calculate an estimate,M1, using a sub-simulation with K1 subpaths. The subpaths are evolved fromtime t1 in path ω1. Repeat this for all possible exercise times creating a seriesof estimates M1, . . . , MN−1.Now calculate Ntn(ω1)
−1 +∑n
i=1 Nti(ω1)−1CFi(ω1) − Mn ∀n = 1, . . . , N and
find the minimum.
4. Repeat the previous step for all paths ωj and average the result to get an
estimate of duality gap ∆0.
With N exercise times, K paths, and K1 subpaths the computational time forcalculating the lower bound is approximately of order K × N × K1 and thereforeextremely slow.
5 Hedging and Sensitivities
The hedging of derivatives is in general at least as important as pricing, and wetherefore dedicate this section to some considerations on the determination of thetwo most important hedge ratios – the delta and the vega. This analysis focusesmainly on the LIBOR market models.
17
5.1 Deltas
Delta denotes the sensitivity of the price with respect to shifts in the initial vectorof forward rates
F (0) = (F0(0), F1(0), . . . . ., FM(0)). (23)
The mth individual delta ratio is defined as follows
∆m =∂V0
∂Fm(0). (24)
One can easily calculate a first order approximation for this number by shifting theinitial term structure and calculating the corresponding shift in the price using themethodology provided in the previous sections. If we denote the shifted price V0
and rate Fm(0) then
∂V0
∂Fm(0)≈ V0 − V0
Fm(0) − Fm(0). (25)
In simple cases the payoff will be a smooth function of the initial curve making this“bump-and-revalue” approach stable enough for practical purposes. However, in ourcase the price of a callable CMS Steepener is not a smooth function of the initialterm structure. The reason is the callable nature of the product. The decision tocall the bond at a point in time is binary and a small shift in the initial rates couldtherefore change this decision. The corresponding price change could be quite largesince the amount of payments on the relevant path will have changed. Anotherreason to avoid using (25) is the computational cost since a new price would haveto be calculated for each perturbation.
A superior method that both alleviates the instability problem as well the com-putational cost is the pathwise approach described in Glasserman and Zhao (1999)and refined in Piterbarg (2004a) and Piterbarg (2004b). The idea is as follows.Recall the forward rate dynamics
dFk(t) =σk(t)φ(Fk(t))(µkdt + dZk(t)), t ≤ Tk−1, ∀ k = 1, . . . , M,
µk =k∑
j=1
1t<Tjρk,jτjσj(t)φ(Fj(t))
1 + τjFj(t).
We can differentiate through this system and calculate dynamics for the delta ratios
18
with respect to rates,
d(∆mFk(t)) =σk(t)φ′(Fk(t))∆mFk(t)(µk(t, Fk(t))dt + dZk(t))
+σk(t)φ(Fk(t))
(
M∑
j=0
∂µk(t, F (t))
∂xj
∆mFj(t)
)
t ≤ Tk−1,∀ k = 1, . . . , M,
where
∂µk
∂xj
∣
∣
∣
∣
xj=Fj(t)
=(φ′(Fj(t))(1 + τjFj(t)) − φ(Fj(t))τj) τjρk,jσj(t)
(1 + τjFj(t))2,
φ(x) =xp,
φ′(x) =pxp−1.
The starting value of this SDE is
∆mFk(0) =
1 for m = k0 for m 6= k
.
An approximation that saves computational time is fixing the drifts at the time zerovalue of the forward rates. This avoids having to recalculate the drift for each timestep. This results in the following SDE,
d(∆mFk(t)) =σk(t)φ′(Fk(t))∆mFk(t)(µk(t, Fk(0))dt + dZk(t))
+σk(t)φ(Fk(t))
(
M∑
j=0
∂µk(t, F (0))
∂xj
∆mFj(t)
)
, (26)
t ≤ Tk−1, ∀ k = 1, . . . , M.
Glasserman and Zhao (1999) show that the errors resulting from this approximationare insignificant.
We can now calculate the delta of the product. Given an estimate of the optimalexercise time η we have seen that the initial bond price can be represented as
V0 = EN
0
[
B−1tη +
η∑
i=1
B−1ti
CFi
]
.
The deltas can now be calculated as follows
∆mV0 =EN
0
[
∆m(B−1tη +
η∑
i=1
B−1ti
CFi)
]
=EN
0
[
∆mB−1tη
]
+ EN
0
[
η∑
i=1
∆m(B−1ti
)CFi)
]
+ EN
0
[
η∑
i=1
B−1ti
∆m(CFi)
]
. (27)
19
Notice that when differentiating through we have ignored that the optimal time ηalso depends on the initial forward rates. Piterbarg (2004a) gives a formal argumentwhy this is reasonable based on the smooth pasting conditions of the optimizationproblem. The first two terms are calculated as follows with t ∈ [tn, tn+1],
∆m(B−1t ) = − B−1
t
n∑
i=1
τi
1 + τiFi(ti−1)∆mFi(ti−1)
− B−1t
t − tn1 + (tn+1 − t)Fn+1(tn)(1 + τnFn+1(tn))
∆mFn+1(tn).
For details see Piterbarg (2004a). The last term in (27) is more difficult. For l > swe have
CFi =
τimax((Sl(ti−1) − Ss(ti−1)) ∗ m, f) for η ≥ i > Tlockout
τiR for 0 ≤ i ≤ Tlockout,
where R is the fixed rate in the lockout period.A standard way to approximate the swap rates is the following formula due to
Rebonato (2002),
Sl(ti) =
β(ti−1+l)∑
j=i−1
wj(ti)Fj(ti) ≈β(ti−1+l)∑
j=i−1
wi(0)Fj(ti)
Ss(ti) =
β(ti−1+s)∑
j=i−1
wj(ti)Fj(ti) ≈β(ti−1+s)∑
j=i−1
wi(0)Fj(ti)
β(t) =m if tm−2 < t ≤ tm−1.
This approximation can be justified by the fact that the variability in the weights isvery small compared to the variation in the forward rates. With this approximationwe can calculate the delta for the long swap rate:
∆m(Sl(ti−1)) =
β(ti−1+l)∑
j=i−1
(∆mwj(0)Fj(ti) + wj(0)∆mFj(ti)) .
From simple calculations we get that
∆mwj(0) = −τm1
1 − τmFm(0)wj(0)
1
1 +∑β(ti−1+l)
k=m
∏kj=i−1
11+τjFj(0)
.
This term is very small especially for large l and can in practice be ignored butwe include it since it comes at little computational cost as it can be calculated
20
before any simulation is performed. We can now evaluate all the relevant terms andcalculate the deltas in equation (26) by Monte Carlo.
Note that the approach described above is not applicable to G2++ model sincethis is not a model of forward rates. In this case we therefore use (25). The standarderror of the delta estimate depends on the size of the rate shift or perturbationFm(0) − Fm(0). Because of the discontinuity this might not be at the smallestpossible level. We have experimented with several shifts and found that a 1 basispoint perturbation gives decent results.
5.2 Vegas
Vega denotes the price sensitivity with respect to the volatility in the market. Inpractice traders not only perform delta hedging but also vega hedging in order to behedged against moves in volatility. One can form a vega hedging portfolio in severalways by using volatility dependent products such as cap, floors and swaptions. Wehave initially chosen to calibrate our models to a large segment of swaptions. How-ever, finding the hedge portfolio of swaptions is difficult since swaption volatilies arenot direct inputs to the versions of the LIBOR market model used in this paper.One could proceed with the naive bump-and-revalue approach by shifting a marketprice and then recalibrating the model. In our case this does not give meaningfulresults since the perturbation will be obscured by calibration error. One way toget meaningful swaption vegas would therefore be to start over and apply a localcalibration procedure (see for example Brigo and Mercurio (2006) or Pietersz andPelsser (2006)) instead of our more global approach. This procedure fits the modelto a small subset of swaptions without error. This subset could for example be themost liquid or the core swaptions. The downside of the local approach is of coursethat you take the chance of fitting the model perfectly to prices that might reflectnoise such as illiquidity and non-synchronous trading.3 This avenue is beyond thescope of our paper and we therefore proceed with calculating the sensitivities of thevolatilities of the individual rates which amounts to finding a vega hedge portfolio ofcaplets. This approach also has the advantage of having a more direct interpretationwith respect to forward rates than the swaption approach.
We proceed in a manner similar to the previous section. We calculate the pricesensitivity of a unit shock to the volatility curve for each rate and we denote eachvega: ∂V0
∂σm. This partial derivative can be calculated using the pathwise approach
in a similar manner as we did for the deltas in the previous section. Differentiating
3An alternative to this is the indirect approach described in Piterbarg (2004b). We have triedthis approach but where not able to get meaningful results.
21
through and fixing the drifts we now get the following SDE:
d(∂
∂σm
Fk(t)) =
[
φ(Fk(t))1m=k + σk(t)φ′(Fk(t))
∂
∂σm
Fk(t)
]
(µk(t, Fk(0))dt + dZk(t))
+σk(t)φ(Fk(t))
(
M∑
j=0
∂µk(t, F (0))
∂xj
∂
∂σm
Fj(t)
)
,
t ≤ Tk−1, ∀ k = 1, . . . , M.
Note that ∂∂σm
Fk(0) = 0 ∀m and
∂µk
∂xj
∣
∣
∣
∣
xj=σj
=(φ′(Fj(t))(1 + τjFj(t)) − φ(Fj(t))τj) τjρk,jσj(t)
(1 + τjFj(t))2
+ 1j=kφ(Fj(t))τjρk,j
(1 + τjFj(t)).
Besides the extra indicator function term this is the same as in the previous section.We can now calculate the vegas of the CMS Steepener bond as follows:
∂
∂σm
V0 =EN
0
[
∂
∂σm
(B−1tη +
η∑
i=1
B−1ti
CFi)
]
=EN
0
[
∂
∂σm
B−1tη
]
+ EN
0
[
η∑
i=1
∂
∂σm
(B−1ti
)CFi
]
+ EN
0
[
η∑
i=1
B−1ti
∂
∂σm
CFi
]
.
(28)
The three terms are calculated with Monte Carlo as in the previous section but with∆m replaced by ∂
∂σm.
Since prices are quoted in terms of Black-76 implied volatility, it is necessary toexpress the vegas in this metric. By using the chain rule of partial differentiationwe have
∂V
∂σm
=∂σBS76
∂σm
∂V
∂σBS76
⇒ ∂V
∂σBS76
=∂V∂σm
∂σBS76
∂σm
(29)
The numerator is calculated in 28. The denominator cannot be calculated in closedform for the CEV model but it can easily be found by numerical differentiation. Forthe standard LMM case we of course have σm = σBS76 so the vegas are given in theproper metric from 28 directly
In the G2++ model there is, in our setup, no proper way to calculate vegas. Apathwise approach as well as a bump-and-revalue approach will be too noisy sinceall volatility parameters in the G2++ model have a global influence.
22
6 Case Study: The Dexia Dannevirke 2005/2016
CMS Steepener
In this section we further investigate the particular issue of ”Dexia Dannevirke”.This was the largest structured bond issue ever seen in Denmark with DKK 2.4billion (about EUR 320 million or USD 425 million) sold. The success of thisparticular issue incited the larger share of all Nordic banks to issue a string ofsimilar products. The bond was issued on January 25, 2005 which we denote timezero. The initial fixed coupon was set at 9.55% and the floating coupons weredetermined as 3 times the floored (at 5bps) difference between the 20 and the 2year EURIBOR swap rate determined in arrears at the previous coupon time. Thefollowing payments CFi were specified,
CFi =
δi9.55% if i ≤ 2δi max [3[S20(ti−1) − S2(ti−1)], 0.05%] if 23 ≥ i > 2,
where δi is determined from a 30/360 day-count fraction. The payments are madeon the dates listed in Table 1. The bond can be called by the issuer at time t2 andonwards. If the bond is called the coupon on the call date is paid as well as theprincipal which we for simplicity assume is 1.
7 Model Calibration
The concept of calibration is to fit the parameters in the model so they are able toreproduce the relevant prices observed in the market with a certain amount of ac-curacy. There is naturally a tradeoff between the number of calibration instrumentsused and the fitting quality. We therefore make a selection of calibration instru-ments that we deem most relevant for the the callable CMS Steepener described inthe previous section.
From Jyske Bank we have obtained market quotes on Euro denominated EU-RIBOR caps and swaptions. All market quotes are in Black-76 implied volatility.The swaptions in the European market are settled annually and the caps are settledeither quarterly or semiannually. Both the LMMs and the G2++ model are markedto market with the zero curve available from Datastream. All rates and quotes arefrom January 25, 2005 – the day the bond was issued.
7.1 G2++
The G2++ model has a limited number of parameters which again limits the amountof calibration instruments. Since the bond we are aiming to price depends mainly on
23
swap rates we choose to calibrate to swaptions only. A subsection of the swaptionmatrix of implied volatilities can be seen in Table 2. The bond price depends amongother things on the twenty year swap rate that sets in 11 years. This means thatswaptions with a timespan that stretches further than 31 years are of little relevanceto the bond price. Excluding this portion of the swaption matrix leaves us with 106datapoints to fit.
The G2++ model has a closed form solution for the swaption price. This formulais shown in Appendix B.1. Following the market convention we translate the modelprices into volatilities by inverting the Black-76 formula. We then minimize thedistance between model volatilities and market volatilities to get the optimal setof parameters, Θ = (a, b, σ, η, ρ), using the following Root-Mean-Square distancemetric,4
Θ = arg min(a,b,σ,η,ρ)
√
√
√
√
√
1
106
∑
1≤i≤10,1≤j≤10,tMi +tTj ≤31
(
σi,jmarket − σi,j
model(a, b, σ, ξ, ρ)
σi,jmarket
)2
.
Here σi,jmarket is the implied volatility of a payer swaption with maturity date tMi and
swap length (tenor) equal to tTj , and σi,jmodel(·) is the corresponding model implied
volatility from inverting formula (31) in the Appendix. Minimization is done in theOx programming language using the Simulated Annealing optimization procedureto get starting values and the standard BFGS optimization routine to refine thesolution. As Table 3 shows the errors are tolerable for smaller tenors but increaseup to 13% for larger tenors and swaption maturities. This can be expected froma model with only five parameters and 106 datapoints to fit. Note from the tablethat we estimate a correlation between the factors equal to −0.891 similar to thecase investigated in Brigo and Mercurio (2006). Since the factors are theoreticalconstructs this number does not have a direct interpretation. But with a valuedifferent from 1 or -1 it does mean that the model is able to pick up some of theinformation on the correlation between rates available in the swaption matrix.
7.2 LIBOR Market Models
The LIBOR Market Model in its currently stated form does not readily lend itselfto calibration. First recall the general market model under the forward numeraire.With tenor structure T1 < · · · < TM we have
dFk(t) = σk(t)φ(Fk(t))dZk(t), t ≤ Tk−1, ∀ k = 1, . . . , M,
4We also tried minimizing prices instead and achieved almost identical results.
24
where
dZi(t)dZj(t) = ρi,jdt.
Since we have semiannual payments in the bond we are currently analyzing, wechoose forward rates with equidistant maturities of 6 months, meaning Tk − Tk−1 =0.5. The furthest maturity we need to deal with in pricing is 31 years. This meansthat TM = 31 and the number of rates is 62. In the log-normal case, ie. whenφ(x) = x, we have only the volatilities σk(t) and the correlations ρi,j to determine.As recommended by Rebonato (2002) and Brigo and Mercurio (2006) we use thefollowing parametric form for the volatilities,
σk(t) = Φk
(
[a(Tk−1 − t) + d] e−b(Tk−1−t) + c)
.
This specification is very tractable. It allows the volatilities to exhibit the typicalhumped shape observed in the market. It is also time-inhomogeneous since the termΦk is forward rate dependent which allows a better initial fit to the market. We dohowever set the Φ’s pairwise equal:
Φk = Φk+1,∀ k odd .
This eases calibration as it reduces the number of parameters to estimate. In un-constrained calibration a common problem is that the parameters oscillate betweenforward rates yielding an unrealistic evolution of the volatility (see Brigo and Mer-curio (2006)). To avoid this we impose a regularity constraint,
|Φk+1 − Φk| < 0.1,∀ k.
The second term, [a(Tk−1 − t) + d] e−b(Tk−1−t) + c, is homogeneous through time anddepends only on time to maturity of the forward rate. This term determines theshape of the volatility structure.The correlations are parametrized through the well-known approach justified inSchoenmakers and Coffey (2000),
ρi,j = exp[− |j − i|M − 1
− (ln(ρ∞)
+ η1i2 + j2 + ij − 3Mi − 3Mj + 3i + 3j + 2M2 − M − 4
(M − 2)(M − 3)
− η2i2 + j2 + ij − Mi − Mj − 3i − 3j + 3M + 2
(M − 2)(M − 3))].
This particular approach yields a correlation matrix that is flexible enough to havethe particular characteristics observed in several empirical studies, see e.g. the survey
25
in Rebonato (2002)). In order to facilitate direct comparison with the G2++ modelwe choose to calibrate to the same swaption data as in the previous section.5
A practical issue often ignored is the fact that the LMM models semiannual rateswhereas the EURIBOR swaptions we calibrate to are settled annually. Throughoutthe paper we therefore use the following adjusted swap rate for a Tα × (Tβ − Tα)-swap.
Sα,β(t) =P (t, Tα) − P (t, Tβ)∑(α−β)/2
i=1 P (t, Tα+2i)=
β∑
i=α+1
wi(ti)Fi(t),
wi(t) =P (t, Ti)
∑(β−α)/2i=1 2τiP (t, Tα+2i)
.
This swap rate is described in Schoenmakers (2002). As before we minimize thedistance between the market quoted volatilities and the model implied volatilitiesusing a root mean square distance metric, yielding the optimal parameters:
Θ = (a, b, c, d, ˜Φkk≥1, η1, η2, ρ∞) =
arg min
√
√
√
√
√
1
106
∑
1≤i≤10,1≤j≤10,tMi +tTj ≤31
(
σi,jmarket − σi,j
model(a, b, c, d, Φk, η1, η2, ρ∞)
σi,jmarket
)2
.
Here σi,jmarket is the Black-76 implied volatility of a payer swaption with maturity date
tMi and swap length (tenor) equal to tTj , and σi,jmodel(·) is the corresponding model
Black-76 volatility from formula (32) in the Appendix. We use the constrainedoptimization algorithm of Lawrence and Tits (2001) to obtain the minimum distanceparameters. In Table 4 we report the corresponding pricing errors and in Table 5the estimated parameters. We obtain an average error of 1.7% which might seemhigh when using 39 parameters to fit 106 datapoints. But 32 of the parameters (theΦ’s) are heavily constrained and have only a local influence.
If we look at the correlation parameters we see that the second parameter η2 hasreverted to zero as in the calibration experiments performed in Schoenmakers andCoffey (2000). We also observe that ρ∞ = 0.41. This parameter is interpreted asthe limiting correlation between rates when distance between maturity is increased.Looking at the 3d-plot of the correlation matrix in Figure 1 we can observe twoimportant features. The first is decorrelation, i.e. the fast reduction in dependencywhen the distance between maturity is increased. The second is the increase in
5The LMM with the current parametrization can also be calibrated to both swaptions and capsbut as described in Rebonato (2002) there are inherent differences between the swaption and thecap market that complicate simultaneous calibration to both markets.
26
interdependency between equidistant rates as maturity is increased. Graphicallythis means that the steepness of the surface when moving away from the diagonalis decreasing.
7.3 The CEV LMM
We also perform a market calibration of the CEV LIBOR market model whichhas φ = xp. As in the previous section we assume the same parametric form forvolatilities and correlation. The difference is that now we have an extra parameterp to determine. This parameter is highly dependent on the skew of the impliedvolatilies of interest rate options. Ideally we would like to determine p from OTM-Swaptions, since swap rates are our main concern. But data for these productsare not readily available as the market has very low liquidity. We therefore followHull and White (2001) and use OTM-Caplets instead.6 We have access to pricesquoted in Black-76 volatilities for 13 different strikes ranging from 1.5% to 10%.The procedure is in two steps. First we calibrate p and (a, b, c, d, Φkk≥1, η1, η2, ρ∞)from OTM-Caplet prices using formula (33) in the Appendix. Secondly we fix pat the value estimated in step 1 and recalibrate (a, b, c, d, Φkk≥1, η1, η2, ρ∞) toswaption implied volatilities by using formula (35) using the values from step 1 asstarting values in the optimization procedure.7 Again optimization is done usingthe standard root mean square distance metric. The errors we get in Table 6 aretolerable and comparable to the LMM case.
8 Numerical Results
8.1 Simulation Details
We use an Euler discretization of the forward LIBOR dynamics under the discretemoney market measure given in the SDE in (11). We choose stepsizes so each couponpayment date/exercise time t1, . . . , t23 are hit. We use 4 Euler steps between eachcoupon date totaling 92 steps per path. In the the pre-simulation step we use 25000antithetic paths (total 50000) to estimate the parameters of the exercise strategy. Inthe pricing step we use another 25000 (50000) paths. The sub-simulations involved
6Caps are settled quarterly for maturities below 2 years and semiannually thereafter. Wetherefore translate quarterly volatilities into semiannual volatilies using the procedure in Brigoand Mercurio (2006). Note also that caplet volatilites are stripped from cap volatilities using thestandard “bootstrap” procedure also described in Brigo and Mercurio (2006).
7Note that we do not have a closed formula for the Black-76 volatility in the CEV case butonly a pricing formula. We therefore plug in the price from formula (35) and invert the Black-76formula.
27
in calculating the lower bound are extremely slow and we therefore estimate thiswith only 25 (50) antithetic paths for 200 (400) sub-simulations. To further increasespeed we have also employed a so-called Runge-Kutta drift approximation in thesub-simulations only. This significantly increases the speed as we avoid having tocontinuously calculate the state dependent drift for each Euler step. The errorsfrom this approximation are in general extremely small (see for example Rebonato(2002) and Glasserman and Zhao (1999)). Zero coupon bond prices setting on non-tenor dates are calculated using the constant interpolation technique described inPiterbarg (2004a).
The short rate dynamics of the G2++ model is simulated in all cases withoutdiscretization error under the terminal forward measure with numeraire P (t, t23).We use the same amount of paths as in the LIBOR Market Models. For details onthis we refer to Brigo and Mercurio (2006).
8.2 Prices
In Table 8 we have calculated the theoretical price of the Dexia Dannevirke bondwith a 10000 bp principal. The value of the call option was calculated by findingthe difference in values to the price of a similar non-callable bond valued using thesame random numbers which can be done at almost zero computational expense.We used a second order polynomial of the core swap rate, the 6m LIBOR rate andthe 20 and 2 year swap rate with all rates setting on the respective exercise datesto calculate the exercise strategy. We have experimented with many other ratessuch as medium term swap rates and other LIBOR rates as well as higher orderpolynomial functions. In all of these cases the duality gap was larger, in some caseshigher than 100 bp. The results of one of these experiments are in Table 9 in theAppendix where we have used a 4th order polynomial function as well as includedmore rates. From this table we can see that the duality gaps have increased withan order of almost 8, and the option prices have been approximately halved. Theseresults corroborate the existing notion in the literature (see for example Piterbarg(2004b)) that basis functions should be kept simple.
We have also supplied a 95% confidence interval which is calculated as follows:
V0 − ∆0 − z1−α/2
√
s2V
K+
s2∆
K1
; V0 + z1−α/2sV√K
,
where sV and s∆ are the standard deviations for the upper bound and the duality gaprespectively. In Table 8 we see that the prices for the different models are statisticallydifferent from each other. The fact that the log-normal model gives higher pricesthan the CEV model is expected since the non-central chi-square distribution of the
28
CEV model gives more weight to the lower end of the distribution of forward rates.The normal distribution of the G2++ model implies even more weight to the lowerend than the CEV model and therefore yields the smallest price of the three.
In Table 10 we have tabulated the probability of exercise at each exercise timepoint. The overall distribution is very similar for the 3 models having at most 28.09%percent chance of early exercise. This is consistent with papers such as Andersenand Andreasen (2001) and Svenstrup (2005) which investigate the exercise strategiesfrom lower factor models in Bermudan swaptions. The only thing that stands out isthe large discrepancy at the final exercise time t22, where the G2++ model has morethan 5 times the higher probability of exercise. One reason for this discrepancy isthat there is less than one month between the last exercise time t22 and maturity att23. The small day-count fraction makes the coupon payment at t23 very small andtherefore the exercise decision is more idiosyncratic and prone to error.
Perhaps one of the most interesting things to observe from this section is theimplied estimate of the profit the issuing bank must have made from selling thesebonds. The bonds where sold at par (10,000bps) and DKK 2.4 billion worth ofbonds where sold. Looking at the lower end of the confidence interval for the G2++model and the upper end for the log normal LMM in Table 8 suggests that theprofit margin for Forbank ranges between 6.8% to 9.07%. In monetary terms thistranslates to a profit ranging from DKK 163.2 million to DKK 217.6 million. Con-sidering that this profit is riskless if a proper hedge portfolio is formed, the amountsare considerable. If we make the somewhat unfair and unscientific extrapolation ofthis result to a USD 50 billion market of CMS spread related products (see Jeffery(2006)) then these type of products have made considerable profits to the issuers.
8.3 Deltas
In figure 2 we show the delta profile for the product for each forward rate in theterm structure. All deltas are in basis points. The LIBOR market model deltas arecalculated using the pathwise approach and the G2++ model deltas are calculatedby the standard bump-and-revalue approach with a 1bp perturbation. The effect ofshifting the initial term structure has two opposite effects. The downwards effect isdue to two reasons. One is simple and due to higher discounting of the coupons; theother is due to the fact that higher initial interest rates make the short CMS ratehigher causing smaller coupons. We can see that the effect begins to shift aroundrates setting in 13 years or more which is natural since these rates only affect thelong CMS rate and not the short one. Overall the delta ratios are very close to eachother for the different models. This would suggest that all models are appropriatefor delta hedging.
29
8.4 Vegas
The vegas in Figure 3 are calculated with the pathwise approach for the LMM andCEV-LMM for the volatility of all forward rates. We notice a somewhat complexvega profile with large differences in the two models. The specific complex pattern ismainly due to the negative vega component in the product. As the volatility of the 2year CMS rate increases the price of the coupons decreases. However, the volatilityof the 2 year CMS rate cannot increase without also affecting the 20 year swap, andhence the effects will offset each other at some points. Looking closer at Figure 3 wesee that rates of low maturity have close to zero vega but as maturity increases thevegas begin to increase as the rates affect more and more coupon payments. Thiseffect is offset by the negative vega component in coupon payments which causes thedip around rates of 7-9 years maturity. The vegas increase again around 11 yearswhich is the maturity date of the bond, but drop as maturity increases further dueto the decreasing effect they have on coupons.
9 Conclusions
This paper has presented a methodology to price and risk manage the so-calledcallable CMS Steepener structured bond. We have used the simple G2++ modelfor the short rate as well as two different LIBOR market models. We have adaptedthe Least-Squares Monte Carlo procedure and shown how looking at the bond inits entirety can avoid a regression step. We have also adapted the procedure ofAndersen and Broadie (2004) to give a downward biased price estimate of issuercallable bonds. This is an important supplement to the upward biased estimateof the LSMC procedure as the gap between the two determines the quality of theexercise strategy. We then proceeded to show how hedge ratios can be calculated.
We have applied the theoretical insights on a particular callable CMS spreadbond. We have found that the different models of the term structure yield signifi-cantly different prices when calibrating to the same data and using the same exercisestrategy. We have also shown that our specific exercise strategy gives duality gapsof a tolerable size. An interesting bi-product of our analysis is an estimate of thesizeable profits the banks must have made from selling this particular bond.
To the authors’ knowledge this paper is the first about these particular productsand we have therefore opted to use relatively simple models of the term structure ofinterest. However, the use of stochastic volatility and/or jumps in the forward ratesare becoming more and more normal in both academia and practice (see Hagan,Kumar, Lesniewski, and Woodward (2002) and Glasserman and Kou (2003)). Ourresults on pricing could easily be extended to these cases. However, in the areaof hedging there are still many unanswered questions in the more advanced model
30
setups.Future work on CMS spread callables could be to evaluate the hedging per-
formance of different models over time. At lot of work has been done recentlyon simpler products such as European and Bermudan swaptions (see Pietersz andPelsser (2006) and the references therein) and the conclusions have mostly beenthat properly calibrated simple models perform very well against more complicatedalternatives. Our results for delta hedging also point in that direction but furtheranalysis are still needed.
31
A Proof of Theorem 1
For any finite submartingale we have
V P0 /N0 = inf
η∈I
EN
0 [N−1tη +
η∑
i=0
N−1ti
CFi)]
= infη∈I
EN
0 [N−1tn +
η∑
i=0
N−1ti
CFi) − Mη + Mη]
≥ infη∈I
EN
0 [N−1tη +
η∑
i=0
N−1ti
CFi) − Mη] + M0
≥EN
0
[
minη∈I
(
N−1tη +
η∑
i=0
N−1ti
CFi − Mη
)]
+ M0. (30)
The first inequality follows from from the optional sampling theorem for submartin-gales (see for example Hoffmann-Jorgensen (1994)). Let us now set Mn = Vtn/Ntn
where M0 = V P0 /N0. We first observe Mn is a submartingale as
Mn =Vtn
Ntn
= min
(
N−1tn +
n∑
i=0
N−1ti
CFi, EN
ti
[
N−1ti
Vti+1
]
)
≤ EN
tn
[
N−1tn+1Vtn+1
]
= EN
tn [Mn+1] .
Plugging this into the dual problem gives us
V D0 ≥ V P
0 + N0EN
0
[
minη∈I
(N−1η +
η∑
i=1
N−1ti
CFi −Vtη
Ntη
)
]
.
The second term in the above equation is positive since the exercise value of thebond will always be larger than or equal to the continuation value when followingan optimal strategy. We therefore have V D
0 ≥ V P0 . Taking the infimum of (30) we
get V D0 ≤ V P
0 which proves the duality relation V D0 = V P
0 attained at the optimalsubmartingale M∗
n = Vtn/Ntn .
B Calibration formulas
B.1 G2++ Model
The formula for a payer swaption is rather involved and it is stated here for com-pleteness. For the proof we refer to Brigo and Mercurio (2006).
32
Consider an interest rate swap with nominal value of 1 and strike K and paymentsat t1 < . . . . . < tn and denote τi = Ti+1 − Ti. Set ci = Kτi for i = 1. . . . . < n − 1and cn = τnK. The price of a payer swaption at t = 0 which gives the right to enterthe swap at t0 < t1 is
PS(0, K) =
∫ ∞
−∞
e−1/2(x−µxσx
)2
σx
√2π
[
Φ(−h1(x)) −n∑
i=0
λi(x)eκi(x)Φ(−h2(x))
]
dx, (31)
where
h1(x) =y − µy
σy
√
1 − ρ2xy
−ρ2
xy(x − µx)
σx
√
1 − ρxy
h2(x) =h1(x) + B(b, t0, ti)σy
√
1 − ρ2xy
λi(x) =ciA(t0, ti)e−B(a,t0,ti)x
κi(x) = − B(b, t0, ti)
[
µy − 1/2(1 − ρ2xy)σ
2yB(b, t0, ti) + ρxyσy
x − µx
σx
]
,
and
µx = −(
σ2
a2+ ρ
σξ
ab
)
[
1 − e−at0]
+σ2
2a2
[
1 − e−2at0]
+ρσξ
b(a + b)
[
1 − e−t0(a+b)]
µy = −(
ξ2
b2+ ρ
σξ
ab
)
[
1 − e−bt0]
+ξ2
2b2
[
1 − e−2bt0]
+ρσξ
a(a + b)
[
1 − e−t0(a+b)]
σx =σ
√
1 − e−2at0
2a
σy =ξ
√
1 − e−2bt0
2b
ρxy =ρσξ
(a + b)σxσy
[
1 − e−(a+b)t0]
.
B.2 LIBOR Market Models
Caplet prices in the log-normal LIBOR market model with tenor structureT = T1 < · · · < TM coincide with the Black(76) formula. The price of a capletpaying at time Ti the difference between the strike K and the forward rate settingat Ti−1 is
Cpl(0, Ti−1, Ti, K, v) =P (0, Ti)τiEi(Fi(Ti−1 − K)+)
=P (0, Ti)τi
[
Fi(0)Φ(d1) − KΦ(d2)]
33
where
d1 =log(Fi(0)/K) + v2/2
v
d2 =log(Fi(0)/K) − v2/2
v
v =
∫ Ti−1
0
σ2i (t)dt.
Swaption prices in the log-normal LMM cannot be derived in closed form. However,a well established approximation due to Rebonato (2002) can be used. Consideragain an interest rate swap with nominal value of 1 and strike K and payments atthe tenor structure T = Tα+1 < . . . . . < Tβ and denote τi = Ti+1 − Ti. The price ofa payer swaption at time t = 0 which gives the right to enter the swap at Tα is
PS(T , K)(0) = Cα,β(t)[
Sα,β(0)Φ(d1) − KΦ(d2)]
,
where
d1 =log(Sα,β(0)/K) + v2
α,β/2
vα,β
d2 =log(Sα,β(0)/K) − v2
α,β/2
vα,β
v2α,β =
β∑
i,j=1
wi(0)wj(0)Fi(0)Fj(0)ρi,j
Sα,β(0)
∫ Tα
0
σi(t)σj(t)dt, (32)
where Cα,β(0) =∑β
i=α+1 τiP (0, Ti) is the present value of a series of basis points alsoreferred to as the swaption numeraire. The weights wi follow from the fact that theswap rate can be written as a weighted sum of forward rates,
Sα,β(t) =
β∑
i=α+1
wi(t)Fi(t).
In this paper the tenor structure of the LIBOR Market Model is semiannual and thisdoes not coincide with the tenor structure of the swaptions in the European marketthat are annually settled. The weights in the above formula is therefore determinedas follows
wi(t) =P (t, Ti)
∑(β−α)/2i=1 2τiP (t, Tα+2i)
.
34
For further details we refer to Schoenmakers (2002).In the extended LMM cap prices can be derived in closed form. With a CEV
parameter 0 < p < 1 we have
Cpl(0, Ti−1, Ti, K) =P (0, Ti)τiEi((Fi(Ti−1) − K)+) (33)
=P (0, Ti)τi
[
Fi(0)(1 − χ2(a, b + 2, c)) − Kχ2(c, b, a)]
, (34)
where
a =K2(1−p)
(1 − p)2v
b =1
1 − p
c =Fi(0)2(1−p)
(1 − p)2v
v =
∫ Ti−1
0
σi(t)2dt,
where χ2(·, x, y) is the distribution function for a non-central χ2-distributed ran-dom variable with non centrality parameter value x and y degrees of freedom. Forstandard approximations of this function we refer to classical textbook of Johnsonand Kotz (1973). Again we use an approximation due to Andersen and Andreasen(2000) to find the swaption value:
PS(T , K) = Cα,β(0)[
Sα,β(0)(1 − χ2(a, b + 2, c)) − Kχ2(c, b, a)]
, (35)
where
a =K2(1−p)
(1 − p)2v
b =1
1 − p
c =Sα,β(0)2(1−p)
(1 − p)2v
v2α,β =
β∑
i,j=1
wi(0)wj(0)Fi(0)pF pj (0)ρi,j
S2pα,β
∫ Tα
0
σi(t)σj(t)dt.
35
References
Amin, A. (2003): “Multi-Factor Cross Currency LIBOR Market Mod-els: Implementation, Calibration and Examples,” preprint, available fromhttp://www.geocities.com/anan2999/.
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38
Tables and Graphs
Table 1: Payment dates
t1 20-06-2005 t13 20-06-2011t2 20-12-2005 t14 20-12-2011t3 20-06-2006 t15 20-06-2012t4 20-12-2006 t16 20-12-2012t5 20-06-2007 t17 20-06-2013t6 20-12-2007 t18 20-12-2013t7 20-06-2008 t19 20-06-2014t8 20-12-2008 t20 20-12-2014t9 20-06-2009 t21 20-06-2015
t10 20-12-2009 t22 20-12-2015t11 20-06-2010 t23 25-01-2016t12 20-12-2010
39
Table 2: Swaption Matrix (of implied volatilities in percent)
Maturity tMi / Tenor tTj 1 2 3 4 5 6 7
1 22.3 21.8 21.1 20.5 19.7 18.6 17.82 22.2 21.3 20.2 19.1 18.1 17.4 16.73 20.6 20.0 19.1 17.9 17.1 16.4 16.04 19.5 18.9 17.9 16.8 16.2 15.7 15.45 18.3 17.7 16.8 15.9 15.4 15.0 14.8
10 15.1 14.5 13.9 13.3 13.0 12.9 12.815 13.0 12.5 12.0 12.0 11.9 11.8 11.820 11.9 11.6 11.6 11.6 11.6 11.6 11.725 11.4 11.4 11.4 11.5 11.5 11.5 -30 11.0 - - - - - -
Maturity tMi / Tenor tTj 8 9 10 15 20 25 30
1 17.1 16.5 16.1 14.5 13.3 12.9 12.62 16.2 15.8 15.4 14.2 13.1 12.7 -3 15.6 15.3 15.1 13.9 13.0 12.6 -4 15.1 14.8 14.6 13.8 12.9 12.5 -5 14.6 14.4 14.3 13.4 12.7 12.4 -
10 12.7 12.6 12.5 13.0 11.4 - -15 11.8 11.8 11.8 11.3 - - -20 11.7 11.7 11.7 - - - -25 - - - - - - -30 - - - - - - -
40
Table 3: Calibration errors (in percent) in G2++
Maturity tMi / Tenor tTj 1 2 3 4 5 6 7
1 -0.41 -0.37 -2.28 -1.92 -1.79 -3.16 -3.352 -0.64 -1.54 -3.47 -4.81 -5.69 -5.32 -5.493 -2.72 -1.72 -2.29 -4.43 -4.85 -5.22 -4.254 -1.10 -0.01 -1.42 -3.92 -3.96 -3.89 -2.855 0.11 0.74 -1.08 -3.37 -3.62 -3.51 -2.19
10 2.25 0.36 -1.79 -4.01 -3.87 -2.30 -0.7815 0.42 -1.40 -3.78 -2.04 -1.09 -0.26 1.3520 -1.66 -2.75 -1.40 -0.24 1.15 2.66 4.7725 -2.76 -1.95 -0.65 1.63 3.11 4.64 -30 -3.91 - - - - - -
Maturity tMi / Tenor tTj 8 9 10 15 20 25 30
1 -3.46 -3.52 -2.73 0.36 1.84 5.77 8.462 -4.99 -4.26 -3.81 0.92 2.25 5.63 -3 -3.67 -2.68 -1.24 2.39 4.32 7.17 -4 -2.02 -1.38 -0.21 5.13 6.32 8.76 -5 -1.00 0.07 1.77 5.48 7.38 10.20 -
10 0.54 1.69 2.74 13.26 6.82 - -15 2.86 4.18 5.49 7.39 - - -20 6.10 7.40 8.66 - - - -25 - - - - - - -30 - - - - - - -
Parameter values are a = 3.47,b = 0.0461 ,σ = 0.0537, ξ = 0.00842 ρ = −0.891 . With datacalibrated to the Swaption implied volatilities on January 28, 2005.
41
Table 4: Calibration errors (in percent) in the log-normal LMM
Maturity tMi / Tenor tTj 1 2 3 4 5 6 7
1 -0.78 -0.58 0.64 2.07 2.26 0.73 0.652 -0.27 1.06 0.56 -0.54 -1.46 -0.83 -0.693 -0.64 0.98 0.88 -1.00 -0.81 -0.60 -0.594 -0.60 0.93 0.37 -0.87 0.09 -0.74 -0.455 -1.01 0.87 1.16 0.50 -0.70 -1.30 -0.35
10 0.42 0.36 -0.39 -0.99 -2.50 -1.88 -0.8415 0.27 0.17 -0.29 -0.66 -0.72 -0.87 0.1220 0.17 0.05 -0.14 -0.37 -1.44 -0.69 0.5725 0.12 -0.09 -0.24 -0.29 0.08 0.80 -30 -0.10 - - - - - -
Maturity tMi / Tenor tTj 8 9 10 15 20 25 30
1 0.70 -0.26 -0.29 -0.26 -2.06 -1.20 -1.102 -1.13 -1.22 -1.35 0.33 -1.57 -1.31 -3 -0.80 -0.35 0.68 1.40 -0.12 -0.34 -4 -0.06 0.26 1.37 3.18 1.08 0.40 -5 0.59 1.70 2.37 2.82 1.24 1.13 -
10 0.39 0.35 0.55 7.20 -3.80 - -15 0.50 0.72 0.49 -2.52 - - -20 0.82 0.69 0.96 - - - -25 - - - - - - -30 - - - - - - -
Average Error=1.74% Max error=7.19%
42
Table 5: Parameters in the log-normal LMM
a 0.033807 η1 0.881421b 0.904988 η2 0c 0.141077 ρ∞ 0.414192d 0.000169
i = Θi i = Θi
1, 2 1.374 33, 34 0.8493, 4 1.473 35, 35 0.7955, 6 1.473 37, 38 0.8847, 8 1.388 39, 40 0.840
9, 10 1.326 41, 42 0.83311, 12 1.260 43, 44 0.79813, 14 1.177 45, 46 0.83015, 16 1.077 47, 48 0.83817, 18 1.007 49, 50 0.88319, 20 1.084 51, 52 0.79921, 22 1.045 53, 54 0.80723, 24 0.985 55, 56 0.81225, 26 0.937 57, 58 0.84327, 28 0.864 59, 60 0.80129, 30 0.957 61, 62 0.77431, 32 0.906
43
Table 6: Calibration errors (in percent) in the Extended/CEV LMM
Maturity tMi / Tenor tTj 1 2 3 4 5 6 7
1 -0.88 -0.45 0.85 2.19 2.27 0.62 0.562 -0.38 1.15 0.66 -0.52 -1.57 -0.90 -0.773 -0.69 1.04 0.93 -1.08 -0.82 -0.63 -0.624 -0.64 0.97 0.33 -0.80 0.13 -0.69 -0.465 -1.02 0.82 1.33 0.64 -0.56 -1.23 -0.34
10 0.47 0.37 -0.36 -0.98 -2.51 -1.93 -0.9215 0.32 0.19 -0.28 -0.62 -0.70 -0.90 0.0720 0.20 0.08 -0.14 -0.38 -1.40 -0.70 0.5325 0.11 -0.07 -0.24 -0.26 0.12 0.79 -30 -0.08 - - - - - -
Maturity tMi / Tenor tTj 8 9 10 15 20 25 30
1 0.60 -0.34 -0.39 -0.30 -2.03 -1.11 -0.972 -1.20 -1.31 -1.45 0.30 -1.53 -1.22 -3 -0.86 -0.43 0.61 1.38 -0.08 -0.25 -4 -0.11 0.22 1.33 3.18 1.13 0.49 -5 0.58 1.69 2.36 2.85 1.32 1.23 -
10 0.30 0.28 0.48 7.17 -3.80 - -15 0.43 0.65 0.45 -2.56 - - -20 0.77 0.65 0.93 - - - -25 - - - - - - -30 - - - - - - -
Average Error=1.76% Max error=7.20%
44
Table 7: Parameters in the extended/CVV LMM
a 0.047104 η1 0.922098b 1.410679 η2 0c 0.092664 ρ∞ 0.395819d -0.01779 p 0.665
i = Θi i = Θi
1, 2 0.610 33, 34 0.4663, 4 0.691 35, 35 0.4525, 6 0.722 37, 38 0.4697, 8 0.698 39, 40 0.468
9, 10 0.683 41, 42 0.45911, 12 0.663 43, 44 0.43513, 14 0.628 45, 46 0.45015, 16 0.580 47, 48 0.45117, 18 0.570 49, 50 0.47919, 20 0.557 51, 52 0.43221, 22 0.570 53, 54 0.43023, 24 0.534 55, 56 0.43525, 26 0.522 57, 58 0.43127, 28 0.469 59, 60 0.44629, 30 0.520 61, 62 0.40831, 32 0.498
45
Table 8: Prices
Model Price Call option Duality gap 95% CLLog-normal LMM 9315.45(2.0) 91.02(1.6) 12.74(1.6) [9297.68 ; 9319.37]
CEV-LMM 9292.54(1.77) 112.29(1.8) 13.79(1.8) [9273.80 ; 9296.01]G2++ 9091.80(0.8) 127.42(1.3) 17.09(2.1) [9070.30 ; 9093.36]
All prices in basis points. Standard errors in parentheses.All prices are calculated with the same basis function which is a second order polynomial of fourrates. The 6m forward LIBOR, core swap rate and 2 and 20 year swap rate. All rates setting oneach exercise date.
46
Table 9: Prices with an alternative stopping strategy
Model Price Call option Duality gap 95% CLLog-normal LMM 9382.49(2.2) 24.02(1.1) 96.88(8.5) [9268.40 ; 9386.81]
CEV-LMM 9344.51(1.9) 46.83(1.4) 91.05(8.1) [9237.15 ; 9348.23]G2++ 9158.98(0.9) 60.24(1.3) 106.09(6.1) [9040.80 ; 9160.74]
All prices are calculated using a basis function consisting of a 4th order polynomial of the 6mforward LIBOR, the core swap rate, the 2 and 20 year swap rates, and the 20 year forwardLIBOR. All rates setting on exercise dates.
47
Table 10: Call time distribution
Times LMM CEV-LMM G2++ Times LMM CEV-LMM G2++t2 0.68% 0.64% 0.54% t13 0.75% 0.84% 0.88%t3 1.13% 1.10% 1.26% t14 0.67% 0.81% 0.99%t4 1.14% 1.12% 1.30% t15 0.84% 0.88% 1.02%t5 1.24% 1.25% 1.32% t16 1.01% 0.99% 1.00%t6 1.08% 1.17% 1.14% t17 1.13% 1.14% 1.04%t7 1.06% 1.14% 1.19% t18 1.38% 1.32% 1.16%t8 0.75% 0.83% 1.09% t19 1.16% 1.38% 1.16%t9 0.74% 0.88% 0.96% t20 1.43% 2.02% 1.46%t10 0.71% 0.79% 1.06% t21 1.88% 2.30% 2.56%t11 0.74% 0.80% 0.98% t22 0.65% 1.00% 5.09%t12 0.67% 0.76% 0.87% t23 79.16% 76.87% 71.91%
48
Figure 1: Implied Correlation
T j
Ti
ρi ,j
510
1520
2530
10
20
30
0.6
0.8
1.0
49
Figure 2: Deltas
-2
-1,5
-1
-0,5
0
0,5
1
0,5
1,5
2,5
3,5
4,5
5,5
6,5
7,5
8,5
9,5
10,5
11,5
12,5
13,5
14,5
15,5
16,5
17,5
18,5
19,5
20,5
21,5
22,5
23,5
24,5
25,5
26,5
27,5
28,5
29,5
30,5
31,5
G2
CEV-LMM
LMM
50
Figure 3: Vegas
-1,00%
-0,50%
0,00%
0,50%
1,00%
1,50%
2,00%
0,5
1,5
2,5
3,5
4,5
5,5
6,5
7,5
8,5
9,5
10,5
11,5
12,5
13,5
14,5
15,5
16,5
17,5
18,5
19,5
20,5
21,5
22,5
23,5
24,5
25,5
26,5
27,5
28,5
29,5
30,5
31,5
CEV-LMM
LMM
51
Chapter II
Fast and Accurate Option Pricing in a
Jump-Diffusion Libor Market Model ∗
David Skovmand,†
University of Aarhus and CREATES
May 1, 2008
Abstract
This paper extends, improves and analyzes cap and swaption approxima-tion formulae for the jump-diffusion Libor Market Model derived in Glasser-man and Merener (2003a).More specifically, the case where the Libor rate follows a log-normal diffusionprocess mixed with a compound Poisson process under the spot measure isinvestigated. The paper presents an extension that allows for an arbitraryparametric specification of the log-jump size distribution, as opposed to thepreviously studied log-normal case. Furthermore an improvement of the exist-ing swaption pricing formulae in the log-normal case is derived. Extensions ofthe model are also proposed, including displaced jump-diffusion and stochas-tic volatility.The formulae presented are based on inversion of the Fourier transform whichis approximated using the method of cumulant expansion. The accuracy ofthe approximations is tested by Monte Carlo experiments and the errors arefound to be at acceptable levels.
∗The author would like to thank Elisa Nicolato for helpful comments†Current affiliation: Aarhus School of Business and the Center for Research in Econometric
Analysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus Schoolof Business, Department of Business Studies, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark,e-mail: [email protected]
1 Introduction
The term Libor Market Model (LMM) is coined with reference to the models de-scribed in the three seminal papers Brace, Gatarek, and Musiela (1997), Miltersen,Sandmann, and Sondermann (1997) and Jamshidian (1997). The novelty of thesethree papers is that they present a unifying no-arbitrage approach to modeling theterm structure of discretely compounded rates such as Libor rates and swap rates.The main assumption in the papers is log-normality of the underlying Libor rate,which stands in opposition to stylized facts observed in financial data such as theimplied volatility smile in interest rate options. Extensions of the log-normal LMMhave since been proposed in order to remedy this deficiency, mirroring a similardevelopment in the equity options literature. One popular approach has been tomodel the Libor rates via a jump-diffusion as proposed in Glasserman and Kou(2003), Glasserman and Merener (2003a), Glasserman and Merener (2003b), andJarrow, Li, and Zhao (2007). All these papers have focused mainly on log-normaljump sizes as originally proposed by Merton (1976). However, the equity optionpricing literature has recently seen several deviations from this assumption. Mostnotable examples are the log-uniformly distributed jumps in Yan and Hanson (2006),Pareto-Beta jump sizes in Ramezani and Zeng (2007), and double exponential dis-tribution investigated in Kou (2002), Kou and Wang (2004), and Kou and Wang(2003). These alternative specifications are motivated by a need for a higher degreeof flexibility, such as asymmetry in the number of positive and negative jumps. Inthe double exponential case the main motivation is the ability to easily price certainexotic options. These motivations also apply to the Libor modelling case, but so farvery little work has been done in an LMM context.If the Libor rates are modeled as affine jump-diffusions (AJD, see Duffie, Pan, andSingleton (2000)) under the rates corresponding forward measures, the extension todifferent jump sizes in the LMM would be completely analogous to the equity case.Indeed the price of a call option on the Libor (a caplet) can be reduced to a singlenumerical integration.This paper investigates the less simple case when rates are specified as a diffusionplus a compound Poisson process, under the discrete risk-neutral measure (spotmeasure), defined by a discretely updated bank account numeraire. Modelling ratesunder the spot measure has the advantage that simulation from the model is simplerand results in a smaller discretization bias as well as variance than the alternativeforward measure specification, as demonstrated in Glasserman and Zhao (2000).The spot measure specification also allows for an easy handling of the dimension-ality of the model, which is not the case for the forward measure specification thatrequires as many jump processes as Libor rates in order for the forward rates tobe AJDs under their respective measures. As the number of rates are not uncom-
54
mon to range in the 60’s this can seriously complicate the use the forward measurespecification. The drawback of the spot measure specification used in this paper, isthat Libor rates are no longer AJDs under their respective forward measures. Thismeans that one has to resort to approximation techniques to price caplets as donein Glasserman and Merener (2003a).This paper derives approximative expressions for caplets and swaptions using differ-ent methods than Glasserman and Merener (2003a). The novelty of the alternativeroute taken in this paper, is that it allows for an arbitrary parametric specificationof the jump size distribution. The method employed, is based on approximating thecharacteristic function describing the jump sizes using a cumulant expansion tech-nique. The derived characteristic function can subsequently be used in the standardFourier inversion pricing methodology introduced by Carr and Madan (1999).However the classic Merton-76 style jump-diffusion is unable to capture impliedvolatility smiles across both maturity and strike. This problem is well known inthe equity option pricing literature, and has recently been documented in a LMMcontext by Jarrow, Li, and Zhao (2007) and Skovmand (2008). Specifically, if shortmaturity smiles are too be matched then the smile flattens too quickly as a functionof maturity due to the fast decline of skewness and kurtosis generated by the jumps.One solution to this problem is to displace the jump-diffusion with a constant, whichresults in skewness even for long maturity rates. In addition kurtosis can be gener-ated in the long maturity rates, by including stochastic volatility. Both extensionsare included and examined in this paper, and their consequences in terms of thederived approximations are discussed.All the proposed approximation formulas are subjected to a number of tests of ac-curacy using Monte Carlo simulations as a benchmark for the true price. Assuminglog-normal jump sizes the accuracy is also compared with the approximations pro-posed in Glasserman and Merener (2003a), and it is found that the proposed formu-las in this paper perform better when pricing swaptions, whereas in the caplet casethe differences are negligible. The case where jump sizes are log-double exponentialis also investigated with and without a displacement factor. In the log-normal aswell as the log-double exponential case the approximation works very well for shortermaturities. For long maturities comparatively larger errors in prices are observed,but in terms of percentages they are only above 1% in very few cases.
2 General Setup
Let P (t, Tk) be the time t zero coupon bond price with maturity at Tk. Consider atenor structure T0 < . . . . . < TK and constant day count fractions δ = Tk − Tk−1.The simply compounded forward LIBOR rates, henceforth referred to as Libor rates,
55
are defined as
Fk(t) = F (t, Tk, Tk+1) =P (t, Tk) − P (t, Tk+1)
δP (t, Tk+1).
2.1 Arbitrage Free Dynamics
The discrete bank account is defined as follows
B0 = 1,
BTk= BTk−1
[1 + δFk−1(Tk−1)], 1 ≤ k < M,
Bt = P (t, Tk)BTk, t ∈ [Tk−1, Tk].
It defines a measure normally referred to as the discrete risk-neutral measure or spotmeasure Q. The dynamics of the Libor rates can be defined under the spot measurein the following manner:
dFk(t)
Fk(t−)= [γ′
kµk − αk] dt + γ′kdW Q(t) + dJk(t), (1)
µk(t) =k∑
j=β(t)
δγjFj(t−)
1 + δFj(t−),
αk(t) =
∫
R
(ex − 1)k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex
)
λf(x)dx,
Jk(t) =
N(t)∑
ℓ=1
(eXℓ − 1),
0 ≤ t ≤ Tk, k = 1, . . . , K,
where W Q(t) is an D-dimensional Brownian motion under Q. γk is a D dimensionalconstant vector. N(t) is a Poisson process independent of W Q(t) with intensity λjumping at the random times τ1 ≤ · · · ≤ τn < t. The Xℓ’s are IID stochastic vari-ables normally referred to as marks with a distribution given by the density functionf(x) and the characteristic function φX(z). β(t) is the index of the first forwardLibor rate that has not expired at t.This specification is nested in the general semi-martingale specification in Jamshid-ian (1999), but this specific model has to the authors knowledge only been studiedin Glasserman and Merener (2003a).A more convenient way to write the jump process Jk(t) is as an integral over a ran-dom measure µ(dx, dt), also referred to as the jump measure, that assigns a mass of
56
one for each pair (Xℓ, τℓ)
N(t)∑
ℓ=1
(eXℓ − 1) =
∫ t
0
∫
R
(ex − 1)µ(dx, dt).
with a corresponding compensator νQ(dx, dt) which has the ability of making
N(t)∑
ℓ=1
h(Xℓ) −∫ t
0
∫
R
h(x)νQ(dx, dt),
a martingale for any function h satisfying standard regularity conditions. In thesimple compound Poisson case above the compensator is deterministic and given as
νQ(dx, dt) = λf(x)dxdt.
3 Caplet Pricing
A caplet with strike K is a claim that pays δ(Fk(Tk)−K)+ at time Tk+1. The priceunder the forward measure Fk with P (0, Tk+1) as numeraire is known from standardtextbooks to be:
Cplk = δP (0, Tk+1)EFk[
(Fk(Tk) − K)+]
.
In order to evaluate this expectation the first step is to switch to the forward measure
Proposition 1. Assume the dynamics are given by equation (1) under the discretebank account measure Q. Changing measures to Fk with P (t, Tk+1) as numeraireyields the following arbitrage free dynamics:
dFk(t)
Fk(t−)=γ′
kdW Fk(t) −∫
R
(ex − 1)νFk(dx, dt) + dJk(t), ∀t ≤ Tk, (2)
where Jk(t) =∑N(t)
ℓ=1 (eXℓ − 1), with compensator
νFk(dx, t) =k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex)
)
λf(x)dx. (3)
Proof. In Glasserman and Kou (2003) it is proved that the dynamics in (2) are freeof arbitrage. (3) follows from Theorem 7 in Jamshidian (1999) or Lemma 4.1 inGlasserman and Merener (2003a) .
57
This proposition shows that changing from the spot measure to the forward measurecomplicates the dynamics of the forward rates. From the compensator in (3) we seethat the jump process is no longer compound Poisson as it has state dependencyin the jump arrival rate as well as the jump size. Unfortunately this means thatan explicit caplet price cannot be derived and one has to resort to approximativemethods.Following Glasserman and Merener (2003a) I therefore seek to approximate thedynamics in equation (2) with the following martingale dynamics:
dFk(t)
Fk(t−)=γkdW (t) −
∫
R
(ex − 1)λ(t)f(x, t)dxdt + d
N(t)∑
ℓ=1
(eXℓ − 1)
(4)
where W (t) is a one dimensional Brownian motion, γk is a scalar, N(t) is a Poissonprocess with time dependent deterministic intensity λ(t) and the Xℓ’s are realiza-tions from a random variable X, with a distribution described by a time dependentdeterministic density function f(x, t).In order to determine λ(t) and f(x, t), the state dependent compensator from (3) isinvestigated.Taylor expanding around the time zero term structure gives us.
νF
k (x, t) = λf(x)×k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex+
δ(1 − ex)
(1 + δFj(0)ex)2[Fj(t) − Fj(0)] + O([Fj(t) − Fj(0)])
)
dx.
(5)
The term1+δFj(0)
1+δFj(0)ex is clearly close to 1, whereas the higher order terms are of
lower orders of magnitude. Ignoring these higher order terms yields the followingapproximate deterministic expression for the compensator
νF
k (x, t) ≈ νk(dx, t) =k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex
)
λf(x)dx.
As shown in the above definition the approximation is akin to the classical trick offreezing the rates at time zero used very frequently in the LMM literature (see forexample Brigo and Mercurio (2006)).Fixing time and integrating the approximate compensator gives us the approximate
58
arrival rate of the jumps:
λ(t) =
∫
R
νk(dx, t)
=
∫
R
k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex
)
λf(x)dx,
and similarly the approximate density
f(x, t) =νk(x, t)
∫
Rνk(dx, t))
=k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex
)
λ
λ(t)f(x).
Finally the diffusion parameter is reduced to a scalar by setting
γk =√
[γ′kγk].
Note that this last step is not approximative. In the absence of jumps (4) is in factequivalent in distribution to the dynamics in (2).Using the dynamics in (4) and the above defined approximations gives us the closedform option price
Proposition 2. Assume the Libor rate follows 1. Then the approximate price of aTk-caplet with strike K is given by
Cplk = P (0, Tk+1)δ
π
∫ ∞
0
e−(iz−α) log(K) φ(z − (α + 1)i)
α2 + α − z2 + i(2α + 1)zdz, (6)
where α > 0 is a tuning parameter satisfying |φ(z − (α + 1)i)| < ∞, ∀z ∈ R+.
The characteristic function φ(z) = E[eiz log(Fk(Tk))] is given by
φ(z) = φcont(z)φjump(z),
with components
φcont(z) = eiz log(Fk(0))− 12iz(z+i)γ2
k , (7)
φjump(z) = exp
(
−izαk + δ
∫
R
k∑
j=1
λ(Tj)(
eizx − 1)
f(x, Tj)dx
)
, (8)
where αk = δ∫
R
∑kj=1(e
x − 1)f(x, Tj)λ(Tj)dx
(PROOF) See appendix
59
3.1 Cumulant Expansion for φjump(z)
The expression in equation (8) is complicated in the sense that its evaluation requiressolving a numerical integral. Evaluating the outer integral in (6) with anotherintegral nested in the integrand is slow, and this effectively means the formula cannotbe used for calibration purposes. Fortunately one can approximate the φjump(z)function very accurately. This can be done by first observing that (8) can be writtenas
φjump(z) = exp
(
−izαk + δk∑
j=1
λ(Tj) (φX(z) − 1)
)
,
where φX(z) =∫
Reizxf(x, t) is the characteristic function of the random variable X
with density f(x, t). The objective is therefore to approximate φX(z), which is doneby writing it as an expansion in terms of the characteristic function for the markdistribution under the spot measure.Let X be the random variable with distribution given by the density f(x) thatdefines the mark distribution under the spot measure in (1). Its correspondingcharacteristic function φX(z) can be written as a cumulant expansion:
φX(z) = E[eizX ] = exp
(
∞∑
n=1
κn(iz)n
n!
)
, (9)
where κn are the cumulants of the distribution which are implicitly defined as thecoefficients of a Taylor expansion of the logarithm of the characteristic function (seefor example Abramovitz and Stegun (1972))
log(φX(z)) =∞∑
n=0
κn(iz)n
n!. (10)
In a similar manner φX(z) can be expressed as a cumulant expansion:
φX(z) = exp
(
∞∑
n=1
κn(t)(iz)n
n!
)
. (11)
Putting equation (9) and (11) together yields
φX(z) = E[eizX ] = exp
(
∞∑
n=1
(κn(t) − κn)(iz)n
n!
)
φX(z).
If X is Gaussian then the above formula is normally referred to as an Edgeworth orGram-Charlier expansion. The above series is a well known method in statistics used
60
to approximate one distribution with another. The underlying statistical theory canbe in Kolassa (2006).f(x) has known cumulants κn, whereas the cumulants of X are easily calculatedvia the recursive formula1
κn(t) = mn(t) −n−1∑
k=1
(
n − 1
k − 1
)
κk(t)mn−k(t), (12)
where mn(t) is the nth moment calculated numerically as
mn(t) =
∫
R
xnf(x, t)dx.
Summarizing the results and assuming a truncation point P , the jump componentin the characteristic function in Proposition 2 can be approximated by
φjump(z)
≈ exp
(
−izαk + δk∑
j=1
λ(Tj)
(
exp
(
P∑
n=1
(κn(Tj) − κn)(iz)n
n!
)
φX(z) − 1
))
. (13)
This approximation significantly increases the speed of the caplet pricing formula.In all cases studied in this paper P is set to 4 which provides more than sufficientaccuracy.
4 Swaption Pricing
A Ta× (Tb−Ta) payer swaption of strike K is a contract that gives the right but notthe obligation to, enter into a Ta× (Tb−Ta) payer swap at time Tb. This means thatif the swaption is exercised the holder will pay the fixed payments of δK and receivethe floating payments of δFi−1(Ti−1) for i = a + 1, . . . , b. The payoff at expirationcan easily be shown to be (see for example Hunt and Kennedy (2004))
Ca,b(Ta)(Sa,b(Ta) − K)+,
where
Ca,b(t) =b−1∑
i=a
δP (t, Ti+1),
Sa,b(t) =P (t, Ta) − P (t, Tb)∑b−1
i=a δP (t, Ti+1),
1The formula is a simple consequence of linking the two relations, κn = ∂nlog(φX(z))
∂zn
∣
∣
∣
z=0
and
mn = (−i)n ∂nφX(z)
∂zn
∣
∣
∣
z=0
61
where is called the Sa,b the swap rate. Ca,b is normally referred to as the PresentValue of a Basis Point(PVBP) and it is used as the numeraire which defines theswaption measure Sa,b.The swap rate can also be represented as a weighted sum of forward rates.
Sa,b(t) =b−1∑
i=a
wi(t)Fi+1(t),
wi(t) =P (t, Ti)
∑b−1j=a δP (t, Tj)
.
Note that the weights themselves are dependent of the forward rates through theirrelation to the zero coupon bonds, making the swap rate a complicated non-linearfunction of the underlying forward rates. A perhaps less misleading expression is
Sa,b(t) =1 −∏b−1
j=a1
1+δFj(t)
δ∑b−1
i=a
∏ij=a
11+δFj(t)
It then follows that the value of a payer swaption under the swaption measure S isgiven by (see again Hunt and Kennedy (2004))
PS(t) = Ca,b(t)ESa,b
t [(Sa,b(Ta) − K)+].
To evaluate this expression the next step is to change the numeraire.
Proposition 3. Assume the dynamics are given by equation (1). Changing measuresfrom the spot measure Q to the swaption measure Sa,b with Ca,b(t) =
∑b−1i=a δP (t, Ti+1)
as the numeraire yields the following arbitrage free dynamics for the forward rates(ignoring drifts)2:
dFk(t)
Fk(t−)=γ′
kdW a,b(t) − [. . . ]dt + dJk(t), (14)
where W a,b(t) is D dimensional Brownian motion under Sa,b and
Jk(t) =
N(t)∑
ℓ=1
(eXℓ − 1),
2The drift in (14) is complicated and in this case irrelevant hence it is left it out. In the casewithout jumps it is given in Brigo and Mercurio (2006) Proposition 6.8.2
62
with state dependent compensator νa,b(dx, t):
νa,b(dx, t) =b−1∑
k=a
wk(t)k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex
)
λf(x)dx,
where wk(t) = P (t,Tk+1)∑b−1
i=a δP (t,Ti+1).
Proof. Follows from Lemma 4.2 in Glasserman and Merener (2003a).
Ignoring the drifts the swap rate dynamics follow using Ito’s lemma for jump-diffusions (see for example Cont and Tankov (2004)):
dSa,b(t) =b−1∑
i=a
∂Sa,b(t−)
∂Fi(t−)Fi(t−)γ′
idW S(t)
+
∫
R
[Sa,b(t−, F (t−)(ex)) − Sa,b(t, F (t−))] µ(dx, dt)
+ [. . . ]dt, (15)
where the swap rate is written as a vector function R+ ×RK → R of the Libor termstructure Sa,b(t, F (t)) where F (t) = (F1(t), . . . , FK(t)).Furthermore it follows from Andersen and Andreasen (2001):
∂Sa,b(t)
∂Fi(t)=
Sa,b(t)δ
1 + δFi(t)
[
P (t, Tb)
P (t, Ta) − P (t, Tb)+
∑bk=i δP (t, Tk)
Ca,b(t)
]
. (16)
Due to the complicated nature of (15) and (16) the swap rate dynamics are highlyintractable. This means that to price a swaption we again have to resort to approx-imative techniques. Using the logic of the previous section the following martingaledynamics are posed as an approximation to the true dynamics in (15):
dSa,b(t)
Sa,b(t−)=γa,bdW (t) −
∫
R
Ha,b(x)λa,b(t)fa,b(x, t)dt
+ d
N(t)∑
ℓ=1
Ha,b(Xℓ)
,
where W (t) is a univariate Brownian motion, N(t) is a Poisson process independentof W (t) with deterministic time dependent intensity λa,b(t) and X1, X2, . . . are IID
stochastic variables drawn from a distribution with time dependent density fa,b(x, t).
63
The diffusion parameter is defined following Andersen and Andreasen (2000)
γa,b =
√
√
√
√
(
b−1∑
j=a
∂Sa,b(0)
∂Fj(0)
Fj(0)
Sa,b(0)γj
)(
b−1∑
j=a
∂Sa,b(0)
∂Fj(0)
Fj(0)
Sa,b(0)γj
)′
.
To derive the jump specific terms λa,b(t) and fa,b(x, t) the rates are again frozen attime zero to get an approximate compensator
νa,b(x, t) ≈ νa,b(x, t) =b−1∑
k=a
wj(0)k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex
)
λf(x).
As before the approximate intensity is found by integrating the above expression
λa,b(t) =
∫
R
b−1∑
k=a
wj(0)k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex
)
λf(x)dx,
and the density is found in a similar manner
fa,b(x, t) =νa,b(x, t)
∫
Rνa,b(y, t))dy
.
For the function Ha,b(x) that translates marks into jump sizes I propose using thefollowing
Ha,b(x) =Sa,b
(
0, F (0)ex)
− Sa,b(0, F (0))
Sa,b(0, F (0)). (17)
For a flat term structure the swap rate is well known to be close to a linear functionof the forward rates (see for example Rebonato (2002)) causing the function in (17)to resemble ex − 1. The formula in Section 5.4 in Glasserman and Merener (2003a)assumes exactly that Ha,b(x) = ex−1, in a slightly different setup. The choice in (17)captures the initial non-linearity in the swap rate vector function Sa,b(t, F (t)) andcan therefore be expected to be more accurate than the Glasserman and Merener(2003a) alternative.Using the approximative swap rate dynamics, swaption pricing is largely equivalentto the caplet as seen in the following proposition.
Proposition 4. Swaption PricingAssume the Libor rate dynamics are given by equation (1). The approximate priceof a Ta × (Tb − Ta) payer swaption with strike K is given by
PS(Ta, Tb) = Ca,b(0)1
π
∫ ∞
0
e(−iz−α) log(K) φ(z − (α + 1)i)
α2 + α − z2 + i(2α + 1)zdz,
64
where α > 0 is a tuning parameter satisfying |φ(z − (α + 1)i)| < ∞.
The characteristic function φ(z) = E[eiz log(Sa,b(Ta))] is given by
φ(z) = φcont(z)φjump(z),
with components
φcont(z) = eiz log(Sa,b(0))− 12iz(z+i)γ2
a,b ,
φjump(z) = exp
(
−izαa,b +
∫
R
a∑
j=0
λ(Tj)δ(eiz log(Ha,b(x)+1) − 1)fa,b(x, Tj)dx
)
,
where αa,b =∫
R
∑aj=1 Ha,b(x)λa,b(Tj)fa,b(x, Tj)dx
Proof. Substituting Fk(t) with Sa,b(t), δP (0, Tk+1) with Ca,b(0) and (ex − 1) withHa,b(x) the proof is identical to Proposition 2
Equivalent to the previous section a cumulant expansion is used in the jump com-ponent of the characteristic function to speed up the evaluation. The derivation isequivalent to the caplet case except it is done in terms of the transformed stochasticvariable log(Ha,b(X) + 1) where X has density fa,b(x, t). Assuming a truncationpoint P this gives us the following approximation
φjump(z)
≈ exp
(
−izαa,b + δ
a∑
j=1
λa,b(Tj)
(
exp
(
P∑
n=1
(κn(Tj) − κn)(iz)n
n!
)
φX(z) − 1
))
,
where moments and cumulants are given by
mn(t) =
∫
R
(
log(Ha,b(x) + 1))n
fa,b(x, t)dx (18)
κn(t) =mn(t) −n−1∑
k=1
(
n − 1
k − 1
)
κk(t)mn−k(t).
5 Displacing the Jump-Diffusion
A simple and popular way generate to implied volatility skews in the Libor rateoptions has been to model the displaced rate Fk(t) + d as a log-normal martingale
65
under the forward measure.3 Specifically with d > 0 the volatility skew, declines interms of moneyness and it is retained for long maturity options, making it a verypowerful extension despite its simplicity. Furthermore mixing a displaced diffusionwith a compound Poisson process has the interesting feature of being able to generatean asymmetric smile in the short end an a skew in the long end of the maturityspectrum.The displacement can easily be applied in this setup, since the displaced model isnested in the general semi-martingale model of Jamshidian (1999). It follows thatthe arbitrage free spot measure dynamics in (1) can be modified to4
dFk(t)
Fk(t−) + d= [γ′
kµk − αk] dt + γ′kdW Q(t) + dJk(t), (19)
µk(t) =k∑
j=β(t)
δγj(Fj(t−) + d)
1 + δFj(t−),
αk(t) =
∫
R
(ex − 1)k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex + d(ex − 1)
)
λf(x)dx,
Jk(t) =
N(t)∑
ℓ=1
(eXℓ − 1).
Jk(t) is a compound Poisson process with intensity λ and mark distribution f(x).Despite its immediate appeal the model has the problematic feature that for d > 0the Libor rates are no longer guaranteed to be positive. Whether this poses anysignificant arguments against the model is unclear5.
5.1 Caplet Pricing
Caplets are priced by first noticing that
Cplk = δP (0, Tk+1)EFk[
(Fk(Tk) − K)+]
= δP (0, Tk+1)EFk
[
(Fk(Tk)(d) − K(d))+
]
,
3In equity option pricing this approach was pioneered by Rubinstein (1983) and extensivelystudied for example in Rebonato (2004). In an LMM context both Rebonato (2002) and Brigoand Mercurio (2006) study it in detail
4Specifically from Jamshidian (1999) Theorem 7 with βi = Φi = Li + d where Li is the Liborrate in that particular setup
5A similar case of non-positivity has frequently been made, to no apparent avail, against theclassic Vasicek (1977)-model for the short interest rate
66
with Fk(Tk)(d) = Fk(Tk) + d and K(d) = K + d. The price can then be seen as a call
option on the displaced rate F(d)k (Tk) with strike K(d). The dynamics of Fk(Tk)
(d)
can then be approximated in the exact same manner as in Section 3. The resultingformula is stated in the appendix.
5.2 Swaption Pricing
Following Section 4 gives us the following the swap rate rate dynamics(ignoringdrifts) under the swaption measure
dSa,b(t) =b−1∑
i=a
∂Sa,b(t−)
∂Fi(t−)(Fi(t−) + d)γ′
idW S(t)
+
∫
R
[Sa,b(t−, (F (t−) + d)(ex − 1)) − Sa,b(t, F (t−))] µ(dx, dt)
+ [. . . ]dt,
where µ(dx, dt) is the jump measure with compensator
νda,b(dx, t) =
b−1∑
k=s
wk(t)k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex + d(ex − 1)
)
λf(x)dx. (20)
This SDE is approximated by the martingale dynamics
dSa,b(t)
Sa,b(t) + d=γd
a,bdW (t) −∫
R
Hda,b(x)λd
a,b(t)fda,b(x, t)dt
+ d
N(t)∑
ℓ=1
Hda,b(Xℓ)
,
where N(t) is a Poisson process with intensity λda,b(t) =
∫
Rνa,b(x, t)dx where νa,b(dx, t)
is an approximation to the compensator in (20) defined as
νa,b(dx, t) =b−1∑
k=s
wk(0)k∏
j=β(t)
(
1 + δFj(0)
1 + δFj(0)ex + d(ex − 1)
)
λf(x)dx.
Furthermore X1, X2, . . . are IID random variables with density fda,b(x, t) =
νa,b(x,t)
λda,b
(t).
Following the non-displaced case the function translating the marks into jumps isdefined as
Hda,b(x) =
Sa,b
(
0, (F (0) + d)(ex − 1))
− Sa,b(0, F (0))
Sa,b(0, F (0)) + d.
67
Finally following Andersen and Brotherton-Ratcliffe (2005) the diffusion parameteris approximated by
γda,b =
√
√
√
√
(
b−1∑
j=a
∂Sa,b(0)
∂Fj(0)
Fi(0) + d
Sa,b(0) + dγj
)(
b−1∑
j=a
∂Sa,b(0)
∂Fj(0)
Fi(0) + d
Sa,b(0) + dγj
)′
As with caplets, the pricing formula is largely the same and given in the appendix.
6 Stochastic Volatility
The displacement factor adds limited flexibility since it generates solely monotonicskews in implied volatility. This is an obvious drawback since non-monotonic smilesare observed in the long maturity spectrum (see Skovmand (2008)). The mostpopular solution to this problem, in equities as well as interest rate models, hasbeen to add stochastic volatility in the diffusion component.Following Andersen and Brotherton-Ratcliffe (2005) who study the pure diffusioncase we can extend the dynamics in (1) to include stochastic volatility as follows
dFk(t)
Fk(t−)= [V (t)γ′
kµk − αk] dt + γ′k
√
V (t)dW Q(t) + dJk(t),
µk(t) =k∑
j=β(t)
δγjFj(t−)
1 + δFj(t−),
αk(t) =
∫
R
(ex − 1)k∏
j=β(t)
(
1 + δFj(t−)
1 + δFj(t−)ex
)
λf(x)dx,
Jk(t) =
N(t)∑
j=1
(eXj − 1),
0 ≤ t ≤ Tk, k = 1, . . . , K,
where V (t) denotes the stochastic volatility process which is assumed to follow themean reverting dynamics posed in Cox, Ingersoll, and Ross (1985)
dV (t) = η(κ − V (t))dt + ǫ√
V (t)dZ(t).
Z(t) is a Brownian motion independent of W Q(t). Fortunately adding stochasticvolatility does not introduce further error in the caplet approximation since theV (t) process is unaffected by a change to the forward measure. The same cannot besaid when pricing swaptions, but the resulting error is well studied and quite small
68
as demonstrated in Andersen and Brotherton-Ratcliffe (2005).Since the extension only affects the diffusion component it follows immediately fromAndersen and Brotherton-Ratcliffe (2005) Lemma 2 that the resulting prices forcaplets and swaptions are found by replacing φcont(z) in Proposition 2 and 4 withelog(Fk(0))I(Tk, z) and elog(Sa,b(0))I(Ta, z) respectively, where
I(t, z) = eA(t,z)+B(t,z)V (0),
with
B(t, z) =κ − D(z)
ǫ2
1 − e−D(z)t
1 − k−D(z)k+D(z)
e−D(z)t,
A(t, z) = κθǫ−2
(
(κ − D(z))t − 2 log
( k−D(z)k+D(z)
e−D(z)t − 1
κ−D(z)κ+D(z)
− 1
))
.
D(z) = 2√
κ2 + z(z + i)ǫ2
Finally it can also be noted that the stochastic volatility component is without a”leverage effect” i.e a correlation between Z(t) and W Q(t). Introducing leverageadds another approximation step since the V (t) process becomes state dependentunder the forward measure as shown in Wu and Zhang (2006) who derive cap andswaption approximations in the pure diffusion case. However, the asymmetric smilenormally generated from the leverage effect (see for example Heston (1993)) is alsoobtainable simply by displacing the dynamics with a constant as done in the previoussection. This approach has the advantage that it does not induce a state dependentvolatility term and hence requires no further approximation. Caplets and swaptionscan then be priced by the applying the same adjustments to the displaced jump-diffusion formulas given in the appendix.
7 Numerical Testing
In this section the formulas are tested using two different examples of jump sizedistribution; the normal and the double exponential distribution. Furthermore thedisplaced diffusion case is investigated, butthe stochastic volatility case is omittedsince it relates purely to the diffusion component, which has already been studiedin Andersen and Brotherton-Ratcliffe (2005).The diffusion component is set to be 2-dimensional, and since the jump componentis the point of interest in the paper the diffusion volatility is chosen to be low withγk = (0.0007, 0.0008)′.The initial term structure of the Libor rates is taken from the EURIBOR market
69
recorded from Datastream on May 5th 2006. The the term structure is plotted inFigure 1 revealing a humped term structure increasing until around the 15 yearmaturity and decreasing for longer maturities.There are two approximative steps in the procedure in the paper. The first isfixing the rates at zero in the state dependent compensator in equation (5), and thesecond is the cumulant expansion approximation in equation (13) and (18). Errorscoming from the latter approximation can be controlled by adding more terms in theexpansion. Furthermore an analysis shows that the maximum percentage pricingerrors from this approximation with P = 4 are around 0.01% for swaptions, andsignificantly smaller for caplets. As this section will show the total approximationerrors are generally at least one order of magnitude bigger, therefore P is kept at 4and the error terms are ignored in the remainder.Another issue with using the Fourier inversion formulas in propositions 2 and 4,is the choice of the tuning parameter α. The optimal choice of α minimizes thetotal variation in the integrand, which causes the numerical integration scheme torequire a minimum number of function evaluations for a desired level of accuracy.Optimality of α is beyond the scope of this paper therefore the tuning parameter issomewhat arbitrarly set at α = 0.75. This allows for the entire caplet price surfacewith 77 prices to be calculated in about 0.5 seconds using Matlab. A number ofpapers exist on this topic for example Lee (2004) and Lord and Kahl (2007) andthese can be referred to if further speed is required.
7.1 Normal Mark Distribution
Recall the the normal distribution has density
f(x) =1
√
2πσ2J
e−
(x−µJ )2
2σ2J ,
with characteristic function
φX(z) = eµJ iz−z2σ2/2.
The cumulants of the normal distribution are κ1 = µJ , κ2 = σ2J and κ3 = κ4 · · · = 0.
The model is tested with two different sets of parameters.
Parameter Set ANORM: µJ = −0.02, σJ = 0.08 and λ = 5.
Parameter Set BNORM: µJ = −0.025, σJ = 0.15 and λ = 1.5.
The first specification has jumps that occur on average 5 times a year, with a jump
70
size negatively biased, and the second specification has jumps that are rarer eventsoccurring only 1.5 times a year on average, but with a bigger magnitude. Both havethe same ATM implied caplet volatility at around 18%.Caplet and swaption prices are found by generating monte carlo simulations underthe spot measure with a standard log-Euler discretization scheme (see Glasserman(2004)) applied to the SDE in equation (1). The jumps are simulated independentlyusing algorithm 6.2 in Cont and Tankov (2004). 500000 paths are simulated usingantithetic sampling for the Brownian motion, and the same seeds are used in bothparameter sets in order to facilitate comparison. In general the simulation is partic-ularly slow because the drift features an integral that has to be reevaluated at eachtime point. Perhaps some of the many approximative techniques already existingfor the Libor Market Model (see Rebonato (2002)) could be applied to speed up thecomputations without loss of accuracy, but this is left for future research.A fixed grid is used in the discretization of equation (1) with a stepsize equal to0.125. Due to the state-dependent drift, discretization error cannot be avoided. Butthe magnitude can be evaluated following Glasserman and Zhao (2000), who arguethat a good measure of discretization error is the percentage differences betweenbond prices from the simulation, and bond prices calculated from the Libor termstructure. In the simulations performed in this paper the percentage differences areof the order 0.001% for the 20 year maturity, and lower in absolute terms for shortermaturities hence discretization biases can be safely ignored.The caplet prices are shown in the Table 1 and 2. The table shows Monte Carloprices as well as the price from Proposition 2(CE), found using the cumulant ex-pansion approximation, and the price calculated using the Glasserman and Merener(2003a) formula(G&M). Except for a few cases in Table 2 the caplet prices revealno noticeable difference between the CE formula, and the G&M-formula. A closeranalysis shows that differences are at the level of 10−5 Bps and therefore withoutany financial relevance.Except for a few cases the differences between the approximative price and the MCprice are not small enough to be explained by Monte Carlo variation, which is per-haps a bit troubling. The approximation appears to deteriorate further as maturityincreases. To evaluate these errors in relative terms, the percentage pricing errorsalong with the half-widths of the 95% confidence limit divided by the MC price isplotted in Figures 2 and 3. While the errors are outside the of the MC confidencelimit, they are still quite small and only outside the 1% for deep OTM prices. As afinal check on the approximation errors, implied volatilies are plotted in Figures 4and 5. For both parameter sets it shows a pronounced volatility smile for the 1 and3 year maturity, and an almost flat volatility curve for the longer maturities. Thedeteriation of the approximation is markedly clearer when looking at implied vol.But even for the longest maturitues the errors levels are well within normal bid-ask
71
spreads typically around 0.005.
The results for Swaption prices are shown in Tables 3 and 4, for the two differentparameter sets. In this case Monte Carlo confidence limits are larger and the errorsfluctuate in and out the 95% interval across maturity and moneyness. The approx-imation errors grow as maturity increases similar to the caplet case. It can also beobserved that the two approximations are indistinguishable for short tenors whichcan be expected since the 1 year tenor swaption behaves essentially like a caplet.But for longer tenors the CE formula is consistently better than the G&M formulain both parameter sets. The differences in the two approximations are investigatedfurther in terms of percentage pricing errors plotted in Figures 8 and 9. Here it canbe observed that in general percentage errors increase heavily as tenor increases.This was also found in the numerical experiments performed by Glasserman andMerener (2003a). The figures show that the CE formula generally performs betterthan the G&M formula in the long tenor case.Finally, implied volatilities are plotted figures 6 and 7. It can be seen that some ofthe larger percentage price errors translate into very small errors in implied volatil-ity. The errors increase when maturity increases, but as in the caplet case they arenot critical around the at-the-money level.
7.2 Double Exponential Mark Distribution
A more flexible jump size distribution can be introduced, by choosing marks follow-ing a double exponential distribution. This distribution has been used with somesuccess in the equity and currency option price literature in Kou (2002), Kou andWang (2003), and Kou and Wang (2004). It was also suggested for the use as amark distribution in the Libor Market Model in Glasserman and Kou (2003), butthey did not further investigate its use.The double exponential distribution has the following density function
f(x) = pη1e−η1x1x≥0 + (1 − p)η2e
η2x1x<0,
here p denotes the probability of an upward jump and (1 − p) the correspondingprobability of a downward jump. The two parameters η1 and η2 control the taildecay of positive and negative axis. Hence the distribution allows for both positiveand negative skewness.For use in our pricing formulas we need the characteristic function given in Kou andWang (2003) Lemma 2.1 as
φX(z) =pη1
η1 − iz+
(1 − p)η2
η2 + iz,
72
This function can be extended to the complex domain with a strip of regularitygiven by−η2 < Im(z) < η1. This means the tuning parameter α has to satisfy −η2 <
−(1 + α) < η1. Recall the nth moment of the distribution is mn = ∂nφX(z)∂zn (0)
yielding
m1 =p
η1
− 1 − p
η2
,
m2 =2p
η12
+ 21 − p
η22
,
m3 =6p
η13− 6
1 − p
η23
,
m4 =24p
η14
+ 241 − p
η24
,
The corresponding cumulants may then be calculated from equation (12). The re-sulting expressions are very lengthy, and is available from the author by request.The formulas are tested using the same diffusion component as used in the normaldistribution case, and with the following parameter set.
Parameter Set DE: p = 0.5 η1 = 1/0.05 and η2 = 1/0.07, with jump intensityλ = 5.
This corresponds to a mean jump size in log differences 5% and 7% for positive andpositive and negative jumps respectively. This parameter set is chosen to approxi-mately match the corresponding level of ATM implied volatility used when testingnormally distributed jump sizes. The caplet and swaption prices can be found intables 5 and 6 and what can be seen is that error levels are similar to the normalcase, and the accuracy goes from being very good in the short maturity spectrum, topoorer in long maturities. The percentage errors and implied vol plotted in figures12, 13 reveal the same pattern.
7.3 Displaced Diffusion
Here the dynamics in (19) are assumed. The dynamics are chosen to have doubleexponential marks and the same diffusion component and term structure of interestrates as in the previous sections.
Parameter Set DEDD: d = 0.04, p = 0.5, η1 = 1/0.03, η2 = 1/0.03 and λ = 5.
Note that these parameters no longer have interpretation of positive and negative
73
mean jump size in log changes because of the displacement factor. But they areagain chosen to have the same level of ATM implied vol as the previous sectionsaround 18%.The caplet and swaption prices are given in tables 7 and 8 and the errors show thatthe approximation fares equally well compared to the previous cases. The impliedvolatilities are graphed in figures 15 and 17 and here the effect of displacing thediffusion is noticeable since the short maturities are practically indistinguishablefrom the previous cases, but in the long run the skew is retained.
8 Summary
This paper derives approximate expressions for caplets and swaptions under the as-sumption that the Libor rates follow a diffusion with compound Poisson jumps underthe spot measure. The model extends the existing pricing methods by allowing foran arbitrary parametric distribution driving the jump sizes. Furthermore, the accu-racy of the derived formulas in the classic log-normal jump size case are comparedto existing approximations by Glasserman and Merener (2003a). The results from aMonte Carlo analysis show that the derived approximations are superior in pricingswaptions with longer tenors. Finally, the paper also extends the framework bydisplacing the jump-diffusion, and allowing for stochastic volatility in the diffusioncomponent.
74
A Proof of Proposition 2
Proof. Defining f(t) = log(Fk(t)) and using Ito’s lemma for jump-diffusions (see forexample Cont and Tankov (2004)) we get
df(t) =
[
−∫
R
(ex − 1)λ(t)f(x, t)dx − 1
2γ2
k
]
dt + γkdW (t)
+
∫
R
(
log[
Fk(t−) + Fk(t−)(ex − 1)]
− log(Fk(t−)))
µ(dx, dt)
=
[
−∫
R
(ex − 1)λ(t)f(x, t)dx − 1
2γ2
k
]
dt + γkdW (t) +
∫
R
xµ(dx, dt),
where µ(dx, dt) is the jump measure corresponding to the compound Poisson processwith compensator νk(x, t) = λ(t)f(x, t).The terminal state can then be written as
f(Tk) =f(0) − αk −1
2γ2
kTk +
∫ Tk
0
γkdW (t) +
∫ Tk
0
∫
R
xµ(dx, dt), (21)
where αk =∫
Rδ∑k
j=1(ex − 1)f(x, Tj)λ(Tj)dx. Since the underlying jump density is
the same between tenor points we can write
∫ Tk
0
∫
R
xµ(dx, dt) = δ
k∑
j=0
N(Tj)∑
n=N(Tj−1)+1
Xjn,
where Xj1 , . . . , X
jn are IID random marks drawn from the distribution with density
f(x, Tj). Defining ∆N(j) = N(Tj) − N(Tj−1) − 1 and observing that ∆N(j) ∼po(λ(Tj)δ), as well as conditioning on the number of jumps in each interval and
75
using their independence yields
E[eiz∫ Tk0
∫
Rxµ(dx,dt)] =
k∏
j=0
E[eiz∑N(Tj)
n=N(Tj−1)+1Xj
n
]
=k∏
j=1
∞∑
n=0
P (∆N(j) = n)E[eizXjn)]n
=k∏
j=1
∞∑
n=0
e−λ(Tj)δ(λ(Tj)δ)
n
n!
(∫
R
eizxf(x, Tj)dx
)n
=k∏
j=1
exp
(
λ(Tj)δ
[∫
R
eizxf(x, Tj)dx − 1
])
= exp
(
δ
∫
R
k∑
j=1
λ(Tj)(
eizx − 1)
f(x, Tj)dx
)
.
From (21) and the above result, it follows that
E[eizf(Tk)] = φcont(z)φjump(z),
where
φcont(z) = eiz log(Fk(0))− 12iz(z+i)γ2
k
φjump(z) = exp
(
−izαk + δ
∫
R
k∑
j=1
λ(Tj)(
eizx − 1)
f(x, Tj)dx
)
.
The formula in (6) then follows from equation (6) in Carr and Madan (1999)
76
B Pricing Caplets with a Displacement
Following the exact same approximation steps as Section 3 gives us the approximatedynamics
dFk(t)
Fk(t−) + d=γkdW (t) −
∫
R
(ex − 1)λd(t)fd(x, t)dx + d
Nd(t)∑
ℓ=1
(eXℓ − 1)
,
0 ≤ t ≤ Tk, k = 1, . . . , K,
where the Nd(t) is a Poisson process with intensity
λ(t) =
∫
R
k∏
j=β(t)
(
1 + Fj(0)
1 + δFj(0)ex + d(ex − 1)
)
λf(x)dx, (22)
and the Xℓ’s are distributed with density
fd(x, t) =
∫
R
k∏
j=β(t)
(
1 + Fj(0)
1 + δFj(0)ex + d(ex − 1)
)
λf(x)dx. (23)
This yields the following proposition
Proposition 5. Assume the Libor rate follows the dynamics 19. Then the approx-imate price of a Tk-caplet with strike K is given by
Cplk = P (0, Tk+1)δ
π
∫ ∞
0
e−(iz−α) log(K+d) φ(z − (α + 1)i)
α2 + α − z2 + i(2α + 1)zdz,
where α is a predetermined dampening constant.The characteristic function φ(z) = E[eiz log(Fk(Tk)+d)] is given by by
φ(z) = φcont(z)φjump(z).
The components of the characteristic function are
φcont(z) = eiz log(Fk(0)+d)− 12iz(z+i)γ2
k
φjump(z) = exp
(
−izαk + δ
∫
R
k∑
j=1
λ(Tj)(
eizx − 1)
f(x, Tj)dx
)
,
where αk = δ∫
R
∑kj=1(e
x − 1)f(x, Tj)λ(Tj)dx with fd(x, Tj) and λd(Tj) defined in(23) and (22) respectively
the cumulant expansion is the same as equation (13) with f(x, t) and λ(t) replacedby fd(x, t) and λd(t)
77
C Pricing Swaptions with a Displacement
Proposition 6. Swaption PricingAssume the Libor rate dynamics are given by equation (19). The price of a Ta ×(Tb − Ta) payer swaption with strike K is given by
PS(Ta, Tb) = Ca,b(0)1
π
∫ ∞
0
e(−iz−α) log(K+d) φ(z − (α + 1)i)
α2 + α − z2 + i(2α + 1)zdz,
where α > 0.5 is a tuning parameter. The characteristic function φ(z) = E[eiz log(Sa,b(Ta))+d]is given by
φ(z) = φcont(z)φjump(z),
where the two components of the characteristic function are
φcont(z) = eiz log(Sa,b(0)+d)− 12iz(z+i)γ2
a,b
φjump(z) = exp
(
−izαa,b +
∫
R
a∑
j=0
λ(Tj)δ(eiz log(Hd
a,b(x)+1) − 1)fd
a,b(x, Tj)dx
)
,
with αa,b =∫
R
∑aj=1 Hd
a,b(x)λda,b(Tj)f
da,b(x, Tj)dx
The cumulant expansion can be carried out as in equation (18) with Ha,b(x), fa,b(x, t)
and λa,b(t) replaced by Hda,b fd
a,b(x, t) and λda,b(t).
78
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81
Tab
le1:
Caple
tP
rices
for
the
Norm
aldistrib
utio
n,Para
mete
rSet
AN
OR
M0.5
0.81
1.21.5
T=
1(MC
)90.86
(0.00082)38.28
(0.00057)12.88
(0.00032)2.61
(0.00015)0.18
(4e-005)T
=1(C
E)
90.8538.27
12.882.61
0.18T
=1(G
&M
)90.85
38.2712.88
2.610.18
T=
3(MC
)90.12
(0.0055)43.29
(0.0039)22.37
(0.0028)10.36
(0.0019)2.89
(0.0011)T
=3(C
E)
90.0943.25
22.3510.35
2.89T
=3(G
&M
)90.09
43.2522.35
10.352.89
T=
7(MC
)84.51
(0.026)47.88
(0.021)31.41
(0.017)20.24
(0.014)10.33
(0.01)T
=7(C
E)
84.4247.79
31.3320.18
10.30T
=7(G
&M
)84.42
47.7931.33
20.1810.30
T=
10(MC
)78.27
(0.048)47.89
(0.04)34.00
(0.034)24.07
(0.03)14.41
(0.024)T
=10(C
E)
78.0947.72
33.8523.95
14.32T
=10(G
&M
)78.09
47.7233.85
23.9514.32
T=
15(MC
)65.68
(0.081)43.84
(0.071)33.60
(0.064)25.92
(0.057)17.81
(0.049)T
=15(C
E)
65.4543.59
33.3725.70
17.63T
=15(G
&M
)65.45
43.5933.37
25.7017.63
T=
20(MC
)53.14
(0.098)37.62
(0.088)30.21
(0.08)24.49
(0.074)18.17
(0.065)T
=20(C
E)
52.8937.35
29.9424.23
17.93T
=20(G
&M
)52.89
37.3529.94
24.2317.93
This
table
show
sth
eca
plet
price
for
diff
erent
matu
ritiesacro
ssm
oney
ness,
X/F
K(0
).M
Cden
otes
the
Monte
Carlo
price
with
the
halfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2,and
G&
Mis
the
caplet
form
ula
deriv
edin
Sectio
n5.3
inG
lasserm
an
and
Meren
er(2
003a).
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
.
82
Tab
le2:
Caple
tP
rice
sfo
rth
eN
orm
aldis
trib
uti
on,Para
mete
rSet
BN
OR
M0.
50.
81
1.2
1.5
T=
1(M
C)
90.8
6(0
.001
1)38
.45
(0.0
0079
)11
.72
(0.0
005)
3.01
(0.0
0029
)0.
42(0
.000
13)
T=
1(C
E)
90.8
738
.47
11.7
43.
010.
43T
=1(
G&
M)
90.8
738
.47
11.7
43.
010.
43T
=3(
MC
)90
.09
(0.0
066)
43.2
0(0
.005
)21
.98
(0.0
038)
10.2
3(0
.002
8)3.
22(0
.001
8)T
=3(
CE
)90
.18
43.2
622
.03
10.2
63.
24T
=3(
G&
M)
90.1
843
.26
22.0
310
.26
3.24
T=
7(M
C)
84.3
8(0
.03)
47.7
0(0
.025
)31
.20
(0.0
21)
20.0
9(0
.018
)10
.37
(0.0
14)
T=
7(C
E)
84.5
447
.82
31.2
820
.16
10.4
3T
=7(
G&
M)
84.5
447
.82
31.2
820
.16
10.4
3T
=10
(MC
)78
.05
(0.0
54)
47.6
8(0
.046
)33
.79
(0.0
41)
23.9
1(0
.036
)14
.36
(0.0
29)
T=
10(C
E)
78.2
247
.78
33.8
723
.98
14.4
1T
=10
(G&
M)
78.2
247
.79
33.8
723
.98
14.4
2T
=15
(MC
)65
.43
(0.0
88)
43.6
2(0
.077
)33
.40
(0.0
7)25
.75
(0.0
64)
17.6
8(0
.055
)T
=15
(CE
)65
.56
43.6
833
.43
25.7
617
.69
T=
15(G
&M
)65
.56
43.6
833
.43
25.7
617
.70
T=
20(M
C)
52.9
7(0
.1)
37.4
8(0
.091
)30
.08
(0.0
84)
24.3
7(0
.077
)18
.07
(0.0
68)
T=
20(C
E)
52.9
937
.45
30.0
224
.30
17.9
9T
=20
(G&
M)
52.9
937
.45
30.0
324
.30
17.9
9
This
table
show
sth
eca
ple
tpri
cefo
rdiff
eren
tm
atu
riti
esacr
oss
money
nes
s,X
/F
K(0
).M
Cden
ote
sth
eM
onte
Carl
opri
cew
ith
the
halfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis
.C
Eden
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
2,and
G&
Mis
the
caple
tfo
rmula
der
ived
inSec
tion
5.3
inG
lass
erm
an
and
Mer
ener
(2003a).
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
.
83
Tab
le3:
Sw
aptio
nP
rices
for
the
Norm
aldistrib
utio
n,Para
mete
rSet
AN
OR
M0.5
11.5
1X
1(MC
)181.69
(0.0021)25.77
(0.0009)0.35
(0.00012)1
X1
181.67(CE
)181.67(G
&M
)25.75(C
E)
25.76(G&
M)
0.35(CE
)0.35(G
&M
)1
X5(M
C)
889.81(0.029)
126.14(0.016)
1.66(0.0019)
1X
5889.71(C
E)
889.71(G&
M)
126.09(CE
)126.37(G
&M
)1.69(C
E)
1.72(G&
M)
1X
10(MC
)1680.92
(0.096)237.72
(0.054)2.95
(0.0055)1
X10
1680.71(CE
)1680.72(G
&M
)237.68(C
E)
239.30(G&
M)
3.06(CE
)3.21(G
&M
)10
X1(M
C)
155.05(0.097)
67.37(0.071)
28.56(0.049)
10X
1154.71(C
E)
154.71(G&
M)
67.07(CE
)67.07(G
&M
)28.39(C
E)
28.39(G&
M)
10X
5(MC
)714.90
(0.55)310.94
(0.42)131.86
(0.3)10
X5
713.29(CE
)713.33(G
&M
)309.61(C
E)
309.73(G&
M)
131.22(CE
)131.33(G
&M
)10
X10(M
C)
1283.37(1.2)
558.39(0.89)
236.32(0.63)
10X
101280.70(C
E)
1280.75(G&
M)
557.03(CE
)557.18(G
&M
)236.62(C
E)
236.77(G&
M)
20X
1(MC
)104.95
(0.19)59.67
(0.16)35.90
(0.13)20
X1
104.45(CE
)104.45(G
&M
)59.15(C
E)
59.15(G&
M)
35.42(CE
)35.42(G
&M
)20
X5(M
C)
474.85(0.9)
270.18(0.74)
162.56(0.59)
20X
5472.90(C
E)
472.84(G&
M)
268.26(CE
)268.13(G
&M
)160.92(C
E)
160.77(G&
M)
20X
10(MC
)837.69
(1.6)476.74
(1.3)286.25
(1)20
X10
835.44(CE
)834.71(G
&M
)475.56(C
E)
474.11(G&
M)
286.33(CE
)284.75(G
&M
)
This
table
show
ssw
aptio
nprices
for
diff
erent
matu
ritiesacro
ssm
oney
ness,
X/F
K(0
).M
Cden
otes
the
Monte
Carlo
price
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
with
the
halfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n4
with
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
G&
Mis
the
caplet
form
ula
deriv
edin
Sectio
n5.4
inG
lasserm
an
and
Meren
er(2
003a).
84
Tab
le4:
Sw
apti
on
Pri
ces
for
the
Norm
aldis
trib
uti
on,Para
mete
rSet
BN
OR
M0.
51
1.5
1X
1(M
C)
181.
68(0
.002
7)23
.44
(0.0
013)
0.84
(0.0
0035
)1
X1
181.
71(C
E)
181.
71(G
&M
)23
.49(
CE
)23
.50(
G&
M)
0.86
(CE
)0.
86(G
&M
)1
X5(
MC
)88
9.73
(0.0
38)
115.
07(0
.021
)3.
92(0
.004
9)1
X5
889.
89(C
E)
889.
90(G
&M
)11
5.30
(CE
)11
5.55
(G&
M)
4.04
(CE
)4.
10(G
&M
)1
X10
(MC
)16
80.7
5(0
.12)
217.
67(0
.071
)6.
88(0
.014
)1
X10
1681
.08(
CE
)16
81.1
1(G
&M
)21
8.03
(CE
)21
9.46
(G&
M)
7.20
(CE
)7.
50(G
&M
)10
X1(
MC
)15
4.63
(0.1
1)66
.96
(0.0
83)
28.4
4(0
.061
)10
X1
154.
95(C
E)
154.
95(G
&M
)67
.11(
CE
)67
.11(
G&
M)
28.5
5(C
E)
28.5
6(G
&M
)10
X5(
MC
)71
3.44
(0.6
1)30
9.22
(0.4
7)13
0.95
(0.3
5)10
X5
714.
46(C
E)
714.
51(G
&M
)30
9.60
(CE
)30
9.73
(G&
M)
131.
52(C
E)
131.
65(G
&M
)10
X10
(MC
)12
81.7
7(1
.2)
555.
82(0
.98)
234.
22(0
.71)
10X
1012
83.0
0(C
E)
1283
.07(
G&
M)
556.
79(C
E)
556.
97(G
&M
)23
6.38
(CE
)23
6.57
(G&
M)
20X
1(M
C)
104.
63(0
.2)
59.4
3(0
.17)
35.7
2(0
.13)
20X
110
4.65
(CE
)10
4.66
(G&
M)
59.3
0(C
E)
59.3
1(G
&M
)35
.53(
CE
)35
.53(
G&
M)
20X
5(M
C)
474.
20(0
.92)
269.
74(0
.76)
162.
12(0
.61)
20X
547
3.89
(CE
)47
3.84
(G&
M)
268.
92(C
E)
268.
81(G
&M
)16
1.26
(CE
)16
1.14
(G&
M)
20X
10(M
C)
838.
00(1
.6)
477.
13(1
.3)
286.
29(1
)20
X10
837.
34(C
E)
836.
66(G
&M
)47
6.75
(CE
)47
5.35
(G&
M)
286.
76(C
E)
285.
23(G
&M
)
This
table
show
ssw
apti
on
pri
ces
for
diff
eren
tm
atu
riti
esacr
oss
money
nes
s,X
/F
K(0
).M
Cden
ote
sth
eM
onte
Carl
opri
cegen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
wit
hth
ehalfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis
.C
Eden
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4w
ith
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
G&
Mis
the
caple
tfo
rmula
der
ived
inSec
tion
5.4
inG
lass
erm
an
and
Mer
ener
(2003a).
85
Tab
le5:
Caple
tP
rices,
Double
Exponentia
lM
ark
s,Para
mete
rSet
DE
0.50.8
11.2
1.5T
=1(M
C)
90.87(0.00094)
38.62(0.00067)
12.96(0.0004)
2.85(0.00021)
0.29(7.7e-005)
T=
1(CE
Fou
rier)90.87
38.6212.96
2.850.29
T=
3(MC
)90.25
(0.0061)43.88
(0.0045)22.98
(0.0033)10.87
(0.0024)3.25
(0.0015)T
=3(C
EFou
rier)90.24
43.8722.97
10.863.25
T=
7(MC
)84.88
(0.029)48.78
(0.024)32.42
(0.02)21.21
(0.017)11.13
(0.013)T
=7(C
EFou
rier)84.84
48.7232.37
21.1711.10
T=
10(MC
)78.72
(0.052)48.87
(0.044)35.11
(0.039)25.19
(0.034)15.40
(0.028)T
=10(C
EFou
rier)78.64
48.7735.02
25.1115.34
T=
15(MC
)66.24
(0.088)44.83
(0.077)34.73
(0.07)27.09
(0.064)18.93
(0.055)T
=15(C
EFou
rier)66.08
44.6634.56
26.9218.78
T=
20(MC
)53.63
(0.1)38.49
(0.094)31.21
(0.087)25.54
(0.08)19.23
(0.071)T
=20(C
EFou
rier)53.51
38.3431.04
25.3719.05
This
table
show
sth
eca
plet
price
for
diff
erent
matu
ritiesacro
ssm
oney
ness,
X/F
K(0
).M
Cden
otes
the
Monte
Carlo
price
with
the
halfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2,and
G&
Mis
the
caplet
form
ula
deriv
edin
Sectio
n5.3
inG
lasserm
an
and
Meren
er(2
003a).
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
.
86
Tab
le6:
Sw
apti
on
Pri
ces,
Double
Exponenti
alM
ark
s,Para
mete
rSet
DE
0.5
11.
51
X1(
MC
)18
1.72
(0.0
024)
25.9
2(0
.001
1)0.
58(0
.000
21)
1X
1(C
E)
181.
7225
.92
0.58
1X
5(M
C)
889.
96(0
.033
)12
6.97
(0.0
19)
2.72
(0.0
03)
1X
5(C
E)
889.
9212
7.02
2.77
1X
10(M
C)
1681
.28
(0.1
1)23
9.49
(0.0
64)
4.81
(0.0
088)
1X
10(C
E)
1681
.12
239.
695.
0010
X1(
MC
)15
6.00
(0.1
)69
.61
(0.0
77)
30.5
6(0
.055
)10
X1(
CE
)15
5.79
69.3
930
.41
10X
5(M
C)
719.
44(0
.59)
321.
46(0
.45)
141.
15(0
.33)
10X
5(C
E)
718.
4432
0.50
140.
6310
X10
(MC
)12
91.9
5(1
.2)
577.
75(0
.96)
253.
15(0
.7)
10X
10(C
E)
1290
.30
577.
0825
3.81
20X
1(M
C)
105.
94(0
.21)
61.6
7(0
.17)
38.0
2(0
.14)
20X
1(C
E)
105.
6961
.32
37.6
420
X5(
MC
)47
9.62
(0.9
5)27
9.54
(0.7
9)17
2.41
(0.6
4)20
X5(
CE
)47
8.69
278.
3317
1.19
20X
10(M
C)
846.
80(1
.7)
493.
88(1
.4)
304.
08(1
.1)
20X
10(C
E)
846.
0249
3.86
304.
97
This
table
show
ssw
apti
on
pri
ces
for
diff
eren
tm
atu
riti
esacr
oss
money
nes
s,X
/F
K(0
).M
Cden
ote
sth
eM
onte
Carl
opri
cegen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
wit
hth
ehalfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis
.C
Eden
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4w
ith
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
G&
Mis
the
caple
tfo
rmula
der
ived
inSec
tion
5.4
inG
lass
erm
an
and
Mer
ener
(2003a).
87
Tab
le7:
Caple
tP
rices,
Double
Exponentia
lM
ark
s,D
ispla
ced
Diff
usio
n,Para
mete
rSet
DED
D0.5
0.81
1.21.5
T=
1(MC
)90.87
(0.00094)38.62
(0.00067)12.96
(0.0004)2.85
(0.00021)0.29
(7.7e-005)T
=1(C
EFou
rier)90.87
38.6212.96
2.850.29
T=
3(MC
)90.25
(0.0061)43.88
(0.0045)22.98
(0.0033)10.87
(0.0024)3.25
(0.0015)T
=3(C
EFou
rier)90.24
43.8722.97
10.863.25
T=
7(MC
)84.88
(0.029)48.78
(0.024)32.42
(0.02)21.21
(0.017)11.13
(0.013)T
=7(C
EFou
rier)84.84
48.7232.37
21.1711.10
T=
10(MC
)78.72
(0.052)48.87
(0.044)35.11
(0.039)25.19
(0.034)15.40
(0.028)T
=10(C
EFou
rier)78.64
48.7735.02
25.1115.34
T=
15(MC
)66.24
(0.088)44.83
(0.077)34.73
(0.07)27.09
(0.064)18.93
(0.055)T
=15(C
EFou
rier)66.08
44.6634.56
26.9218.78
T=
20(MC
)53.63
(0.1)38.49
(0.094)31.21
(0.087)25.54
(0.08)19.23
(0.071)T
=20(C
EFou
rier)53.51
38.3431.04
25.3719.05
This
table
show
sth
eca
plet
price
for
diff
erent
matu
ritiesacro
ssm
oney
ness,
X/F
K(0
).M
Cden
otes
the
Monte
Carlo
price
with
the
halfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2,and
G&
Mis
the
caplet
form
ula
deriv
edin
Sectio
n5.3
inG
lasserm
an
and
Meren
er(2
003a).
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
.
88
Tab
le8:
Sw
apti
on
Pri
ces,
Double
Exponenti
alM
ark
s,D
ispla
ced
Diff
usi
on,Para
mete
rSet
DED
D0.
51
1.5
1X
1(M
C)
181.
78(0
.002
7)26
.45
(0.0
012)
0.69
(0.0
0018
)1
X1(
CE
)18
1.79
26.4
20.
681
X5(
MC
)89
0.09
(0.0
36)
125.
37(0
.019
)2.
83(0
.004
1)1
X5(
CE
)89
0.13
125.
292.
801
X10
(MC
)16
81.3
2(0
.11)
230.
38(0
.062
)4.
49(0
.011
)1
X10
(CE
)16
81.3
423
0.39
4.50
10X
1(M
C)
158.
44(0
.079
)65
.16
(0.0
51)
22.2
7(0
.03)
10X
1(C
E)
158.
3765
.11
22.2
910
X5(
MC
)72
9.62
(0.4
4)29
8.68
(0.2
9)10
1.12
(0.1
7)10
X5(
CE
)72
9.25
298.
6310
1.43
10X
10(M
C)
1309
.30
(0.9
2)53
4.71
(0.6
3)17
9.69
(0.3
8)10
X10
(CE
)13
08.8
853
5.57
181.
3520
X1(
MC
)10
9.40
(0.1
4)57
.85
(0.1
)29
.08
(0.0
72)
20X
1(C
E)
109.
2757
.77
29.0
620
X5(
MC
)49
6.09
(0.6
8)26
2.95
(0.5
)13
2.41
(0.3
5)20
X5(
CE
)49
5.57
262.
8413
2.68
20X
10(M
C)
878.
82(1
.2)
468.
11(0
.92)
236.
65(0
.65)
20X
10(C
E)
878.
7546
9.55
239.
06
This
table
show
ssw
apti
on
pri
ces
for
diff
eren
tm
atu
riti
esacr
oss
money
nes
s,X
/F
K(0
).M
Cden
ote
sth
eM
onte
Carl
opri
cegen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
wit
hth
ehalfw
idth
ofth
e95%
confiden
celim
itin
para
nth
esis
.C
Eden
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4w
ith
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
G&
Mis
the
caple
tfo
rmula
der
ived
inSec
tion
5.4
inG
lass
erm
an
and
Mer
ener
(2003a).
89
0 5 10 15 20 25 300.036
0.038
0.04
0.042
0.044
0.046
0.048
Maturity
Rat
e
Figure 1: EURIBOR Term StructureThis figure shows the EURIBOR forward rates with 6 month tenor taken from Datastream the 6th of May 2006.This term structure is used in all numerical experiments.
90
0.5
11.
5−0
.2
−0.1
5
−0.1
−0.0
50
0.050.
1
0.15
T=1
Mon
eyne
s(X
/F)
Pct. Error
0.5
11.
5−0
.050
0.050.
1
0.15
T=3
Mon
eyne
s(X
/F)
Pct. Error
0.5
11.
5−0
.2
−0.10
0.1
0.2
0.3
0.4
0.5
T=7
Mon
eyne
s(X
/F)
Pct. Error
0.5
11.
5−0
.2
−0.10
0.1
0.2
0.3
0.4
0.5
0.6
T=10
Mon
eyne
s(X
/F)
Pct. Error
0.5
11.
5−0
.4
−0.20
0.2
0.4
0.6
0.81
1.2
T=15
Mon
eyne
s(X
/F)
Pct. Error
0.5
11.
5−0
.50
0.51
1.5
T=20
Mon
eyne
s(X
/F)
Pct. Error
C
E95
% C
L
Fig
ure
2:C
aple
tP
rice
Perc
enta
ge
err
ors
,N
orm
alM
ark
s,Para
mete
rSet
AN
OR
MT
his
figure
show
sth
eca
ple
tpri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
the
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
the
diff
eren
tm
atu
riti
esacr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
2.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
91
0.51
1.5−2.5 −2
−1.5 −1
−0.5 0
0.5T=1
Moneynes(X
/F)
Pct. Error
0.51
1.5−0.6
−0.5
−0.4
−0.3
−0.2
−0.1 0
0.1T=3
Moneynes(X
/F)
Pct. Error
0.51
1.5−0.6
−0.5
−0.4
−0.3
−0.2
−0.1 0
0.1
0.2T=7
Moneynes(X
/F)
Pct. Error
0.51
1.5−0.4
−0.3
−0.2
−0.1 0
0.1
0.2
0.3T=10
Moneynes(X
/F)
Pct. Error
0.51
1.5−0.4
−0.3
−0.2
−0.1 0
0.1
0.2
0.3
0.4T=15
Moneynes(X
/F)
Pct. Error
0.51
1.5−0.4
−0.2 0
0.2
0.4
0.6T=20
Moneynes(X
/F)
Pct. Error
C
EFourier
95% C
L
Figu
re3:
Caple
tP
ricePerce
nta
ge
erro
rs,N
orm
alM
ark
s,Para
mete
rSet
BN
OR
MT
his
figure
show
sth
eca
plet
price
percen
tage
error
defi
ned
as
100
times
the
simula
tedprice
min
us
the
CE
price
div
ided
by
the
simula
tedprice.
The
errors
are
plo
ttedfo
rth
ediff
erent
matu
ritiesacro
ssm
oney
nessX
/F
K(0
).M
Cprices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
92
0.5
11.
50.
17
0.18
0.190.
2
0.21
0.22
0.23
0.24
0.25
T=1
Mon
eyne
s(X
/F)
Implied Vol.
0.5
11.
50.
175
0.18
0.18
5
0.19
0.19
5
0.2
T=3
Mon
eyne
s(X
/F)
Implied Vol.
0.5
11.
50.
178
0.18
0.18
2
0.18
4
0.18
6
0.18
8
0.19
0.19
2
0.19
4T=
7
Mon
eyne
s(X
/F)
Implied Vol.
0.5
11.
50.
18
0.18
2
0.18
4
0.18
6
0.18
8
0.19
0.19
2T=
10
Mon
eyne
s(X
/F)
Implied Vol.
0.5
11.
50.
18
0.18
2
0.18
4
0.18
6
0.18
8
0.19
0.19
2T=
15
Mon
eyne
s(X
/F)
Implied Vol.
0.5
11.
50.
182
0.18
4
0.18
6
0.18
8
0.19
0.19
2
0.19
4T=
20
Mon
eyne
s(X
/F)
Implied Vol.
MC
CE
Four
ier
95%
CL
Fig
ure
4:C
aple
tIm
plied
Vola
tility
,N
orm
alM
ark
s,Para
mete
rSet
AN
OR
M
This
figure
show
sth
eca
ple
tim
plied
vola
tility
.M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
2.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
93
0.51
1.50.16
0.18
0.2
0.22
0.24
0.26T=1
Moneynes(X
/F)
Implied Vol.
0.51
1.50.175
0.18
0.185
0.19
0.195
0.2
0.205T=3
Moneynes(X
/F)
Implied Vol.
0.51
1.50.178
0.18
0.182
0.184
0.186
0.188
0.19
0.192
0.194T=7
Moneynes(X
/F)
Implied Vol.
0.51
1.50.18
0.182
0.184
0.186
0.188
0.19
0.192T=10
Moneynes(X
/F)
Implied Vol.
0.51
1.50.181
0.182
0.183
0.184
0.185
0.186
0.187
0.188
0.189T=15
Moneynes(X
/F)
Implied Vol.
0.51
1.50.183
0.184
0.185
0.186
0.187
0.188
0.189T=20
Moneynes(X
/F)
Implied Vol.
M
CC
EFourier
95% C
L
Figu
re5:
Caple
tIm
plie
dV
ola
tility,N
orm
alM
ark
s,Para
mete
rSet
BN
OR
M
This
figure
show
sth
eca
plet
implied
vola
tility.M
Cprices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
94
0.5
11.
5
0.16
0.180.
2
0.22
0.24
1 X
1
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
5
0.16
0.180.
2
0.22
0.24
1 X
5
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
5
0.16
0.180.
2
0.22
0.24
1 X
7
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
5
0.16
0.180.
2
0.22
0.24
1 X
10
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
17
0.18
0.190.
25
X 1
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
17
0.18
0.190.
25
X 5
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
17
0.18
0.190.
25
X 7
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
17
0.18
0.190.
25
X 10
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.18
5
0.19
0.19
515
X 1
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
215
X 5
Mon
eyne
s(X/
F)Implied Vol.
0.5
11.
50.
18
0.190.
215
X 7
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
215
X 1
0
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
220
X 1
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
220
X 5
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
220
X 7
Mon
eyne
s(X/
F)
Implied Vol.
0.5
11.
50.
18
0.190.
220
X 1
0
Mon
eyne
s(X/
F)
Implied Vol.
MC
CE
G&M
95%
CL F
igure
6:Sw
apti
on
Implied
Vola
tility
,N
orm
alM
ark
s,Para
mete
rSet
AN
OR
MT
his
figure
show
sth
esw
apti
on
implied
vola
tility
.T
he
titl
efo
rea
chsu
bgra
ph
den
ote
sM
atu
rity
×Ten
or.
MC
pri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
95
0.51
1.50.1
0.2
0.31 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.17
0.18
0.19
0.25 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.17
0.18
0.19
0.25 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.17
0.18
0.19
0.25 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.17
0.18
0.19
0.25 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19515 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19515 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19515 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19515 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19520 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19520 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.19520 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.220 X 10
Moneynes(X/F)
Implied Vol.
MC
CE
G&M
95% C
LFigu
re7:
Sw
aptio
nIm
plie
dV
ola
tility,N
orm
alM
ark
s,Para
mete
rSet
BN
OR
MT
his
figure
show
sth
esw
aptio
nim
plied
vola
tility.T
he
titlefo
rea
chsu
bgra
ph
den
otes
Matu
rity×
Ten
or.
MC
prices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n4.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
96
0.5
11.
5−0
.50
0.5
1 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−4−202
1 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
0−5051
X 7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
0−5051
X 10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
5 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
5 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
−0.50
0.5
5 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−2−101
5 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1012
15 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
15 X
5
Mon
eyne
s(X/
F)Pct. Err
0.5
11.
5−0
.50
0.51
15 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
15 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1012
20 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1012
20 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
20 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
20 X
10
Mon
eyne
s(X/
F)
Pct. Err
CE
G&M
95%
CL
Fig
ure
8:Sw
apti
on
Pri
cePerc
enta
ge
err
ors
,N
orm
alM
ark
s,Para
mete
rSet
AN
OR
MT
his
figure
show
sth
esw
apti
on
pri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
Matu
rity
×Ten
or
acr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
97
0.51
1.5−4 −2 0 2
1 X 1
Moneynes(X/F)
Pct. Err0.5
11.5
−5 0 51 X 5
Moneynes(X/F)
Pct. Err
0.51
1.5−10 −5 0 5
1 X 7
Moneynes(X/F)
Pct. Err
0.51
1.5−10 −5 0 5
1 X 10
Moneynes(X/F)
Pct. Err
0.51
1.5−1
−0.5 0
0.55 X 1
Moneynes(X/F)
Pct. Err
0.51
1.5−2 −1 0 1
5 X 5
Moneynes(X/F)
Pct. Err
0.51
1.5−2 −1 0 1
5 X 7
Moneynes(X/F)
Pct. Err
0.51
1.5−4 −2 0 2
5 X 10
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.515 X 1
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.515 X 5
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.515 X 7
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.515 X 10
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.5 120 X 1
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.5 120 X 5
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.5 120 X 7
Moneynes(X/F)
Pct. Err
0.51
1.5−0.5 0
0.520 X 10
Moneynes(X/F)
Pct. Err
CE
G&M
95%C
L
Figu
re9:
Sw
aptio
nP
ricePerce
nta
ge
erro
rs,N
orm
alM
ark
s,Para
mete
rSet
BN
OR
MT
his
figure
show
sth
esw
aptio
nprice
percen
tage
error
defi
ned
as
100
times
the
simula
tedprice
min
us
CE
price
div
ided
by
the
simula
tedprice.
The
errors
are
plo
ttedfo
rM
atu
rity×
Ten
or
acro
ssm
oney
nessX
/F
K(0
).M
Cprices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n4.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
98
0.5
11.
5−2−10123
x 10
−3T=
1
Mon
eyne
s(X
/F)
Bps Error
0.5
11.
5−0
.01
−0.0
050
0.00
5
0.01
0.01
5T=
3
Mon
eyne
s(X
/F)
Bps Error
0.5
11.
5−0
.04
−0.0
20
0.02
0.04
0.06
T=7
Mon
eyne
s(X
/F)
Bps Error
0.5
11.
5−0
.06
−0.0
4
−0.0
20
0.02
0.04
0.06
0.080.
1T=
10
Mon
eyne
s(X
/F)
Bps Error
0.5
11.
5−0
.1
−0.0
50
0.050.
1
0.150.
2
0.25
T=15
Mon
eyne
s(X
/F)
Bps Error
0.5
11.
5−0
.15
−0.1
−0.0
50
0.050.
1
0.150.
2T=
20
Mon
eyne
s(X
/F)
Bps Error
CE
Four
ier
95%
CL
Fig
ure
10:C
aple
tP
rice
Perc
enta
ge
err
ors
,D
ouble
Exponenti
alD
istr
ibuti
on,Para
mete
rSetA
NO
RM
This
figure
show
sth
eca
ple
tpri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
the
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
the
diff
eren
tm
atu
riti
esacr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
2.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
99
0.51
1.50.16
0.18
0.2
0.22
0.24
0.26T=1
Moneynes(X
/F)
Implied Vol.
0.51
1.50.18
0.185
0.19
0.195
0.2
0.205
0.21T=3
Moneynes(X
/F)
Implied Vol.
0.51
1.50.184
0.186
0.188
0.19
0.192
0.194
0.196
0.198
0.2T=7
Moneynes(X
/F)
Implied Vol.
0.51
1.50.186
0.188
0.19
0.192
0.194
0.196
0.198
0.2T=10
Moneynes(X
/F)
Implied Vol.
0.51
1.50.188
0.19
0.192
0.194
0.196
0.198
0.2T=15
Moneynes(X
/F)
Implied Vol.
0.51
1.50.188
0.19
0.192
0.194
0.196
0.198T=20
Moneynes(X
/F)
Implied Vol.
MC
CE
Fourier95%
CL
Figu
re11:
Caple
tIm
plie
dV
ola
tility,D
ouble
Exponentia
lD
istributio
n,Para
mete
rSet
AN
OR
M
This
figure
show
sth
eca
plet
implied
vola
tility.M
Cprices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
100
0.5
11.
5−0
.50
0.5
1 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−4−202
1 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−4−202
1 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−505
1 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.20
0.2
5 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.20
0.2
5 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
5 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
−0.50
0.5
5 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
15 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
15 X
5
Mon
eyne
s(X/
F)Pct. Err
0.5
11.
5−0
.50
0.5
15 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
15 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
20 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.51
20 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
20 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
20 X
10
Mon
eyne
s(X/
F)
Pct. Err
CE
95%
CL
Fig
ure
12:
Sw
apti
on
Pri
cePerc
enta
ge
err
ors
,D
ouble
Exponenti
al
Dis
trib
uti
on,
Para
mete
rSet
AN
OR
MT
his
figure
show
sth
esw
apti
on
pri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
Matu
rity
×Ten
or
acr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
4.
The
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
101
0.51
1.50.1
0.2
0.31 X 1
Moneynes(X/F)
Implied Vol.0.5
11.5
0.1
0.2
0.31 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.2
0.215 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.2
0.215 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.2
0.215 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.2
0.215 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.215 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.215 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.215 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.215 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.50.18
0.19
0.220 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.19
0.195
0.2
0.20520 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.19
0.195
0.2
0.20520 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.19
0.195
0.2
0.20520 X 10
Moneynes(X/F)
Implied Vol.
MC
CE
95% C
L
Figu
re13:
Sw
aptio
nIm
plie
dV
ola
tility,D
ouble
Exponentia
lD
istributio
n,Para
mete
rSet
AN
OR
MT
his
figure
show
sth
esw
aptio
nim
plied
vola
tility.T
he
titlefo
rea
chsu
bgra
ph
den
otes
Matu
rity×
Ten
or.
MC
prices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n4.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
102
0.5
11.
5−0
.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.10
0.1
T=1
Mon
eyne
s(X/
F)Pct. Error
0.5
11.
5−0
.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.10
0.1
T=3
Mon
eyne
s(X/
F)
Pct. Error
0.5
11.
5−0
.5
−0.4
−0.3
−0.2
−0.10
0.1
0.2
T=7
Mon
eyne
s(X/
F)
Pct. Error
0.5
11.
5−0
.5
−0.4
−0.3
−0.2
−0.10
0.1
0.2
T=10
Mon
eyne
s(X/
F)
Pct. Error
0.5
11.
5−0
.3
−0.2
−0.10
0.1
0.2
T=15
Mon
eyne
s(X/
F)
Pct. Error
0.5
11.
5−0
.4
−0.3
−0.2
−0.10
0.1
0.2
0.3
T=20
Mon
eyne
s(X/
F)
Pct. Error
C
E95
% C
L
Fig
ure
14:
Caple
tP
rice
Perc
enta
ge
err
ors
,D
ouble
Exponenti
al
Dis
trib
uti
on,
Dis
pla
ced
Diff
usi
on,
Para
mete
rSet
DED
DT
his
figure
show
sth
eca
ple
tpri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
the
diff
eren
tm
atu
riti
esacr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
5in
the
appen
dix
.T
he
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
103
0.51
1.50.18
0.2
0.22
0.24
0.26
0.28
0.3T=1
Moneynes(X
/F)
Implied Vol.
0.51
1.50.17
0.18
0.19
0.2
0.21
0.22
0.23
0.24T=3
Moneynes(X
/F)
Implied Vol.
0.51
1.50.16
0.17
0.18
0.19
0.2
0.21
0.22T=7
Moneynes(X
/F)
Implied Vol.
0.51
1.50.16
0.17
0.18
0.19
0.2
0.21
0.22T=10
Moneynes(X
/F)
Implied Vol.
0.51
1.50.16
0.17
0.18
0.19
0.2
0.21
0.22T=15
Moneynes(X
/F)
Implied Vol.
0.51
1.50.16
0.17
0.18
0.19
0.2
0.21
0.22T=20
Moneynes(X
/F)
Implied Vol.
MC
CE
95% C
L
Figu
re15:
Caple
tIm
plie
dV
ola
tility,D
ouble
Exponentia
lD
istributio
n,D
ispla
ced
Diff
usio
n,Para
me-
ter
Set
DED
DT
his
figure
show
sth
eca
plet
implied
vola
tility.M
Cprices
are
gen
erated
usin
gth
esim
ula
tion
schem
ein
the
Appen
dix
alo
ng
with
the
95%
confiden
celim
it.C
Eden
otes
the
price
calcu
lated
usin
gth
eappro
xim
atio
nin
Pro
positio
n2.
The
cum
ula
nt
expansio
nis
cut
off
at
P=
4and
the
tunin
gpara
meter
for
the
num
ericalin
tegra
tion
isset
at
α=
0.7
5.
104
0.5
11.
5−1012
1 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−2−101
1 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−2−101
1 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−4−202
1 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.20
0.2
5 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
5 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
−0.50
0.5
5 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
−0.50
0.5
5 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.20
0.2
15 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
15 X
5
Mon
eyne
s(X/
F)
Pct. Err0.
51
1.5
−0.50
0.5
15 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−1
−0.50
0.5
15 X
10
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
20 X
1
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
20 X
5
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−0
.50
0.5
20 X
7
Mon
eyne
s(X/
F)
Pct. Err
0.5
11.
5−2−101
20 X
10
Mon
eyne
s(X/
F)
Pct. Err
CE
95%
CL
Fig
ure
16:Sw
apti
on
Pri
cePerc
enta
ge
err
ors
,D
ouble
Exponenti
alD
istr
ibuti
on,D
ispla
ced
Diff
usi
on,
Para
mete
rSet
DED
DT
his
figure
show
sth
esw
apti
on
pri
ceper
centa
ge
erro
rdefi
ned
as
100
tim
esth
esi
mula
ted
pri
cem
inus
CE
pri
cediv
ided
by
the
sim
ula
ted
pri
ce.
The
erro
rsare
plo
tted
for
Matu
rity
×Ten
or
acr
oss
money
nes
sX/F
K(0
).M
Cpri
ces
are
gen
erate
dusi
ng
the
sim
ula
tion
schem
ein
the
Appen
dix
alo
ng
wit
hth
e95%
confiden
celim
it.
CE
den
ote
sth
epri
ceca
lcula
ted
usi
ng
the
appro
xim
ati
on
inP
roposi
tion
6in
the
appen
dix
.T
he
cum
ula
nt
expansi
on
iscu
toff
at
P=
4and
the
tunin
gpara
met
erfo
rth
enum
eric
alin
tegra
tion
isse
tat
α=
0.7
5.
105
0.51
1.50.1
0.2
0.31 X 1
Moneynes(X/F)
Implied Vol.0.5
11.5
0.1
0.2
0.31 X 5
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 7
Moneynes(X/F)
Implied Vol.
0.51
1.50.1
0.2
0.31 X 10
Moneynes(X/F)
Implied Vol.
0.51
1.5
0.16
0.18
0.2
0.22
0.24
5 X 1
Moneynes(X/F)
Implied Vol.
0.51
1.50.16
0.18
0.2
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Chapter III
Alternative Specifications for the Levy Libor
Market Model: An Empirical Investigation∗
David Skovmand†
University of Aarhus and CREATES
May 1, 2008
Abstract
This paper introduces and analyzes specifications of the Levy MarketModel originally proposed by Eberlein and Ozkan (2005). An investigationof the term structure of option implied moments shows that the Brownianmotion and homogeneous Levy processes are not suitable as modelling de-vices, and consequently a variety of more appropriate models is proposed.Besides a diffusive component the models have jump structures with low orhigh frequency combined with constant or stochastic volatility. The modelsare subjected to an empirical analysis using a time series of data for Euri-bor caps. The results of the estimation show that pricing performances areimproved when a high frequency jump component is incorporated. Specifi-cally, excellent results are achieved with the 4 parameter Self-Similar VarianceGamma model, which is able to fit an entire surface of caps with an averageabsolute percentage pricing error of less than 3%.
∗The author would like to thank Elisa Nicolato, Thomas Kokholm and Peter Løchte Jørgensen,for useful comments
†Current affiliation: Aarhus School of Business and the Center for Research in EconometricAnalysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus Schoolof Business, Department of Business Studies, Fuglesangs Alle 4, DK-8210 Aarhus V, Denmark,e-mail: [email protected]
1 Introduction
The Libor Market Model formulated in the seminal papers by Miltersen, Sand-mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997), and Jamshid-ian (1997) was a breakthrough in the world of academics as well as practitioners. Itprovided a theoretical framework for the market practice of using the classic Blackand Scholes (1973) formula for pricing options on Libor rates, such as caps andswaptions. However, the model was almost immediately extended due to its in-ability to fit the volatility smile already known to be present in options on Liborrates. The extensions have mirrored the development in the equity option pricingliterature and they include Libor market models with stochastic volatility (Andersenand Brotherton-Ratcliffe (2005), Hagan, Kumar, Lesniewski, and Woodward (2002),and Wu and Zhang (2006)) and/or jumps generated by a compound Poisson process(Jarrow, Li, and Zhao (2007) and Glasserman and Kou (2003) ).In spite of these advances it is still an open question how to optimally price liquidinterest rate options in the cross-section of both maturity and strike. The strikedimension alone is well fitted by for example the SABR model of Hagan, Kumar,Lesniewski, and Woodward (2002), which has gained popularity to the point wheretraders have begun to quote smiles in terms of its parameters. But the failure of theSABR model is manifest since practitioners must use maturity dependent parame-ters in order to fit the entire surface.More advanced models have also been developed such as the Levy process driven Li-bor Market Model of Eberlein and Ozkan (2005), Eberlein and Kluge (2007), Kluge(2005), and the more general semi-martingale model of Jamshidian (1999). Thesepapers generalize previous approaches by allowing high frequency jump processes,as opposed to the classic compound Poisson jump-diffusion framework, originallyproposed in Merton (1976). High frequency jump models have already been tremen-dously successful in pricing equity options, as shown in Carr, Geman, Madan, andYor (2002), Carr, Geman, Madan, and Yor (2003), and Carr, Geman, Madan, andYor (2007). But whether the high frequency jump components are relevant as mod-eling devices in interest rate models has, to the authors knowledge, not yet beenthoroughly investigated from an empirical point of view.This paper analyzes different specifications of jump processes coupled with stochas-tic volatility in a Libor Market Model, and sheds light on their comparative abilityto fit a time series of Euribor cap prices in both the maturity and strike dimensions.The different models are formulated in a unified framework which can be seen as anextension of Eberlein and Ozkan (2005). The framework also has the advantage ofallowing for a fast and accurate pricing of caps through Fourier inversion techniquesin the spirit of Carr and Madan (1999).The specific choice of driving processes is motivated by an investigation of the term
108
structure of higher moments such as variance, skewness and kurtosis implied fromthe data. The analysis is inspired by Konikov and Madan (2002) who examine equityoptions and find that kurtosis and negative skewness are increasing as a function ofmaturity. The cap market investigated in this paper displays similar patterns in theterm structure of implied moments and therefore the driving processes are selectedaccording to their capability of capturing these stylized features.The chosen Libor market models are divided in three categories. 1) The ClassicModels , where the building block is a displaced diffusion with stochastic volatil-ity and/or compound Poisson jumps. This class can be seen as fusing the approachtaken in Andersen and Brotherton-Ratcliffe (2005) and Glasserman and Kou (2003).A similar specification was also analyzed in a slightly different setup in Skovmand(2008). 2) The Self-Similar Additive Models , where the driving process is an in-homogeneous Levy process with the added feature of self-similarity. This processclass has been introduced and thoroughly studied in Sato (1991) and Sato (1999).The utility of self-similar additive processes in the context of equity option pricinghas been convincingly demonstrated in Carr, Geman, Madan, and Yor (2007) andGalloway (2006). The concrete specification used in this paper to describe Liborrates is a self-similar additive extension of the well known Variance Gamma modelof Madan, Carr, and Chang (1998). 3) Time-Changed Levy Models , where the driv-ing process is a homogeneous Levy process, specifically a Variance Gamma process,subordinated, in order to incorporate stochastic volatility, to an integrated Cox,Ingersoll, and Ross (1985) process. This modelling framework was first studied inCarr, Geman, Madan, and Yor (2003) and further analyzed for example in Carr andWu (2004), although still in the context of equity options.Each model class is characterized by a different jump structure. In the classic mod-els, jumps are big and infrequent whereas the last two model classes have an infinityof small jumps, resulting in a behavior similar to a diffusion process. Whether anactual diffusion component is still relevant in this setup, is explored by mixing theinfinite activity models with a standard Brownian motion, as well as a Brownianmotion with stochastic volatility.For each different model a daily recalibration is performed and the pricing errorsare scrutinized in the maturity and strike dimension, as well as across the sampleperiod. The models are ranked in terms of their average absolute percentage pricingerrors (APE), both in- and out-of-sample.In order to carry out formal tests, a sample-wide estimation based on the General-ized Method of Moments is implemented, and the results are used to perform theDiebold and Mariano (1995) test for superior performance.The empirical investigation first documents the smile in the Euribor cap marketand provides a more general description of the data. The models are then taken tothe data and the results show that the Time-Changed Levy Models and Self-Similar
109
Additive Models outperform the classic models, with average absolute percentageerrors of 2.5%, 2.8% compared to 4-4.5% for the classic specifications.Mixing a stochastic volatility diffusion component with either a self-similar additiveor time-changed Levy jump component has a small (≈ 0.1− 0.2%) but statisticallysignificant effect on performances compared to the pure jump case. On the otherhand including a constant volatility diffusion component is only significant for thetime-changed Levy models. Finally, perhaps the most striking result is the impres-sive performance of the Self-Similar Additive Model with no diffusion component.This is the most parsimoneous model studied in the paper, with only 4 process spe-cific parameters. Despite its simplicity it is able to capture an entire price surfaceconsisting of 97 prices, with a performance comparable to the best model specifi-cation, which is a 9 parameter time-changed Levy process mixed with a stochasticvolatility diffusion component.The paper is structured in the following manner. First, the Generalized Libor Mar-ket Model framework is presented. Then the dataset is explored and the smile andother stylized features are documented in the Euribor cap dataset. Subsequently, ananalysis of the term structure of moments is performed and the specific processes aremotivated, and further analyzed in detail. Finally, the empirical methodology is de-scribed, and the results are presented, followed by a discussion of their implications.The paper ends with a conclusion.
2 The General Libor Market Model
Let P (t, Tk) be the time t zero coupon bond price with maturity at Tk. For a tenorstructure T0 < . . . . . < TK+1 = T ∗ and constant day count fractions δ = Tk − Tk−1,the simply compounded forward Libor rates, or more succinctly Libor rates, aredefined as
Fk(t) = F (t, Tk, Tk+1) =P (t, Tk) − P (t, Tk+1)
δP (t, Tk+1)(1)
0 ≤ t ≤ Tk, k = 1, . . . , K.
From (1) it follows that the Libor rate Fk(t) can be seen as a price process scaledby the numeraire P (t, Tk+1). Therefore absence of arbitrage implies that for eachrate Fk(t) there exists an associated probability measure Fk+1, termed the forwardmeasure, under which Fk(t) is a martingale. Of particular relevance is the finalforward measure or terminal measure denoted by
F∗ := FK+1.
In what follows the entire Libor market model will be specified along the lines ofEberlein and Ozkan (2005) by defining the evolution of the terminal rate under F∗
110
and subsequently using a backward induction procedure to derive the dynamics ofthe remaining rates in the term structure. More precisely the explicit constructioncan be carried using the following inputs. The initial term structure Fk(0) fork = 1, . . . , K; a series of rate specific constants λ1, . . . , λK allowing for flexibility inmodelling the volatilities across maturity; finally the dynamics for the terminal rateFK(t) under the terminal measure F∗ which are given as follows
FK(t) = FK(0) exp
(∫ t
0
bK(s)ds + λK
∫ t
0
dLF∗
s
)
. (2)
The driving source of randomness is the stochastic process LF∗
t defined as
LF∗
t =
∫ t
0
c(s)dW F∗
s + J∗t , (3)
where W F∗
t is a 1-dimensional Brownian motion and c(t) is a positive process inde-pendent of W F
∗
t satisfying standard regularity conditions and∫ T ∗
0
c(s)2ds < ∞ F∗ a.s. (4)
The process J∗t is a purely discontinuous martingale obtained by a time change as
follows
J∗t = X(Y (t)),
where X(t) is an additive martingale process1, and Y (t) is the integrated process
Y (t) =
∫ t
0
y(s)ds, (5)
with y(t) being a strictly positive process independent of X(t) and W F∗
t .Denoting with µ(dt, dx) and νF
∗
(dt, dx) the random measure of jumps and its com-pensator respectively, J∗
t can be rewritten as
J∗t =
∫ t
0
∫
R
x(µ − νF∗
)(ds, dx),
νF∗
(dt, dx) = y(t)k(t, x)dtdx ∀ t ≤ TK ,
where k(t, x) is the deterministic Levy system associated with the additive processX(t).For simplicity it is also assumed that
∫ T ∗
0
∫
R
(1 ∧ |x|)k(t, x)dxdt < ∞,
1An additive process has independent but possibly non-stationary increments
111
implying that the jump process J∗t has finite variation. However most of the concrete
specifications in this paper will display infinite activity i.e∫
R
k(t, x)dx = ∞, ∀ t ≤ TK .
In other words the process J∗t jumps an infinite number of times on any finite time
interval.Another technical assumption that ensures the martingale property of FK(t)
EF∗
[exp(uJ∗T ∗)ds] < ∞ (6)
for |u| < (1 + ǫ)M where ǫ > 0 and M are constants such that∑K
j=1 λj < M .
To complete the description of the terminal rate FK(t) the drift term bK(t) is definedas
bK(t) = −1
2c(t)2 −
∫
R
(
eλKx − 1 − λKx)
νF∗
(t, dx). (7)
Under the regularity conditions in (6) and (4) the above definition only ensures thatFK(t) is a martingale if the time-change Y (t) integrated process is deterministic. Fora stochastic Y (t) defined in (5), the X(Y (t)) process lacks independent incrementswhen conditioning on the filtration including Y (t) implying that the FK(t) processis in general not a martingale. This means the associated market model is exposedto the possibility of dynamic arbitrage strategies as noted in Carr, Geman, Madan,and Yor (2003). One can argue whether the ability to create arbitrage strategiesbased on continuous observation of the Y (t) process is realistic in practice, as Y (t)is far from being directly observable. Moreover, following the same arguments asCarr, Geman, Madan, and Yor (2003) one can show that the the terminal rate FK(t)possesses the so called martingale marginals property i.e there exists a process, whichis a martingale and exhibits the same marginal distributions as FK(t) in (2), albeitdefined on a different filtration.
We can now proceed to specify the remaining rates Fk(t) for k = 1, . . . , K −1 undertheir own forward measures Fk+1. The connection between the terminal measure andthe remaining forward measures can be established through the backward inductionprocedure described in Eberlein and Ozkan (2005). Assuming that the forwardmeasures Fj+1 for j = k + 1, . . . , K and correspondingly the martingale forwardrates Fj(t) have been constructed, the (k + 1)-th forward measure is then definedvia the Radon-Nikodym derivative
dFk+1
dF∗=
K∏
j=k+1
1 + δFj(Tk+1)
1 + δFj(0)=
P (0, T ∗)
P (0, Tk+1)
K∏
j=k+1
(1 + δFj(Tk+1)).
112
The dynamics of the Libor rate Fk(t) under Fk+1 are then defined as
Fk(t) = Fk(0) exp
(∫ t
0
bk(s)ds + λk
∫ t
0
dLFk+1
s
)
, (8)
with the process LFk+1
t given by
LFk+1
t =
∫ t
0
c(s)dW Fk+1
s +
∫ t
0
∫
R
x(µ − νFk+1
)(ds, dx),
where W Fk+1
s is an Fk+1-Brownian motion and νFk+1
denotes the Fk+1-compensatorof the random measure of jumps µ. Using Girsanov’s theorem for general semi-martingales (see for example Jacod and Shiryaev (1987)) it follows that the connec-tion between the Brownian motions is
dWFk+1
t = dW ∗t −
K∑
j=k+1
δc(t)Fj(t−)
1 + δFj(t−)dt. (9)
while the relation between the compensators is given by2
νFk+1(dt, dx) =K∏
j=k+1
(
1 +δ(eλjx − 1)Fj(s)
1 + δFj(s)
)
νF∗
(dt, dx). (10)
Finally the martingale condition (with the discussed caveats) is secured by settingthe drift as
bk(t) = −1
2c(t)2 −
∫
R
(
eλkx − 1 − λkx)
νFk+1
(t, dx). (11)
For completeness the dynamics of k-th Libor rate under the terminal measure F∗
can be expressed using (9) and (10) as follows
Fk(t) = Fk(0) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
∀k = 1, . . . , K, (12)
with
b∗k(t) = −K∑
j=k+1
δc(t)Fj(t−)
1 + δFj(t−)
− 1
2c(t)2 −
∫
R
(
λkx − (eλkx − 1)K∏
j=k+1
(
1 +δ(eλjx − 1)Fj(t−)
1 + δFj(t−)
)
)
νF∗
(t, dx).
(13)
2This result is also derived in a different setting in Jamshidian (1999)
113
The derivation of expression (13) is given in the appendix.
The model can be extended to nest the displaced diffusion framework studied rig-orously in Rebonato (2002) and Brigo and Mercurio (2006) for the continuous case.Carrying out the steps outlined above one may then obtain the following dynamicsfor the Libor rates under the terminal measure
Fk(t) + α = (Fk(0) + α) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
∀k = 1, . . . , K, (14)
with
b∗k(t) = −K∑
j=k+1
δc(t)(Fj(t−) + α)
1 + δFj(t−)
− 1
2c(t)2 −
∫
R
(
λkx − (eλkx − 1)K∏
j=k+1
(
1 +δ(eλjx − 1)(Fj(t−) + α)
1 + δFj(t−)
)
)
νF∗
(t, dx).
(15)
From (12) and (14) it can be observed that the driving process for all rates underthe terminal measure is the L∗
t process defined in (3). The overall result is a 1factor model for the entire term structure implying that all Libor rates are perfectlycorrelated. This obviously unrealistic implication is immaterial to this study sinceonly caps are investigated and these are unaffected by correlation.3
2.1 Caplet Pricing
A cap is a portfolio of call options on Libor rates. A Tk+1-cap with strike K pays(Fi(Ti) − K) at Ti+1 for i = 1, . . . , k. The total time t < T1 value of the cap istherefore
Cap(t, Tk+1, K) =k∑
i=1
δP (t, Ti+1)EFi+1 [(Fi(Ti) − K)+].
The individual payments, referred to as caplets, have value:
δP (t, Ti+1)EFi+1 [(Fi(Ti) − K)+].
3Correlation, can be easily included by using a multidimensional Brownian motion, and this hasbeen the topic of a large part of the LMM literature (see Brigo and Mercurio (2006) and Rebonato(2002) for an overview in the pure diffusion case. Dependence structure with jumps is a moredelicate issue has to the authors knowledge only been considered in LMM context in Belomestnyand Schoenmakers (2006))
114
By investigating the dynamics of Fk(t) under its own forward measure given in (8),it is easily concluded that a closed form expression for the above expectation isnot possible to derive in general since the compensator is state dependent via itsrelationship in (10). Fortunately an accurate approximation is derived in Kluge(2005).4
Proposition 1. Given a Libor Market Model described by (14) the approximatedprice of a Caplet with strike K and maturity Tk+1 is given by
Cpl(0,Tk+1, K) =
δP (0, Tk+1)K + α
π
∫ ∞
0
Re[( K + α
Fk(0) + α
)R+iu 1
(R + iu)(R + 1 + iu)
×φ(
− ifk + (iR − u)λk, Tk
)
× φ(
− ifk − iλk, Tk
)R+iu
φ(
− ifk, Tk
)−(R+1+iu)]du, (16)
where
fk :=K∑
j=k+1
δ(Fk(0) + α)
1 + δFk(0)λj,
and φ(u, t) is the characteristic function of the driving process L∗t
φ(u, t) = EF∗
[exp(uiL∗t )] .
R is a properly chosen constant such R < −1 and φ(
− ifk + (iR − u)λk, Tk
)
< ∞
The caplet pricing formula in the above proposition is very efficient since it onlyrequires the calculation of a one-dimensional numerical integral.
3 Exploring the Data
The primary dataset is on Euribor caps and it is retrieved through Reuters; theterm structure information is collected from Datastream. The Euribor is the Euro-equivalent of the standard Libor rate, but in order to avoid confusion both will bereferred to as Libor.The data is recorded daily in the sample but for computational reasons the analysisis restricted to each Wednesday in the sample (If missing, Thursday). The datasetcovers the period from May 7th, 2003, to November 10th, 2004, resulting in 80 days
4Kluge (2005) derives the formula without a displacement nevertheless the proof can be triviallyextended to encompass this slightly more general case
115
of data. Euribor caps are given for a range of maturities 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,12, 15, and 20 years. It is also given for a large range of strikes, specifically 1.75,2.0, 2.25, 3.0, 3.5, 4.0, 4.5, 5.0, 6.0, 7.0, 8.0, and 10.0 percent, and on June 20th,2003, the 1.5 strike is added and 4.5 strike is removed.Ideally, one would like to investigate prices for caplets instead of caps since theseprovide more fundamental information on the distributional properties of the indi-vidual rates. Therefore a bootstrapping algorithm is implemented as described inPiza (2005), to extract caplet prices from cap prices. Unfortunately, Euribor capsfor the 1 and 2 year maturity are struck on rates with 3 month to expiry, whereas theremaining maturities are struck on the 6 month rate. This complicates the use of abootstrapping algorithm, so for simplicity it is assumed that the implied volatilitiesfor the 3 month rates are the same as the 6 month. This is not as simplifying as itmight seem. In markets where caps on both underlying 6 month and 3 month ratesare observed (such as the Danish, Norwegian and Swiss markets) the volatilities areclose to identical and differences are normally well within bid-ask spreads.Since the rates change every day so does the moneyness (K/Fk(0)) of the option.For computational reasons it is convenient to have constant moneyness through timetherefore caplet prices are interpolated between strikes, for each day in the sample.Following Jarrow, Li, and Zhao (2007) and Li and Zhao (2006) the interpolation isdone using local cubic polynomials to retain the structure of the prices. Moneynesslevels are restricted to 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3 except for the 1 year capletswhere due to liquidity reasons only 0.9, 1 and 1.1 are included and for the 2 year1.3 and 0.6 are excluded. This leaves 7760 option prices for the entire data with 97prices each Wednesday.The average price level in basis points is drawn in Figure 1 and it shows a fairlysmooth and consistent surface with prices increasing in moneyness but a clear non-monotonic behavior in the maturity dimension.The corresponding average implied volatilities are found by inverting the Black-Scholes formula, and the results are drawn in Figure 2. The surface shows a slighthump in the volatility surface peaking at 2 years and declining for all other ma-turities. This is consistent with other studies, such as Rebonato (2002) who per-forms a thorough investigation and provides some economic explanations of thisphenomenon.Figure 3 is a 2-dimensional version of Figure 2. Here we clearly observe a smile within-the-money (ITM) and out-of-the money (OTM) options having higher impliedvolatility than the at-the-money level. But the smile is more like a smirk since itis tilted to the right, with ITM being bigger than the corresponding OTM level.Figure 4 shows the term structure of volatility as it evolves through time for threedifferent levels of moneyness. The long maturity volatility appears close to beingconstant whereas the short maturities show an upward increasing trend, as well as
116
being significantly more erratic. Again this is true for the three different levels ofmoneyness.Finally, in Figure 5 the term structure of 6 month Libor rates throughout the sampleis plotted. The figure shows a persistent increasing term structure with a decreas-ing overall level and slope. It also shows long rates being somewhat volatile in thebeginning of the sample.
4 Implied Features of the market
In light of the model described in Section (2) the main question is : What kind offeatures should the distribution of L∗
t have? Essentially the objective is to generate arealistic implied volatility/price surface. Looking again at Figure 2 there are severaleffects that have to be accounted for. The first is the hump in implied volatilityaround the 2 year maturity. This effect can easily be generated with a maturitydependent scaling (the λk’s), as shown in the next section. Another more delicateissue is the smile and its behavior across maturity. In order to generate the smilethe distribution of the underlying has to exhibit positive excess kurtosis meaningthat extreme moves have to occur more often than what is predicted by a normaldistribution. Furthermore there is asymmetry the smile or smirk prevalent in Figure3 which implies that a negatively skewed distribution is needed.Non-Gaussian models are of course well established in the literature (see variouspapers cited in the introduction) but the majority of them suffer from the fact thatthey cannot price options in both the maturity and strike dimension simultaneously.One of the simplest examples of a non-Gaussian model with this ”flaw” is the Vari-ance Gamma (VG) model of Madan, Carr, and Chang (1998).The Variance Gamma process is constructed by time changing a Brownian motionwith drift with respect to a gamma process. So the law of the VG process is givenby
XV G(t) = θGνt + σW (Gν
t ),
where Gνt is a gamma process with mean rate 1 and variance ν and W (t) is a
Brownian motion. The process has variance, skewness and excess kurtosis equal to
(variance) µ2 = t(θ2ν + σ2),
(skewness) γ1 =1√t
(2θ3ν2 + 3σ2θν)
(θ2ν + σ2)3/2,
(excess kurtosis) γ2 =1
t
3σ2ν + 12σ2θ2ν2 + 6θ4ν3
(θ2ν + σ2)2.
117
Skewness is controlled by the θ parameter and kurtosis is partly controlled by theν parameter. What can also be observed is that variance increases linearly as afunction of time, and skewness and excess kurtosis decreases with a rate of 1/
√t
and 1/t respectively. This means that for long maturities the distribution generatedby the model is essentially Gaussian. Furthermore Konikov and Madan (2002) showthat all homogeneous Levy processes converge to Gaussianity in this manner. Thisstands in contrast to Figure 3 that shows a smile with an almost constant slopeacross maturity.In order to asses the scope of this problem an analysis similar to Konikov and Madan(2002) of the term structure of moments is performed. A separate Variance GammaLibor market model with L∗
t = XV G(t) is calibrated to each option maturity in thesample. For each maturity the variance, skewness, and kurtosis is calculated usingthe above formulas and the results are averaged within four different periods of thesample. The three series are then plotted against maturity in Figure 6-8.Looking first at the variance in Figure 6 we see a clear concave function of variancein time; a behavior different from the linear scaling implied by a Levy process.The skewness in Figure 7 starts at a level close to zero but is increasing in absoluteterms; the complete opposite of a Levy process. The excess kurtosis is plottedin Figure 8 and it shows the same behavior as the skewness but with oppositesign. What can be inferred from these graphs is that a model based on a singlehomogeneous Levy process is in direct conflict with the data and therefore notappropriate as a modeling device.A minimum criterium for a ”good” model is therefore different than linear scalingof variance as well as the the ability to retain skewness and excess kurtosis in timei.e be able delay the convergence to Gaussianity. The next section presents 3 classesof models with these features.
5 Specifying the Driving Process
In this section three classes of models all nested in the general specification in Section2 are studied. The driving stochastic processes are defined and their characteristicfunctions are given, along with their domain of existence which is essential for acorrect numerical implementation of the pricing formula in Proposition 1.Besides setting the L∗
t process, there is also considerable freedom in specifying theconstants λk for k = 1, . . . , K which as described in Section 2 determines the volatil-ity structure. For simplicity these are chosen as λk = 1 for k = 2, . . . , K and allowingonly λ1 to be determined by the data. This allows for the investigated models togenerate the observed hump across maturity in the implied volatility surface. Thischoice is primarily motivated by simplicity and could easily be extended to more
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realistic, but also less parsimonious structures (see Brigo and Mercurio (2006) foran overview in the log-normal case).
5.1 Classic models
In this setup a standard log-normal process is perturbed by 1) a displacement factorα, 2) stochastic volatility and 3) a compound Poisson jump component with normallydistributed jump sizes.
5.1.1 DDSV
The first model studied is characterized by a displaced diffusion and stochasticvolatility hence the name DDSV. The Libor rates have dynamics given under theterminal measure F∗ by
Fk(t) + α = (Fk(0) + α) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
, (17)
where b∗k(s) is given in expression (15) and the driving process L∗t is defined as
L∗t =
∫ t
0
√
y(s)dW ∗s ,
where y(t) follows the classic mean reverting diffusion
dy(t) = κ(η − y(t))dt + ǫ√
y(t)dZ(t). (18)
where Z(t) is a Brownian motion independent of W ∗t . The properties of the square-
root process y(t) are studied in Cox, Ingersoll, and Ross (1985) and Heston (1993).In particular it is well known that if the mean reversion speed κ is positive then y(t)is stationary and ergodic implying that the process L∗
t will converge to Gaussianity.However, it is shown in the appendix that L∗
t displays positive excess kurtosis givenby5
γ2 =3V ar(
∫ t
0y(s)ds)
(
ηt + (y(0) − η) 1κ(1 − e−κt)
)2 . (19)
allowing for more flexibility in the term structure of moments, in particular permit-ting a lower decay rate of convergence to Gaussianity compared to the homogeneousLevy process.
5An explicit, but less intuitive, expression for the kurtosis is derived in Das and Sundaram(1999)
119
The assumption that W ∗t is independent of Z(t) means that the L∗
t process is sym-metric due to the symmetry of the Brownian motion. This is also referred to asabsence of the popularly coined ”leverage” effect meaning that negative moves ofthe interest rate are followed by an increase in the volatility. There is empiricalevidence suggesting that this assumption is reasonable under the physical measure.For example Chen and Scott (2004) find that the correlation between short ratesand volatility is very small but this may or may not apply to the martingale measuredistribution studied in this paper.6 In any case, from an option pricing perspectivea leverage effect is important only because it generates skewness in the martingalemeasure distribution of the underlying which results in an asymmetric smile. Inthe DDSV model in (17) skewness is added directly in the rates through the dis-placement factor α with positive values corresponding to negative skewness. Theskewness in the rates will in fact persist as L∗
t converges to Gaussianity, retainingan implied volatility skew – even for long maturities as shown in Rebonato (2002).The downside of this model is that the Libor rates are not guaranteed to be positivebut whether this poses any other than purely theoretical problems is questionable.The characteristic function φSV (u, t) of L∗
t is given by (see for example in Cox,Ingersoll, and Ross (1985))
φSV (u, t) := E[exp(iuL∗t )] = A exp(y(0)B), (20)
x =iu2
2,
D =√
κ2 − 2ǫ2ix,
A =exp
(
ηκ2tǫ2
)
(
cosh(Dt/2) + κD
sinh(Dt/2))
2ηκ
ǫ2
,
B =2ix
κ + D coth(Dt/2).
5.1.2 Strip of regularity for φSV
In order to calculate the price in Proposition 1 it is necessary to evaluate the char-acteristic function in the complex domain. In doing this we must ensure that thecharacteristic function is only evaluated in its strip of regularity
z ∈ C | |φSV (z, t)| < ∞.
Unfortunately the strip of regularity cannot be derived explicitly, but instead onecan set up parameter restrictions.
6In fact Jarrow, Li, and Zhao (2007) find that there is evidence that points toward correlationbetween rates and volatility
120
From Andersen and Piterbarg (2005) Proposition 3.1 it follows that for each z =z1 + iz2 there exists a critical time T when |φSV (z, T )| = ∞.with E = (κ2 − ǫ2(z2 + z2
2)) the explosion times are given as
1. If E ≥ 0 or z2 ∈ [0, 1] then
T = ∞.
2. If E < 0 then
T = 41√E
(
π + arctan
(
−0.5
√E
κ
))
.
This means that for each z2 the parameters have to satisfy
E ≥ 0 or T ∗ < 41√E
(
π + arctan
(
−0.5
√E
κ
))
.
These restrictions ensure that the characteristic function is well defined up to thelongest maturity T ∗ = 20 years in the sample of caplet prices.
5.1.3 DDSVJ
The DDSVJ model extends the previous DDSV model by adding a jump componentdescribed by a compound Poisson process. This extension allows the model togenerate more kurtosis and therefore steeper smiles in the short end of the maturityspectrum. Since a compound Poisson process is a Levy process it will normallyconverge to Gaussianity faster than the stochastic volatility component, resulting inonly secondary effects on the shape of the smile for longer maturities.As before the rates are specified as
Fk(t) + α = (Fk(0) + α) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
,
and jumps are introduced by setting
L∗t =
∫ t
0
√
y(s)dW ∗s +
N(t)∑
j=1
Xj,
where N(t) is a Poisson process with intensity λCP and the Xj’s are IID randomvariables drawn from a normal distribution with mean µCP and variance σ2
CP .
121
The L∗t process can be rewritten in the random measure notation used in Section 2
as
L∗t =
∫ t
0
√
y(s)dW ∗s +
∫ t
0
∫
R
x(µ(ds, dx) − kCP (x)dsdx),
where the Levy density kCP is given by
kCP (x) =λCP√2πσCP
exp
(−(x − µCP )2
σ2CP
)
.
Using standard calculations (see for example Cont and Tankov (2004)) the charac-teristic function for L∗
t is given by
E[exp(iuL∗t )] = φDDSV J(u, t)
= φSV (u, t) × exp[
tλCP
(
exp(−σ2CP u2/2 + iuµCP ) − 1
)
)]
.
The second exponential term is non-explosive ∀u ∈ C so the parameter restrictionsare µCP ∈ R, σCP > 0 and those in Section 5.1.2.
5.2 Self-Similar Additive Models
A process is called self-similar if it has marginal laws that obey
L(ta)d= aγL(t) ∀a > 0,
where γ is termed the exponent of self-similarity andd= denotes equality in law.
The Brownian motion is a trivial example of a self-similar process with γ = 0.5.Processes that are both self-similar and additive have been studied in Sato (1991)and are therefore often referred to as Sato processes. These processes are verymuch related to the concept of self-decomposability. A random variable X is self-decomposable if for any constant c ∈ [0, 1] there exists an independent randomvariable X(c) such that X can be decomposed as
Xd= cX + X(c).
X is also self-decomposable if the corresponding Levy density has the form h(x)x
whereh(x), is increasing for negative x and decreasing for positive x. h(x) is referred toas the self-decomposability characteristic for the random variable X.Sato (1991) showed that a law is self-decomposable if and only if it is the law atunit time of an additive and self-similar process. This means that a Sato processcan be constructed from any self-decomposable law X.
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A very desirable feature of these processes is that skewness and kurtosis are constantthrough time and variance scales with t2γ as discussed in Carr, Geman, Madan, andYor (2007). In terms of matching the implied variance in Figure 6 a scaling of t2γ
is indeed promising since the concave behavior observed in the figure resembles thecase with γ < 0.5. For skewness and kurtosis the constant implication is far fromthe increasing patterns seen in Figure 7 and 8, but it is nevertheless closer to theobserved behavior than the decreasing patterns of the homogeneous Levy process.Motivated by its superior performance and flexibility demonstrated in Carr, Geman,Madan, and Yor (2007) this paper will focus on the Sato process associated withthe Variance Gamma (VG) law.Recall that the VG distribution is obtained as the unit time law of the VG processpreviously introduced in Section 4, i.e
XV G = XV G(1).
Carr, Geman, Madan, and Yor (2002) show that the Levy density of the VG law is
kV G(x) = 1x<0C exp(Gx)
|x| + 1x>0C exp(−Mx)
|x| , (21)
with
C =1
ν,
G =
(√
θ2ν2
4+
σ2ν
2− θν
2
)−1
,
M =
(√
θ2ν2
4+
σ2ν
2+
θν
2
)−1
,
and characteristic function given by
φV G(u) =
(
GM
GM + (M − G)iu + u2
)C
. (22)
From (21) one can notice that the VG law is self-decomposable with self-decomposabilitycharacteristic given by
h(x) = 1x<0C exp(Gx) + 1x>0C exp(−Mx).
Therefore for a given exponent of self-similarity γ a Variance Gamma Self-SimilarSelf-Decomposable (VGSSD) process XV GSSD(t) satisfying
XV GSSD(t)d= tγXV G
123
can be constructed. Its characteristic function φV GSSD(u, t) is given by
φV GSSD(u, t) = φV G(utγ) =
(
GM
GM + (M − G)iutγ + u2t2γ
)C
. (23)
For z ∈ R we see that φV GSSD(iz, t) is well defined only when
GM − (M − G)ztγ − z2t2γ > 0
⇔ −G
tγ< −z <
M
tγ,
Again z can be considered fixed and the parameters can be restricted according tothe above expression.The process XV GSSD(t) can also be written in terms of random measures as
XV GSSD(t) =
∫ t
0
∫
R
x(µ(dx, dt) − kV GSSD(t, x)dxdt),
where kV GSSD(t, x) is the Levy system
kV GSSD(t, x) = 1x<0h′( x
tγ
) 1
t1+1γ− 1x<0h
′(x
tγ)
1
t1+1γ
as derived in Theorem 1 in Carr, Geman, Madan, and Yor (2007).Three different models are now constructed using the VGSSD process with the Liborrate dynamics given as in Section 2 by
Fk(t) = Fk(0) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
.
5.2.1 VGSSD
The first self-similar additive model has no diffusion component and is simply definedas
L∗t = XV GSSD(t)
with characteristic function φV GSSD(u, t) given in (23)
5.2.2 VGSSDC
This model is driven by the VGSSD process and a Brownian motion with a constantdiffusion parameter. It is defined as
L∗t =
∫ t
0
cdW ∗s + XV GSSD(t).
124
It follows from standard calculations that the above process has characteristic func-tion
φV GSSDC(u, t) = exp(−c2tu2/2)φV GSSD(u, t).
5.2.3 VGSSDSV
Here stochastic volatility is added in the diffusion component yielding
L∗t =
∫ t
0
√
y(s)dW ∗s + XV GSSD(t),
where y(t) is a CIR process defined in (18). The above process has characteristicfunction
φV GSSDSV (u, t) = φSV (u, t)φV GSSD(u, t),
with φSV (u, t) defined in (20) subject to regularity conditions described Section5.1.2.
5.3 Time-Changed Levy processes
As discussed earlier, the standard Levy processes suffer from the fact that theyconverge to Gaussianity when maturity increases. One way to postpone this central-limit-theorem effect is to time-change the Levy process with respect to an increasingprocess. The time-change is also referred to as adding stochastic volatility to a Levyprocess (see Carr, Geman, Madan, and Yor (2003)). Taking XV G(t) from Section 4and a rate of time change y(s) given by a CIR process independent of XV G(t), weconstruct the new process XV GSV as follows
XV GSV (t) = XV G(Y (t)).
where Y (t) =∫ t
0y(s)ds.
The characteristic function follows from standard calculations (see for exampe Carr,Geman, Madan, and Yor (2003))
φV GSV (u, t) = φSV (−i log(φV G(u))) ,
with φSV and φV G defined in (20) and (22) respectively.
The strip of regularity follows from the arguments in Section 5.1.2:
125
For z = z1 + iz2 we have |φV GSV (z)| < ∞ if the following two conditions are met
1. G < −z2 < M .
2. For w = Im(−i log(φV G(z1 + iz2, 1)) and E = (κ2 − ǫ2(w + w2))
E ≥ 0 or T ∗ < 41√E
(
π + arctan
(
−0.5
√E
κ
))
.
The VGSV process can also be defined using the random jump measure µ as
XV GSV (t) =
∫ t
0
∫
R
x(µ(dx, dt) − kV G(x)y(t)dxdt),
where kV G(x) is the Levy density for the Variance Gamma law defined in (21).Several models can now be built using the VGSV process.
5.3.1 VGSV
The first model is a pure jump process simply defined as
L∗t = XV GSV (t)
where the above process has characteristic function φV GSV (u, t).
5.3.2 VGSVC
Adding a Brownian motion with a constant diffusion parameter yields
L∗t =
∫ t
0
cdW ∗s + XV GSV (t),
where the above process has characteristic function
φV GSV C(u, t) = exp(−c2tu2/2)φV GSV (u, t).
5.3.3 VGSVD
In this model a stochastic volatility process for the diffusion component is alsoincluded. Specifically
L∗t =
∫ t
0
√
y(s)dW ∗s + XV GSV (t), (24)
126
where y(t) is again described by a CIR process
dy(t) = κ(η − y(t))dt + ǫ√
y(t)dZt.
y(t) is assumed independent of W ∗t and XV GSV (t). The characteristic function of
L∗t in (24) is then given by
φV GSV D(u, t) = φSV (u, t)φV GSV (u, t),
where φSV is equal to φSV with κ, η, ǫ and y(0) replaced by κ, η, ǫ and y(0).
6 Estimation Methodology
The problem of estimating an option pricing model is that we have observations ofthe underlying interest rate process as well as the prices of options on the underlying.The underlying is observed under the physical measure but caplets are priced underthe forward measure. Reconciling the differences between the physical measure anda martingale measure is a technically daunting task (see for example Jones (2003)for the equity case) and requires many assumptions on the structure of the riskpremiums that arise when moving from one measure to another. In this paper thefocus is exclusively on modeling the martingale measure, and therefore the relationto the physical measure is ignored. This means that the underlying Libor rate termstructure will only appear as an input to the option pricing formula. This approachis similar to well known studies of the risk-neutral distribution such as Bakshi, Cao,and Chen (1997) and Huang and Wu (2004) who study S&P 500 index options andequity options respectively.This section is divided into two different subsections. In Section 6.1 the single daycalibration approach is described. The purpose of this avenue is to investigate pa-rameter stability and pricing performance, on single day data alone.Throughout the paper the parameters are estimated by minimizing the sum ofsquared percentage pricing errors (SSE), whether on a daily or sample-wide ba-sis. For each day t = 1, . . . , T = 80 we observe moneyness mi and maturity τi fori = 1, . . . , M = 97. The percentage pricing error is defined as
ui,t =C(t, mi, τi) − C(t, mi, τj, Θ)
C(t, mi, τi). (25)
The sum of squared percentage errors for time t is defined as
SSEt = u′tut =
M∑
i=1
(
C(t, mi, τi) − C(t, mi, τj, Θ)
C(t, mi, τi)
)2
,
127
where C() and C() denote model and market price respectively. Θ is the vector ofparameters in the model.The SSE is appealing as an objective function for two reasons. First, taking percent-ages means that the procedure weighs pricing errors evenly across moneyness andmaturity. Second, the quadratic nature of the SSE penalizes larger errors therebycreating less variability in the percentage error matrix. This approach is also prefer-able to an objective function based on absolute deviations since that would favor themore expensive long maturity options. Naturally, there are other metrics that wouldachieve the same objectives, for example an SSE using implied volatilities instead ofprices. This would perhaps be more appropriate due to the market practice of quot-ing prices and bid-ask spreads in terms of their implied volatilities. Unfortunatelythis approach is not computationally feasible in a larger dataset such as this onedue to implied volatility not having a closed form representation, in the generalizedLevy models studied in this paper.
6.1 Daily Recalibration
For each day and for each model, the following optimization problem is solved
Θ = arg min ut(Θ)′ut(Θ), t = 1, . . . , T.
The starting value of the volatility process y(0) is normalized to 1 for identificationreasons. The optimization procedure used is the Nelder and Mead (1964)-algorithmwith starting values from the previous day. For the first day in the sample, parametervalues are found using the randomized global search algorithm CMAES by Hansen,Muller, and Koumoutsakos (2003).
6.2 Sample Wide Analysis
A different way of evaluating the model performance is by a sample-wide estimationwhich tests the models ability to simultaneously fit the entire time-series of data.Specifically this procedure allows us to perform a more formal test of superior per-formance.The estimation is performed by solving the optimization problem:
Θ = arg min1
T
T∑
t=1
SSEt(Θ).
Again the starting value for the y(0) process is normalized to 1. Alternatively, thevolatility process y(t) could have been treated as a latent variable and estimatedusing a filtering approach (see for example Pan (2002)). Filtering out the volatility
128
essentially creates a new parameter to estimate for each day in the sample, giv-ing much more flexibility to the models where the stochastic volatility componentaccounts for most of the randomness. This effectively biases any performance com-parison toward these models hence this procedure is avoided, and the starting valueof the volatility process(y(0)) is kept fixed.The covariance matrix of Θ can be estimated using the classical robust estimatorof White (1980) which is trivially extended to the present nonlinear setting (see forexample in Mittelhammer, J., and Miller (2000) Section 15.4.1)
cov(Θ) =
[
T∑
t=1
(
∂ut(Θ, yt)
∂Θ
∂ut(Θ, yt)
∂Θ
′) ∣∣
∣
Θ=Θ
]−1
×[
T∑
t=1
(
∂ut(Θ, yt)
∂ΘSSEt
∂ut(Θ, yt)
∂Θ
′) ∣∣
∣
Θ=Θ
]
×[
T∑
t=1
(
∂ut(Θ, yt)
∂Θ
∂ut(Θ, yt)
∂Θ
′) ∣∣
∣
Θ=Θ
]−1
. (26)
The gradient vectors in the above expression can be derived in closed form for allthe models studied in this paper. This is done by inserting the pricing formula fromProposition (1) in the expression for ut in (25) and differentiating with respect tothe parameters to calculate the gradient vectors. These results are rather lengthyand available from the author upon request.To compare the performance of two different models, a t-test is performed based onthe sample differences of the sum of squared errors. Defining dt = SSEi
t − SSEjt as
the difference in errors for the ith and jth model and setting d = 1T
∑Tt=1 dt we get
a test statistic :
S =d
stdev(d).
Diebold and Mariano (1995) show that S is approximately standard normal underthe null hypothesis of equal mean squared percentage errors. The denominator isadjusted using the Newey and West (1987) correction for autocorrelation in errors:
stdev(d) =
√
√
√
√
T∑
i=1
d2t + 2
q∑
v=1
[
1 − v
q + 1
] T∑
t=v+1
(dtdt−v),
where q is the number of lags determined through the optimal selection procedureof Andrews (1991)
129
7 Results
7.1 Results: Daily recalibration
The overall pricing performance for each model is reported as the average absolutepercentage pricing error(APE) across both time, maturity and strike.
APE =1
TM
T∑
t=1
M∑
i=1
∣
∣
∣
∣
∣
C(t, mi, τi) − C(t, mi, τj, Θ)
C(t, mi, τi)
∣
∣
∣
∣
∣
.
As a benchmark case the pure log-normal model of Brace, Gatarek, and Musiela(1997), Miltersen, Sandmann, and Sondermann (1997) and Jamshidian (1997) withL∗
t =∫ t
0cdW ∗
s , is estimated for each day. As expected the log-normal model has arather high APE of 20.02% (s.e 0.25%) with c = 0.134.The results for all the models specified in the previous section can be found in Tables1 to 3. The first thing to note is a considerably better pricing performance for allthe different specifications. The lowest APE is found with the VGSVD model withan APE of 2.457%. This is not surprising as it is by far the most flexible model witha whopping 9 parameters. However the simpler time-changed Levy and self-similaradditive models have only a slightly worse performance so it appears that little isgained when increasing the complexity. In terms of performance the classic models,DDSV and DDSVJ are ranked the lowest with APE’s of 4.65% 3.91% respectively.These levels are, however, still lower than the 5% mark often considered to be anupper threshold of a tolerable APE.Figure 9 plots the daily APE across time, and here it can be seen that movingfrom DDSV to DDSVJ has a clear significant change in overall performance. Thesechanges are not at all apparent when moving between specifications in the self-similar additive or time-changed Levy models. In fact adding a constant or stochas-tic volatility diffusion component seems to increase the performance very little.Turning to the parameters and starting with the classic models in Table 1 it is firstnoticed that α is of a considerable size in the DDSV case. This confirms empiricalresults in section (4), that the interest rate distribution is heavily skewed to the left.α falls slightly when adding the compound Poisson jumps as this process also adds tothe left-skewness by having a mean jump µCP of negative size. The average intensityλCP = 1.5 corresponds to an infrequent jump approximately every 8 months. Thevolatility of volatility appears very low at 0.005. However one must be careful not tointerpret the values in log-normal terms. Ignoring stochastic volatility and jumps,Rebonato (2004) show that as α increases the process converges to an arithmeticBrownian motion. Gaining intuition from this limit case, we would indeed expectthe parameters to be of a lower magnitude since as α increases the parameters beginto determine variance in the level of rates as opposed to the variance of log differ-
130
ences.Moving to the self-similar additive models in Table 2 it is again seen that consider-able skewness is needed to capture the caplet prices with an average M being morethan twice the size of the average G. A constant volatility diffusion componentappears to be unnecessary as c is very small whereas adding stochastic volatilityhas a slightly bigger impact. The stochastic volatility component has a volatility ofvolatility ǫ of 0.09 meaning that it accounts for a significant part of the randomness.Finally the γ parameter is fairly constant, both across models and time, with anaverage value around 0.09-0.11 corresponding to a ”smaller than linear” scaling ofvariance with time which was also observed in Figure 6.Finally, moving to the time-changed Levy models in Table 3 it can first be seen inVGSVC that the constant volatility diffusion component is of slightly higher mag-nitude than in VGSSDC. It also seems to have a stabilizing effect as the standarderrors have decreased compared to VGSV. Adding stochastic volatility in the diffu-sion component in the VGSVD model has an impact mainly in the short run sincethe mean reversion speed is very fast.
7.1.1 Error Analysis
In order to investigate how the error is distributed across time, the daily APE isplotted as a function of time in Figure 9. Here we see that the self-similar additivemodels and time-changed Levy models are very similar within their respective modelclasses. This is not the case for the classic models since going from DDSV to DDSVJhas a noticeable effect. Moving along the time dimension the errors appear fairlystationary without any particular critical days.In Tables 4 to 6 the average percentage errors are given across moneyness andmaturity, but I refrain from taking absolute values in order to detect any systematicover- or underpricing.Starting with the classic models in Table 4 we see that if we move from low to highmoneyness there is a tendency to heavily underprice ITM caplets, slightly overpriceATM and then again underprice deep OTM options. This pattern indicates theclassic model’s overall inability to create enough curvature in the smile.Looking further at the self-similar additive models in Table 5, we see an overallimprovement especially for out-of-the money pricing errors which have decreased toalmost a third of the level in the classic models.The same effect is observed for the time-changed Levy models in Table 6 but witha slightly lower overall absolute error level.
131
7.1.2 Out-Of Sample Performance
As a robustness check an out-of-sample analysis is performed. Since an interest ratemodel of the kind investigated in this paper is mainly used for hedging purposes it isnot only relevant how the model replicates todays prices but also future prices. Foreach day in the sample the calibrated parameters are used to calculate the pricingerror today, 1 week, 1 month, 3 months, and 6 months ahead. The results areaveraged and reported in Table 7. The out-of-sample pricing performance rankingof the models is almost the same as its in-sample counterparts, with the only changebeing the VGSSD model faring slightly better than the VGSSDC model. The errorsare very similar across models for the 6 month horizon, but what is perhaps the mostrelevant to option traders is the 1 week and 1 month ahead pricing errors since thesecorrespond to the most often used rebalancing frequencies of a hedge portfolio. Inthese two horizons the gain from using the time-changed Levy or self-similar additivemodels over the classic models is clear.
7.2 Results: Sample-wide estimation
The results of the sample-wide estimation procedure is shown for the three differentclasses of models in Tables 8-10. Overall the parameters are also largely of thesame magnitude as in the previous section, and one can repeat the interpretationtherein. One slight difference however is in DDSVJ model which for the sample wideanalysis has a significantly higher frequency of jumps with λCP = 15.97 comparedto an average of 1.5 in the daily recalibration case. The higher frequency causesthe jump component to generate higher skewness but the effect is offset by a lowerdisplacement coefficient α. This could indicate that the two sources of skewness inthe DDSVJ model are not fully identified by the data.The benefit of doing a sample wide analysis is that it allows us to perform theDiebold-Mariano test for superior pricing performance. The results can be found inTable 11. A negative (i, j)’th value means that model i is superior to model j andcritical values, for the null hypothesis of equal performance, are values smaller than-1.645 and larger than 1.645. We can see that both the self-similar additive modelsand the time-changed Levy models outperform the classic models in a statisticallysignificant manner. Comparing the self-similar additive models and time-changedLevy models we see that the latter outperforms the former in all cases. One canalso note that adding a constant volatility diffusion component has a significantimpact on performance for the time-changed Levy models but not for the self-similaradditive models.
132
8 Conclusion
In this paper a theoretical and empirical analysis is performed concerning the pric-ing of caplets. By investigating the term structure of moments the paper providesan actual desideratum for the behavior of the driving process in an extended LevyLibor Market Model. The models presented in this paper can be considered a firstattempt at achieving the posed goals. The empirical analysis in the paper showthat high frequency jump models outperform traditional diffusion or finite activityjump-diffusion based models, in pricing caplets. It also shows that adding stochasticvolatility diffusion component to an infinite activity jump model improves the per-formance at a statistically significant but economically negligible level. Furthermorethe Self-Similar Additive Variance Gamma model is shown to have an impressiveperformance despite its parsimony.High frequency jump models no doubt have the potential of becoming as popular ininterest rate modeling as they have recently become in stock price modeling. How-ever, the popularity is contingent on the ease of which the model can price not justcaps but also swaptions. Clearly the perfect correlation implication of the 1 factorapproach taken in this paper is insufficient as swaptions are correlation sensitiveproducts. Several factors, with a well specified dependence structure, would have tobe added to the existing framework in order to get realistic prices. Needless to saythis is a non-trivial task left for future research.
133
Appendix A: Deriving the drift under the Terminal
Measure
Proof. Inserting (9) and (10) in (7) and (3) you get
bk(t) = − 1
2c(t)2 −
∫
R
(
eλkx − 1 − λkx)
K∏
j=k+1
(
1 +δ(eλjx − 1)Fj(t−)
1 + δFj(t−)
)
νF∗
(t, dx),
and
LFk+1
t = −∫ t
0
K∑
j=k+1
δc(s)Fj(s)
1 + δFj(s)ds +
∫ t
0
c(s)dW F∗
s +
∫ t
0
∫
R
x(µ − νFk+1
)(ds, dx).
Using (10) and∫ t
0
∫
Rx(µ − νF
k+1)(ds, dx) =
∫ t
0
∫
Rx(µ − νF
∗
+ νF∗ − νF
k+1)(ds, dx)
the above equation can be written as
LFk+1
t = −∫ t
0
K∑
j=k+1
δc(s)Fj(s)
1 + δFj(s)ds
+
∫ t
0
∫
R
x
(
1 −K∏
j=k+1
(
1 +δ(eλjx − 1)Fj(s)
1 + δFj(s)
)
)
νF∗
(ds, dx) + L∗t .
Note that the second term is finite because of the assumption of finite variation inthe jump process.A new drift can then be defined as
b∗k(t) = −K∑
j=k+1
δc(t)Fj(t−)
1 + δFj(t−)
− 1
2c(t)2 −
∫
R
(
λkx − (eλkx − 1)K∏
j=k+1
(
1 +δ(eλjx − 1)Fj(t−)
1 + δFj(t−)
)
)
νF∗
(t, dx),
and the Libor rate process under the Terminal measure can be written as
Fk(t) = Fk(0) exp
(∫ t
0
b∗k(s)ds + λk
∫ t
0
dL∗s
)
∀k = 1, . . . , K
134
Appendix B: Excess Kurtosis of a Brownian motion
with CIR Stochastic Volatility
Defining
Z =
∫ t
0
√
y(s)dWs.
The excess kurtosis is defined as
γ2 =E[(Z − E[Z])4]
E[(Z − E[Z])2]2− 3 (27)
Since the Brownian motion has mean zero we get
E[Z] = 0, (28)
and from the Ito isometry we get
E[Z2] = E[(
∫ t
0
√
y(s)dWs)2] =
∫ t
0
E[y(s)]ds.
Using Cox, Ingersoll, and Ross (1985) equation 19 we have that E[y(s)] = η+(y(0)−η)e−κs. Inserting this in the above gives us
E[Z2] = ηt + (y(0) − η)1
κ(1 − e−κt). (29)
Since the fourth moment of the normal distribution is 3 it follows that
E[Z4] = E[(
∫ t
0
y(s)ds)23] = 3V ar(
∫ t
0
y(s)ds) (30)
Inserting (28) (29) and (30) in (27) we get the result in equation (19)
135
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139
Table 1: Classic Models: Single day calibrationΘ DDSV DDSVJα 0.5956 0.4362
(0.01384) (0.01831)κ 0.9813 2.636
(0.01878) (0.6734)η 0.000115 1.09e-005
(1.413e-005) (3.587e-006)ǫ 0.005712 0.004863
(0.0004473) (0.0003668)λ(1) 0.9275 0.5836
(0.01079) (0.01474)λCP - 1.544
(0.4126)µCP - -0.002502
(0.006779)σCP - 0.01048
(0.003175)APE 0.04656 0.03908
(0.00036653) (0.00040048)
This table shows the average parameter values for the classic models. Parameters are estimated by minimizing the
SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE denotes the
average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis
140
Table 2: Self-similar Additive Models: Single day calibrationΘ VGSSD VGSSDC VGSSDSVC 1.315 1.304 1.243
(0.01666) (0.01886) (0.02253)G 2.848 2.833 2.556
(0.06433) (0.06728) (0.07757)M 6.173 6.139 6.819
(0.06266) (0.06798) (0.161)γ 0.07221 0.07022 0.05995
(0.006818) (0.007003) (0.007732)λ1 0.504 0.5031 0.517
(0.004906) (0.004926) (0.006627)c - 0.00247 -
(0.001018) (-)κ - - 1.664
(0.3125)η - - 0.1134
(0.06212)ǫ - - 0.09344
(0.009094)APE 0.02871 0.02867 0.02691
(0.0005743) (0.0005681) (0.00057461)
This table shows the average parameter values for the self-similar additive models. Parameters are estimated by
minimizing the SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE
denotes the average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis
141
Table 3: Time-Changed Levy Models: Single day calibrationΘ VGSV VGSVC VGSVDC 4.464 2.812 2.673
(0.9362) (0.1534) (0.165)G 4.285 3.888 2.793
(0.1708) (0.1114) (0.09465)M 7.467 7.078 14.05
(0.1544) (0.09654) (0.9767)κ 1.287 1.197 1.537
(0.1088) (0.09786) (0.09272)η 0.0141 0.007875 0.0002765
(0.001885) (0.00154) (7.928e-005)ǫ 0.4633 0.478 0.5014
(0.04138) (0.02326) (0.02886)λ1 0.6833 0.6891 0.7202
(0.009835) (0.009266) (0.01259)c - 0.01549 -
(0.002142) (-)κ - - 2.432
(0.07716)η - - 0.001761
(0.0001926)ǫ - - 0.3886
(0.01588)APE 0.02551 0.0253 0.02457
(0.0005838) (0.00058789) (0.00070581)
This table shows the average parameter values for the time-changed Levy models. Parameters are estimated by
minimizing the SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE
denotes the average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis
142
Tab
le4:
Avera
ge
Perc
enta
ge
Err
ors
ofC
lass
icM
odels
Mon
eynes
s/M
aturi
ty0
12
34
56
78
910
1215
20Pan
elA
:A
vera
gePer
centa
geE
rror
sof
DD
SV
0.6
--
0.18
00.
172
0.17
00.
167
0.16
00.
155
0.14
10.
129
0.11
10.
078
0.04
40.
7-
0.13
30.
137
0.12
50.
122
0.12
20.
121
0.11
90.
107
0.09
60.
082
0.04
80.
015
0.8
-0.
098
0.08
30.
066
0.06
30.
067
0.06
80.
071
0.06
50.
050
0.04
50.
010
-0.0
160.
90.
008
0.05
90.
031
0.01
40.
013
0.01
70.
019
0.02
40.
021
0.00
90.
010
-0.0
22-0
.043
1-0
.010
0.02
4-0
.004
-0.0
18-0
.017
-0.0
15-0
.014
-0.0
06-0
.007
-0.0
15-0
.010
-0.0
39-0
.054
1.1
0.00
20.
005
-0.0
20-0
.029
-0.0
24-0
.022
-0.0
23-0
.012
-0.0
13-0
.017
-0.0
09-0
.034
-0.0
421.
20.
000
-0.0
01-0
.021
-0.0
22-0
.013
-0.0
10-0
.012
-0.0
01-0
.001
-0.0
010.
011
-0.0
10-0
.017
1.3
--
-0.0
12-0
.007
0.00
60.
009
0.00
90.
020
0.02
10.
023
0.03
70.
020
0.01
2Pan
elB
Ave
rage
Per
centa
geE
rror
sof
DD
SV
J0.
6-
-0.
128
0.13
60.
139
0.13
90.
136
0.13
40.
124
0.11
60.
104
0.07
70.
046
0.7
-0.
062
0.10
60.
107
0.10
50.
105
0.10
30.
103
0.09
40.
086
0.07
80.
050
0.02
00.
8-
0.04
30.
071
0.06
40.
058
0.05
90.
056
0.05
90.
055
0.04
20.
043
0.01
5-0
.010
0.9
0.00
10.
019
0.03
40.
024
0.01
70.
015
0.01
20.
015
0.01
30.
003
0.00
9-0
.017
-0.0
351
-0.0
05-0
.002
0.00
8-0
.001
-0.0
07-0
.013
-0.0
19-0
.013
-0.0
15-0
.021
-0.0
11-0
.034
-0.0
451.
10.
001
-0.0
11-0
.003
-0.0
08-0
.013
-0.0
20-0
.028
-0.0
20-0
.022
-0.0
24-0
.011
-0.0
30-0
.034
1.2
--0
.011
-0.0
03-0
.003
-0.0
03-0
.009
-0.0
18-0
.011
-0.0
11-0
.010
0.00
7-0
.008
-0.0
091.
3-
-0.
003
0.00
90.
013
0.00
80.
001
0.00
90.
009
0.01
20.
031
0.02
00.
018
This
table
show
sth
eaver
age
caple
tpri
cing
erro
racr
oss
money
nes
sand
stri
ke
when
per
form
ing
daily
reca
libra
tion
each
Wed
nes
day
from
May
7th
2003
to
Novem
ber
10th
2004
143
Tab
le5:
Avera
ge
Perce
nta
ge
Erro
rsofSelf-sim
ilar
Additiv
eM
odels
Mon
eyness/M
aturity
01
23
45
67
89
1012
1520
Pan
elA
:A
veragePercen
tageE
rrorsof
VG
SSD
0.6-
-0.037
0.0470.055
0.0580.059
0.0640.062
0.0630.065
0.0520.043
0.7-
-0.0210.036
0.0450.048
0.0490.049
0.0520.049
0.0480.049
0.0310.019
0.8-
-0.0200.034
0.0380.037
0.0370.032
0.0350.034
0.0250.030
0.005-0.008
0.9-0.020
-0.0160.032
0.0350.031
0.0250.017
0.0170.015
0.0060.012
-0.017-0.029
1-0.004
-0.0150.029
0.0330.028
0.0170.004
0.0050.002
-0.0040.002
-0.027-0.035
1.10.002
-0.0200.020
0.0260.022
0.009-0.006
-0.004-0.007
-0.0090.000
-0.023-0.024
1.2-
-0.0280.009
0.0170.017
0.004-0.011
-0.009-0.010
-0.0090.006
-0.010-0.003
1.3-
-0.000
0.0120.014
0.002-0.011
-0.008-0.009
-0.0050.013
0.0040.015
Pan
elB
Average
Percen
tageE
rrorsof
VG
SSD
SV
0.6-
-0.036
0.0470.055
0.0580.059
0.0640.062
0.0620.064
0.0500.040
0.7-
-0.0210.038
0.0470.050
0.0510.051
0.0540.051
0.0490.049
0.0300.016
0.8-
-0.0150.039
0.0420.041
0.0410.037
0.0400.039
0.0300.034
0.007-0.009
0.9-0.027
-0.0090.034
0.0330.029
0.0250.018
0.0200.019
0.0110.018
-0.012-0.027
1-0.009
-0.0090.025
0.0230.018
0.008-0.002
0.0020.001
-0.0030.005
-0.023-0.033
1.10.003
-0.0130.014
0.0140.010
-0.001-0.013
-0.008-0.009
-0.0100.001
-0.022-0.024
1.2-
-0.0170.006
0.0090.007
-0.003-0.016
-0.010-0.010
-0.0070.009
-0.008-0.005
1.3-
-0.002
0.0080.009
0.000-0.011
-0.005-0.004
0.0010.020
0.0080.014
This
table
show
sth
eavera
ge
caplet
pricin
gerro
racro
ssm
oney
ness
and
strike
when
perfo
rmin
gdaily
recalib
ratio
nea
chW
ednesd
ay
from
May
7th
2003
to
Novem
ber
10th
2004
144
Tab
le6:
Avera
ge
Perc
enta
ge
Err
ors
ofT
ime-C
hanged
Levy
Models
Mon
eynes
s/M
aturi
ty0
12
34
56
78
910
1215
20Pan
elA
:A
vera
gePer
centa
geE
rror
sof
VG
SV
0.6
--
0.04
00.
047
0.05
30.
057
0.05
80.
064
0.06
30.
065
0.06
90.
060
0.05
40.
7-
-0.0
060.
038
0.04
00.
041
0.04
20.
043
0.04
80.
047
0.04
70.
051
0.03
70.
029
0.8
-0.
000
0.03
10.
027
0.02
30.
023
0.02
00.
026
0.02
70.
020
0.02
90.
008
0.00
00.
9-0
.015
0.00
60.
024
0.01
60.
010
0.00
5-0
.001
0.00
30.
004
-0.0
020.
008
-0.0
17-0
.024
10.
005
0.00
60.
018
0.01
10.
004
-0.0
05-0
.014
-0.0
10-0
.010
-0.0
13-0
.003
-0.0
28-0
.033
1.1
0.00
00.
000
0.01
10.
008
0.00
3-0
.007
-0.0
18-0
.013
-0.0
12-0
.013
-0.0
01-0
.022
-0.0
231.
2-
-0.0
080.
004
0.00
50.
005
-0.0
04-0
.016
-0.0
10-0
.008
-0.0
050.
012
-0.0
04-0
.001
1.3
--
-0.0
010.
006
0.00
90.
001
-0.0
08-0
.002
0.00
00.
006
0.02
60.
015
0.02
1Pan
elB
Ave
rage
Per
centa
geE
rror
sof
VG
SV
D0.
6-
-0.
027
0.03
30.
038
0.04
10.
041
0.04
40.
041
0.04
20.
044
0.04
00.
035
0.7
--0
.017
0.02
80.
030
0.02
90.
029
0.02
90.
032
0.03
00.
030
0.03
60.
029
0.02
00.
80.
000
0.00
10.
033
0.02
70.
022
0.02
00.
015
0.01
90.
020
0.01
40.
024
0.00
9-0
.001
0.9
-0.0
230.
015
0.03
50.
027
0.02
00.
013
0.00
50.
006
0.00
70.
001
0.01
1-0
.010
-0.0
211
-0.0
010.
010
0.02
60.
022
0.01
60.
005
-0.0
06-0
.003
-0.0
04-0
.008
0.00
1-0
.020
-0.0
291.
10.
001
-0.0
020.
012
0.01
40.
011
0.00
1-0
.012
-0.0
09-0
.010
-0.0
110.
000
-0.0
19-0
.020
1.2
--0
.012
0.00
10.
008
0.01
00.
001
-0.0
12-0
.008
-0.0
09-0
.007
0.00
7-0
.007
-0.0
031.
3-
--0
.006
0.00
50.
012
0.00
5-0
.005
-0.0
01-0
.001
0.00
20.
018
0.00
80.
014
This
table
show
sth
eaver
age
caple
tpri
cing
erro
racr
oss
money
nes
sand
stri
ke
when
per
form
ing
daily
reca
libra
tion
each
Wed
nes
day
from
May
7th
2003
to
Novem
ber
10th
2004
145
Table 7: Out-of-Sample ErrorsModel Today 1 week 1 month 3 months 6 monthsDDSV 0.0466 0.0501 0.0565 0.0648 0.072DDSVJ 0.0391 0.0436 0.0502 0.0583 0.0664VGSSD 0.0287 0.0331 0.0423 0.0476 0.057VGSSDC 0.0287 0.0332 0.0424 0.0478 0.0572VGSSDSV 0.0269 0.0317 0.0415 0.0466 0.0561VGSV 0.0255 0.0315 0.0406 0.0444 0.0552VGSVC 0.0253 0.0313 0.0399 0.0437 0.0547VGSVD 0.0246 0.0296 0.0385 0.0436 0.0522
This table shows the average caplet pricing error. The SSE is minimized for every day in the sample and then the
error is recorded 1 week to 6 months ahead and averaged across each day in the sample
Table 8: Classic Models: Sample Wide EstimationΘ DDSV DDSVJα 0.5395 0.3415
(0.001916) (6.652e-005)κ 0.9348 0.8601
(0.03145) (0.004776)η 0.0001027 4.257e-012
(0.001979) (0.0003069)ǫ 0.005296 0.008384
(0.1526) (0.1964)λ(1) 0.9942 0.8589
(0.01337) (0.0002484)λCP - 15.97
(1.795)µCP - -0.003644
(0.008604)σCP - 0.0004898
(0.0004087)APE 0.06064 0.05717
This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized
Method Moments for the entire sample. Standard errors are denoted in parenthesis
146
Table 9: Self-similar Additive Models: Sample Wide EstimationΘ VGSSD VGSSDC VGSSDSVC 1.333 1.333 1.257
(0.005949) (0.003396) (0.04389)G 2.847 2.847 2.336
(0.01039) (0.004659) (0.09208)M 6.277 6.277 7.821
(0.01169) (0.004226) (0.1773)γ 0.07492 0.07492 0.05086
(0.0002244) (0.0001615) (0.005208)λ1 0.491 0.491 0.5114
(0.003487) (0.002321) (0.0006514)c - 8.044e-009 -
(31.18) (-)κ - - 1.183
(0.004361)η - - 4.918e-011
(7.244e-005)ǫ - - 0.1453
(0.01066)APE 0.04912 0.04912 0.04751
This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized
Method Moments for the entire sample. Standard errors are denoted in parenthesis
147
Table 10: Time-Changed Levy Models: Sample Wide EstimationΘ VGSV VGSVC VGSVDC 2.383 2.069 2.23
(0.01169) (0.02239) (0.1605)G 4.1 3.67 3.245
(0.009628) (0.02409) (0.1821)M 7.515 7.083 15.01
(0.007561) (0.02217) (0.3459)κ 0.8159 0.8251 0.9738
(0.009497) (0.0007558) (0.003811)η 0.007727 0.0005804 6.786e-008
(0.0002186) (0.001099) (0.0001121)ǫ 0.4651 0.4688 0.4997
(0.00139) (0.01958) (0.04134)λ1 0.7093 0.7102 0.7759
(0.0005323) (0.0001673) (0.001157)c - 0.02905 -
(0.002724) (-)κ - - 2.327
(0.003144)η - - 0.001577
(3.863e-005)ǫ - - 0.4655
(0.001141)APE 0.0474 0.04713 0.04522
This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized
Method Moments for the entire sample. Standard errors are denoted in parenthesis
148
Tab
le11
:D
iebold
-Mari
ano
Test
Resu
lts
ΘD
DSV
DD
SV
JV
GSSD
VG
SSD
CV
GSSD
SV
VG
SV
VG
SV
CV
GSV
DD
DSV
0-
--
--
--
DD
SV
J-1
8.2
0-
--
--
-V
GSSD
-3.2
1-1
.74
0-
--
--
VG
SSD
C-3
.21
-1.7
4-0
.006
210
--
--
VG
SSD
SV
-3.9
-2.4
4-1
2-1
20
--
-V
GSV
-4.1
8-2
.77
-11.
2-1
1.2
-5.4
80
--
VG
SV
C-4
.23
-2.8
3-1
2-1
2-6
.38
-15.
60
-V
GSV
D-4
.63
-3.2
4-1
5.9
-15.
9-1
0.2
-9.6
6-8
.75
0
This
table
show
sth
eD
iebold
-Mari
ano
test
stati
stic
s.T
he
test
stati
stic
sare
crit
icalat
a5%
level
sfo
rth
enull
hypet
his
of
equalper
form
ance
when
smaller
than
-1.6
45
and
larg
erth
an
1.6
45.
The
entr
ys
inth
eta
ble
indic
ate
model
ivs.
model
jin
the
(i,j)’
then
try.
Aneg
ati
ve
valu
ein
dic
ate
sth
at
model
iis
super
ior
tom
odel
j
149
0.4
0.6
0.8
1
1.2
1.4 05
1015
20
0
20
40
60
80
100
120
140
160
180
MaturityMoneynes(X/F)
Pric
e
Figure 1: Average Caplet PricesThis figure shows the average Euribor caplet price in basis points for our weekly sample covering May 7th 2003 toMay 4th 2005. Prices are interpolated in terms of strike to get a common moneyness throughout the sample
150
0.5
1
1.50 2 4 6 8 10 12 14 16 18 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Moneynes(X/F)
Maturity
Imp
lied
vo
l
Figure 2: Average Caplet Implied VolatilitiesThis figure shows the average Euribor caplet implied vol for our weekly sample covering May 7th 2003 toNovember 10th 2004. Volatilities are found by inverting the Black-Scholes formula for the prices in Figure 1.
151
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Moneynes(X/F)
Imp
lied
Vo
l.
Figure 3: Average Caplet Implied VolatilitiesThis figure shows the average Euribor caplet implied vol for our weekly sample covering May 7th 2003 toNovember 10th 2004. Volatilities are found by inverting the Black-Scholes formula for the prices in Figure 1. Thetwo curves slightly shorter than the rest are the 1 and 2 year maturity. The remaining curves are declining withrespect to maturity.
152
May03Dec03
Jul040 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
Time
ITM X/F=0.6
Im
plie
d V
ol.
May03Dec03
Jul040 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
Time
ATM X/F=1
Im
plie
d V
ol.
May03Dec03
Jul040 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
Time
Maturity
OTM X/F=1.3
Im
plie
d V
ol.
Figure 4: ITM/ATM/OTM Caplet Implied Volatilies
This figure shows the in-the-money, at-the-money and out-of-money term structure of Euribor caplet implied volfor our weekly sample covering May 7th 2003 to November 10th 2004. Volatilities are found by inverting theBlack-Scholes formula for the prices.
153
May03Aug03
Dec03Apr04
Jul04Nov04
05
1015
20
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
MaturityTime
6m
LIB
OR
ra
te
Figure 5: Term Structure of the 6 month Libor RateThis figure shows the shows the 6 month Libor rate for all maturities from 1 to 20 years for our weekly samplecovering May 7th 2003 to November 10th 2004.
154
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7Average Implied Variance
Time
Var
ianc
e
2003−05−07 to 2003−09−172003−09−24 to 2004−02−042004−02−11 to 2004−06−232004−06−30 to 2004−11−10
Figure 6:
This figure shows the average implied variance as a function of time backed out from the caplet dataset. Variancesare backed out for each week in the sample using Variance Gamma model for each time to maturity. The variancesare then averaged across the four different periods in the legend.
155
0 2 4 6 8 10 12 14 16 18 20−3
−2.5
−2
−1.5
−1
−0.5
0Average Implied Skewness
Time
Ske
wne
ss
2003−05−07 to 2003−09−172003−09−24 to 2004−02−042004−02−11 to 2004−06−232004−06−30 to 2004−11−10
Figure 7:
This figure shows the average implied skewness as a function of time backed out from the caplet dataset. Theskewness’s are backed out for each week in the sample using a Variance Gamma model for each time to maturity.They are then then averaged across the four different periods in the legend.
156
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
14Average Implied Kurtosis
Time
Exc
ess
Kur
tosi
s
2003−05−07 to 2003−09−172003−09−24 to 2004−02−042004−02−11 to 2004−06−232004−06−30 to 2004−11−10
Figure 8:
This figure shows the average implied excess kurtosis as a function of time backed out from the caplet dataset.The kurtosis’s are backed out for each week in the sample using a Variance Gamma model for each time tomaturity. They are then then averaged across the four different periods in the legend.
157
Ma
y0
3D
ec0
3Ju
l04
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
Cla
ssic
Mo
de
ls
← D
DS
V
← D
DS
VJ
Tim
e
Percentage Pricing Error
Ma
y0
3D
ec0
3Ju
l04
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
Se
lfsim
ilar A
dd
itive
Mo
de
ls
← V
GS
SD
← V
GS
SD
C
← V
GS
SD
SV
Tim
e
Percentage Pricing Error
Ma
y0
3D
ec0
3Ju
l04
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6T
ime
−C
ha
ng
ed
Le
vy M
od
els
← V
GS
V←
VG
SV
C←
VG
SV
D
Tim
e
Percentage Pricing Error
Figu
re9:
Daily
Reca
libra
tion:
Perce
nta
ge
Erro
rs
This
figure
show
sth
eavera
ge
percen
tage
error
for
each
day
inth
esa
mple
coverin
gM
ay
7th
2003
toN
ovem
ber
10th
2004
158
SCHOOL OF ECONOMICS AND MANAGEMENT UNIVERSITY OF AARHUS - UNIVERSITETSPARKEN - BUILDING 1322
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Policy Issues. 1999-6 Peter Raahauge, Dynamic Programming in Computational Economics. 1999-7 Torben Dall Schmidt, Social Insurance, Incentives and Economic Integration. 1999 Jørgen Vig Pedersen, An Asset-Based Explanation of Strategic Advantage. 1999 Bjarke Jensen, Five Essays on Contingent Claim Valuation. 1999 Ken Lamdahl Bechmann, Five Essays on Convertible Bonds and Capital Structure
Theory. 1999 Birgitte Holt Andersen, Structural Analysis of the Earth Observation Industry. 2000-1 Jakob Roland Munch, Economic Integration and Industrial Location in Unionized
Countries. 2000-2 Christian Møller Dahl, Essays on Nonlinear Econometric Time Series Modelling. 2000-3 Mette C. Deding, Aspects of Income Distributions in a Labour Market Perspective. 2000-4 Michael Jansson, Testing the Null Hypothesis of Cointegration. 2000-5 Svend Jespersen, Aspects of Economic Growth and the Distribution of Wealth. 2001-1 Michael Svarer, Application of Search Models. 2001-2 Morten Berg Jensen, Financial Models for Stocks, Interest Rates, and Options: Theory
and Estimation. 2001-3 Niels C. Beier, Propagation of Nominal Shocks in Open Economies. 2001-4 Mette Verner, Causes and Consequences of Interrruptions in the Labour Market. 2001-5 Tobias Nybo Rasmussen, Dynamic Computable General Equilibrium Models: Essays
on Environmental Regulation and Economic Growth.
2001-6 Søren Vester Sørensen, Three Essays on the Propagation of Monetary Shocks in Open Economies.
2001-7 Rasmus Højbjerg Jacobsen, Essays on Endogenous Policies under Labor Union
Influence and their Implications. 2001-8 Peter Ejler Storgaard, Price Rigidity in Closed and Open Economies: Causes and
Effects. 2001 Charlotte Strunk-Hansen, Studies in Financial Econometrics. 2002-1 Mette Rose Skaksen, Multinational Enterprises: Interactions with the Labor Market. 2002-2 Nikolaj Malchow-Møller, Dynamic Behaviour and Agricultural Households in
Nicaragua. 2002-3 Boriss Siliverstovs, Multicointegration, Nonlinearity, and Forecasting. 2002-4 Søren Tang Sørensen, Aspects of Sequential Auctions and Industrial Agglomeration. 2002-5 Peter Myhre Lildholdt, Essays on Seasonality, Long Memory, and Volatility. 2002-6 Sean Hove, Three Essays on Mobility and Income Distribution Dynamics. 2002 Hanne Kargaard Thomsen, The Learning organization from a management point of
view - Theoretical perspectives and empirical findings in four Danish service organizations.
2002 Johannes Liebach Lüneborg, Technology Acquisition, Structure, and Performance in
The Nordic Banking Industry. 2003-1 Carter Bloch, Aspects of Economic Policy in Emerging Markets. 2003-2 Morten Ørregaard Nielsen, Multivariate Fractional Integration and Cointegration. 2003 Michael Knie-Andersen, Customer Relationship Management in the Financial Sector. 2004-1 Lars Stentoft, Least Squares Monte-Carlo and GARCH Methods for American
Options. 2004-2 Brian Krogh Graversen, Employment Effects of Active Labour Market Programmes:
Do the Programmes Help Welfare Benefit Recipients to Find Jobs? 2004-3 Dmitri Koulikov, Long Memory Models for Volatility and High Frequency Financial
Data Econometrics. 2004-4 René Kirkegaard, Essays on Auction Theory.
2004-5 Christian Kjær, Essays on Bargaining and the Formation of Coalitions. 2005-1 Julia Chiriaeva, Credibility of Fixed Exchange Rate Arrangements. 2005-2 Morten Spange, Fiscal Stabilization Policies and Labour Market Rigidities. 2005-3 Bjarne Brendstrup, Essays on the Empirical Analysis of Auctions. 2005-4 Lars Skipper, Essays on Estimation of Causal Relationships in the Danish Labour
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of Benefits. 2005-6 Marianne Simonsen, Essays on Motherhood and Female Labour Supply. 2005 Hesham Morten Gabr, Strategic Groups: The Ghosts of Yesterday when it comes to
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Credit Risk. 2006-1 Peter Sandholt Jensen, Essays on Growth Empirics and Economic Development. 2006-2 Allan Sørensen, Economic Integration, Ageing and Labour Market Outcomes 2006-3 Philipp Festerling, Essays on Competition Policy 2006-4 Carina Sponholtz, Essays on Empirical Corporate Finance 2006-5 Claus Thrane-Jensen, Capital Forms and the Entrepreneur – A contingency approach
on new venture creation 2006-6 Thomas Busch, Econometric Modeling of Volatility and Price Behavior in Asset and
Derivative Markets 2007-1 Jesper Bagger, Essays on Earnings Dynamics and Job Mobility 2007-2 Niels Stender, Essays on Marketing Engineering 2007-3 Mads Peter Pilkjær Harmsen, Three Essays in Behavioral and Experimental
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Investments 2007-5 Peter Tind Larsen, Essays on Capital Structure and Credit Risk
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2008-2 Jie Zhu, Essays on Econometric Analysis of Price and Volatility Behavior in Asset
Markets 2008-3 David Glavind Skovmand, Libor Market Models - Theory and Applications
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