ALM -Interest Rate Risk Management

1

INTEREST RATE RISK MANAGEMENT:DEVELOPMENTS IN INTEREST RATE TERM

STRUCTURE MODELING FOR RISK MANAGEMENTAND VALUATION OF INTEREST-RATE-DEPENDENT

CASH FLOWSAndrew Ang* and Michael Sherris†

ABSTRACT

This paper surveys the main concepts and techniques of recent developments in the modelingof the term structure of interest rates that are used in the risk management and valuation ofinterest-rate-dependent cash flows. These developments extend the concepts of immunizationand matching to a stochastic interest rate environment. Such cash flows include the cash flowson assets such as bonds and mortgage-backed securities as well as those for annuity products,life insurance products with interest-rate-sensitive withdrawals, accrued liabilities for defined-benefit pension funds, and property and casualty liability cash flows.

1. INTRODUCTION

The aim of this paper is to discuss recent develop-ments in interest rate term structure modeling andthe application of these models to the interest raterisk management and valuation of cash flows that aredependent on future interest rates. Traditional ap-proaches to risk management and valuation are basedon the concepts of immunization and matching ofcash flows. These ideas were pioneered in the actu-arial profession by the British actuary Frank Reding-ton (1952). Interest rates have long been recognizedas important to the risk management of insurance li-abilities. Recent developments have incorporated astochastic approach to modeling interest rates. Anumber of actuaries were early pioneers in this area,including John Pollard (1971), Phelim Boyle (1976,1978), and Harry Panjer and David Bellhouse (1980,1981).

Only a small number of actuaries have been ac-tively involved in international developments in theseareas over more recent times. These have included

*Andrew Ang is a doctoral student at the Graduate School of Busi-ness, Stanford University, Stanford, Calif. 94325.

†Michael Sherris, A.S.A., F.I.A., F.I.A.A., is Associate Professor in Ac-tuarial Studies, School of Economics and Financial Studies, Mac-quarie University, Sydney, NSW 2109, Australia.

Boyle (1992), Reitano (1991a, 1991b, 1992, 1993),Sharp (1988), Sherris (1994), Shiu (1987, 1988,1990), and Tilley (1992, 1993), amongst others. Ac-ademics and practitioners in the finance and invest-ment area have dominated the theoretical devel-opments and the practical implementation of thetechniques to financial risk management. In someways the actuarial profession has been lucky since itcan draw on a wealth of research ideas in this area.The formation of the AFIR (Actuarial Approach forFinancial Risks) Section of the International ActuarialAssociation has also allowed many of the results ofthis research in the financial economics literature toreach a wider actuarial audience.

Many actuarial problems can use the recent devel-opments in term structure modeling. These rangeacross all areas of actuarial practice: life insurance,general insurance, and superannuation. The stochas-tic approach to term structure modeling is readily ap-plied to the interest rate risk management andvaluation of any set of fixed or interest rate dependentcash flows.• Annuity Products. Term certain annuities and the

expected cash flows under life annuity products canbe treated as fixed cash flows. Guarantees on thepayments of life annuity products represent a formof interest rate option. These products require a sto-chastic approach for risk management and valuation.

2 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2

• Insurance Products with Interest-Rate-SensitiveLapses. Many insurance products have cash flowsthat are dependent on future interest rates not onlybecause their cash flows depend on interest ratesbut also because the decrement rates that deter-mine the timing of payment of the cash flows de-pend on interest rates. For example, thewithdrawals on an annual premium life productcould increase when interest rates rise.

• Property and Casualty Insurance Company Out-standing Claims Reserves. The expected claimspayments for general insurance companies are oftentreated as risk-free cash flows and are discounted topresent values using risk free rates. To the extentthat the amount or timing of these liability cashflows depend on interest rates, a stochastic ap-proach can incorporate interest rate uncertainty inthe valuation.

• Accrued Liabilities of Defined-Benefit PensionFunds. These liabilities can be treated as expectedfuture cash flows based on service to date and cur-rent salary. These cash flows can be valued and ap-propriate investments selected for them using themodels in this paper.In many actuarial problems the cash flows depend

on other variables apart from interest rates. For ex-ample, indexed annuities depend on future inflation,and claims payments for long-tail liability business forproperty and casualty insurance companies also de-pend on economic inflation. In these cases the ap-proach in this paper needs to be extended to realinterest rates. This is not covered in this paper, butthere are examples of extensions to the approach cov-ered in this paper that allow for inflation; see, for ex-ample, Brown and Schaefer (1994) and Pennachi(1991).

The techniques in this paper have largely been de-veloped for asset applications. These range from bondvaluation to complex interest-rate-related securitiessuch as structured notes.• Bond Valuation. Fixed-interest securities can be

valued by using a stochastic interest rate model.The parameters of these stochastic models aresometimes determined by ‘‘fitting’’ them to tradedbond prices for government securities. These are re-ferred to as ‘‘arbitrage-free’’ models.

• Interest Rate Options. A stochastic model is re-quired to value nonfixed cash flows. The basictraded security that is dependent on future interestrates is the interest rate option. These instrumentsinclude bond options, options on interest rate fu-tures, and swaptions.

• Mortgage-Backed Securities. These securities havecash flows that depend on the history of interestrates. This is because the cash flows are influencedby the level of prepayments on the underlying mort-gages. The amount of these prepayments dependson the history of interest rates. Where the cashflows depend on the history of interest rates, thecash flows are said to be path-dependent. Prepay-ments can be modeled in a similar manner to dec-rement rates in insurance products. As an example,Ang (1994) develops and fits prepayment models us-ing Australian data.

• Structured Notes. A recent example of innovationin financial markets is the structured note. This in-strument is basically a package of a straight bond ornote along with a number of embedded interest rateoptions. The embedded interest rate options usuallytake the form of caps, floors, or collars.

2. TRADITIONAL APPROACH TOINTEREST RATE RISK

The traditional approach to interest rate risk manage-ment and valuation developed and often used by ac-tuaries assumes a single interest rate. This rate is usedto value cash flows and to determine the sensitivityof the value of the cash flow to interest rate changes.This approach is based on the assumption that theyield curve is flat and interest rates change in a par-allel and deterministic manner. Historical data indi-cate that yield curves can take a variety of shapesincluding upward sloping, downward sloping (in-verse), and even humped. This is demonstrated in thehistorical data for Australian Government Securityyields from January 1972 to October 1994 presentedin Figure 1. It can also be observed from this data thatyield curve changes are not parallel. It should also beclear that the process driving interest rate changes isnot deterministic.

2.1 Yield Curves

It is common practice to fit yield curves to marketyields to maturity to describe the relationship be-tween yields and term to maturity. The actual yieldsto maturity are not smooth because of coupon, li-quidity, and taxation effects. These effects are re-moved by fitting spot or zero-coupon yields that areconsistent with the yields to maturity. It is best to fita curve to the discount function, which is the valueof the zero-coupon bonds for differing maturities,rather than the spot yields. The discount function

INTEREST RATE RISK MANAGEMENT 3

FIGURE 1GRAPH OF AUSTRALIAN GOVERNMENT BOND YIELDS TO MATURITY

curve should be smooth. A number of techniques canbe used to fit these yields. These techniques are sim-ilar to those used by actuaries in fitting smooth curvesto mortality rates. Two common techniques are theuse of splines and of mathematical formulas:• Splines. This involves fitting a series of polynomial

curves over the data points, with the points wherethe curves join arbitrarily chosen. A specified de-gree of continuity is imposed on the derivatives. Theabscissa values that define the segments are calledknots. Cubic splines, the most common approach,use a series of quadratics fitted to the data by leastsquares; see Steeley (1991) for an example.

• Formulas. Parameters in a formula may be fitted tothe series of interest rates by using methods suchas least squares and maximum likelihood. Formulassuch as that in Nelson and Siegel (1987) have beenderived with a theoretical justification.Yield curves can also be fitted directly to spot or

forward rates as well as yields to maturity. The aimof yield curve fitting is often to derive a zero-couponyield curve. Zero-coupon yields can be used to valueany series of cash flows by discounting each cashflow using the appropriate zero-coupon yield,whereas yields to maturity and forward rates do nothave such a simple application for valuation of ar-bitrary fixed cash flows. Yields to maturity areneeded by traders in trying to predict interest ratemovements and bond price variations. Forward in-terest rates are often more informative for interestrate futures and options traders. Forward rates arethe basis of the most recent methods developed byfinancial economists for constructing stochastic

interest rate pricing models as in Heath, Jarrow andMorton (1990a, 1990b, 1992).

The information contained in the set of currentyields to maturity, spot rates, and forward rates isequivalent, because they are related by definition.The discrete time relationships between the forwardrates, spot rates and bond prices are:

t2TP(t, T) 5 [1 1 Y(t, T)] for t , T

P(t, T)f(t, T, T 1 1) 5 2 1

P(t, T 1 1)

P (t, T ) 5

1

[1 1 f (t, t, t 1 1)][1 1 f (t, t 1 1, t 1 2)]...[1 1 f (t, T 2 1, T )]

where

P(t, T) 5 the price at time t of a zero-couponbond maturing at time T with yieldto maturity Y(t, T) and maturity(T2t) years

f(t, m, n) 5 the forward rate of interest impliedin the zero-coupon bond yieldcurve at time t applying from timem to time n years, t,m,n

f(t, T, T11) is the one-period forward rate attime t, applying from time T totime T11, t,T.

2.2 Arbitrage-Free Conditions

If arbitrage opportunities are not to exist in a deter-ministic economy, then P(t, T)5P(t, s)P(s, T) fort,s,T. This is referred to as the principle of no-ar-bitrate in financial economics. If this condition doesnot hold for all s, then it is possible to generate adollar at time T at a different cost to the zero-couponbond price at time t by rolling over short-term bondsat the deterministic forward rates.

This result implies that f(s, T, T11)5 f(t, T, T11)for all s, t≤T. The forward rate applying to a particulartime interval (but not time to maturity) must be con-stant through time in a deterministic setting if arbi-trage opportunities are not to exist. The yield curvedoes not have to be flat, but the forward rates for anyfixed time interval must be constant through time.Bond prices will be known but need not be constant,but they must obey the no-arbitrage condition if mar-kets are to be in equilibrium. This constant propertyof the forward rate curve in a deterministic model is


one reason why it is often easier to model forwardrates in a stochastic model.

2.3 Expectations Hypothesis

Various theories have been developed to explain therelationships between interest rates of varying termswhen interest rates are assumed not known with cer-tainty. These are based on assumptions about expec-tations of future interest rates in the economy. Oneversion of the expectations hypothesis assumes thatthe expected one-period interest rate is the same forinvestments of all maturities. There is also a versionof the expectations hypothesis in which expected fu-ture one-period interest rates are equal to the currentimplied forward interest rates. In an economy with nouncertainty, this result will hold. Under other theoriessuch as the liquidity premium hypothesis and the pre-ferred habitat, the expectations hypothesis is modi-fied to include a risk premium that reflects liquidityor maturity preferences in the economy.

The various forms of the expectations hypothesisare discussed in Ingersoll (1987, Chapter 18). Underthe local expectations hypothesis, the expected hold-ing period return on bonds of all maturities over thenext time period are assumed to be equal. Thus underthe local expectations hypothesis the expected one-period return on a bond is equal to the prevailing oneperiod spot rate so that:

P(t, T) 5 E[R(t)R(t 1 1) . . . R(T 2 1)]21

R(t) 5 1 1 the one-period spot interest rate at timet, and if continuous compounding is assumed

T

P(t, T) 5 E exp 2* r(s)ds@ ~ !#t

where r(s) is the continuous compounding instanta-neous spot interest rate.

Hence under the local expectations hypothesis, allbond prices can be derived from the one-period spotrate in discrete time or the instantaneous spot rate incontinuous time. This rate is referred to as the ‘‘short’’rate. Similar valuation formulas are derived from sto-chastic term structure models under the assumptionsof no arbitrage between bonds of different maturities.These models use a ‘‘risk-adjusted’’ short rate in thevaluation formula for bonds. These results will be re-viewed under the discussion of stochastic interest ratemodels.

2.4 Interest Rate Risk Management

Interest rate risk management in the traditional de-terministic approach aims to manage variations in as-set and liability values on the assumption that interestrates undergo small deterministic changes. The orig-inal approach developed in the actuarial literature byRedington (1952) effectively assumes that all cashflows are fixed and not dependent on interest rates.This excludes any asset or liability for which the tim-ing or amount of the cash flows depends on the levelof interest rates. It is also assumed that all cash flowsare valued using the same interest rate. This meansthat the yield curve is assumed to be flat across allmaturities. Finally, the yield to maturity curve is as-sumed to move in a parallel fashion. This approachhas been expanded to non-flat yield curves (Fisherand Weil, 1971), to non-parallel shifts in the yieldcurve (Shiu, 1990), and to allow for stochastic interestrates and interest-rate-dependent cash flows (Boyle,1978). It is important to note that a model based onparallel yield curve shifts will not be arbitrage-free.Given that bond prices are convex in their yields, itis possible to construct a portfolio of bonds requiringzero initial investment that will provide a non-nega-tive return if all yields change by the same amount.

Traditional interest rate risk management uses theduration and convexity of cash flows as the mainmeasures of risk.

The use of duration as a measure of the effect ofinterest rate changes on the value of fixed-interest se-curities originated in the concepts of the average timeto receipt of the cash flows of the security, which wasreferred to as ‘‘duration’’ by Macaulay (1938), ‘‘aver-age period’’ by Hicks (1939), ‘‘weighted average time pe-riod’’ by Samuelson (1945), and the ‘‘mean term of thevalue of the cash flows’’ by Redington (1952). The textby McCutcheon and Scott (1986, page 232) uses theterminology ‘‘discounted mean term.’’ Nearly all partic-ipants in financial markets use the term ‘‘duration.’’

Interest rate risk is traditionally measured by thederivative of the security value with respect to theinterest rate. This is based on a first-order approxi-mation to the price change using a Taylor series ex-pansion of the price for a small change in yield aroundthe current price. Such a definition is readily appliedto interest rate derivative securities for which the du-ration is not well defined. Minus the first derivative ofthe security value with respect to the interest rate asa proportion of the current value is referred to as themodified or effective duration of the cash flows.


If the Taylor series approximation is taken to sec-ond derivatives, then the second-order term that re-sults is referred to as the convexity. The price at yieldto maturity i1, P(i1), can be expressed as a Taylor se-ries in terms of the price evaluated at the currentyield to maturity i0, P(i0), and the price derivatives asfollows:

(i 2 i )1 0P(i ) 5 P(i ) 1 P ' (i )1 0 01!2(i 2 i )1 01 P" (i ) 1 z z z02!

If the price at i0 is deducted from both sides and bothsides are divided by this price, then we have

P(i ) 2 P(i ) (i 2 i ) P ' (i )i 0 1 0 05P(i ) 1! P(i )0 0

2(i 2 i ) P" (i )1 0 01 1 z z z2! P(i )0

This can be used to develop a second-order approxi-mation

2P(i ) 2 P(i ) (i 2 i )i 0 1 0' 2(i 2 i ) MD 1 C1 0P(i ) 20

where MD is the modified duration of the security andconvexity is defined as

P"(i )0C 5P(i )0

Another measure of interest rate risk that is oftenused in practice is referred to as M2. This terminologyappears to have originated in the paper by Fong andVasicek (1984). This measure was called the ‘‘spread’’of the values of a set of cash flows about the meanterm by Redington (1952). The M2 of a set of assetcash flows is equivalent to a discounted variance inthe same sense that the duration of a set of cash flowsis the discounted mean term. In fact, the M2 of a setof asset cash flows can be described as a weightedvariance of time to receipt of the cash flows aroundthe duration of the cash flows, the weights being thesame as used to calculate the duration.

Modern practice allows for nonparallel yield curveshifts in the deterministic model using the conceptsof multivariate duration and convexity analyzed byReitano (1991a, 1991b, 1992a, 1992b). These multi-variate duration measures are similar to the key ratedurations of Ho (1992). The term structure is dividedinto separate maturity ranges sometimes referred toas ‘‘buckets.’’ The sensitivity of values is determined

separately for changes in the yield curve in eachbucket and all other buckets remain unchanged.These approaches give no recognition to the correla-tion structure of yield curve changes across the ma-turity spectrum. To do this requires a stochasticmodel. These deterministic multivariate approachesare not arbitrage-free as Shiu points out in his dis-cussion of Reitano (1992b).

The concepts underlying immunization in a deter-ministic model extend to a stochastic multifactor in-terest rate model. An example is Jarrow and Turnbull(1994).

2.5 Matching and Optimisation

Portfolios of bonds that match a specific set of liabil-ities can be selected by using linear programming(Shiu 1988; Kocherlakota, Rosenbloom and Shiu1988, 1990). The matched portfolio of bonds is cho-sen to minimize the cost of the bond portfolio suchthat the asset cash flows can always meet the liabilitycash flows. A carry-forward of positive cash balancesfrom an excess of asset cash flows over liability cashflows can be included. This methodology can also beapplied to select assets with a minimum cost thathave a duration that matches the liability cash flows.

The portfolio of bonds can also be selected by min-imizing the risk (M2) of the difference between theasset and liability cash flows subject to the constraintthat the present value of the bonds equals the presentvalue of the liabilities and the duration of the bondsequals the duration of the liabilities. This approachrequires the spread of asset cash flows to exceed thespread of the liability cash flows and no options in thecash flows (Shiu 1990). Once again this approach ex-tends naturally to a multifactor stochastic interestrate model. In the stochastic case the sensitivities tothe random factors are matched for the assets andliabilities.

Introducing risk requires an approach similar to theportfolio selection problem for any investor. The re-lation between matching of cash flows and portfolioselection models is discussed in Sherris (1992). Port-folio selection models often use quadratic program-ming to select assets that minimize the variance ofsurplus, defined as the accumulated excess of assetcash flows over liability cash flows, for a given ex-pected value of surplus on a terminal date. This ap-proach to matching and portfolio selection forinterest-sensitive cash flows uses a stochastic modelfor interest rates.


3. STOCHASTIC TERM STRUCTUREMODELS

Most actuaries would be unfamiliar with models, suchas Ho and Lee; Hull and White; Black, Derman, andToy (BDT); and Heath, Jarrow, and Morton (HJM),that are commonly used by brokers and investmentbanks as the basis for the interest rate risk manage-ment of their bond and interest rate option portfolios.These models use a stochastic approach to termstructure modeling.

An early example of the stochastic approach to in-terest rate risk management is found in the prize-win-ning paper by Boyle (1978). The stochastic approachhas seen rapid development in recent years probablybecause of the increase in volatility of interest rates,but a major factor would be the significant increasein desktop computing power. The techniques used arethe same as those underlying option-pricing. Termstructure models and option-pricing models are ex-amples of a more general contingent claims pricingtechnique.

Early models used an equilibrium asset pricing ap-proach to determining bond prices. More recently, thearbitrage-free approach has found more favor. Themore recent approach to term structure models isbased on martingale methods. For further references,readers are directed to the following recently pub-lished books. Jarrow (1996) covers interest rate mod-els in the HJM framework mostly in a discrete timesetting. Jarrow and Turnbull (1996) and Ritchken(1996) are recent textbooks that contain an excellentcoverage of the modern approach to interest rate de-rivative valuation. For a comprehensive coverage ofthe theoretical tools and the various models used instochastic term structure models, readers are referredto Rebonato (1996).

3.1 General Contingent Claims Pricing

In contingent claims pricing stochastic models wereinitially developed in continuous time by deriving astochastic partial differential equation to representchanges in the random factors, or state variables,driving the model, or in discrete time by stochasticdifference equations. In continuous time the factor isassumed to follow a diffusion process. For a factor Xthe form of such a process for the instantaneouschange in X is:

dX 5 µ(X,t)dt 1 s(X,t)dZ

where dZ is the increment in a standard Wiener pro-cess.

The term µ(X,t) is called the drift, and s(X,t) is thediffusion term, which is also often referred to as thevolatility. The drift represents the expected change inthe series and the diffusion provides the randomchange.

In discrete time the process is usually approxi-mated using the first-order Euler approximation:

=X(t 1 Dt) 2 X(t) 5 µ(X,t)Dt 1 s(X,t) Dtε t

where εt;N(0,1). Higher-order approximation methodshave been developed (Kloeden and Platen 1992).Some commonly used forms of these processes as-sume that the difference in the variable X is station-ary. Such a process is said to have a ‘‘unit root’’ andbe difference stationary. Other commonly used formsof these processes assume that the variable X itself isstationary as in the ‘‘mean-reverting’’ models used forinterest rates.

The result required from contingent claims pricingin term structure modeling is the partial differentialequation for security values assuming that marketsare complete and arbitrage-free. If V(X, t) denotes thevalue of a security that is a function of X and t, thenthe partial differential equation (pde) that must besatisfied by V is:

V [µ(X,t) 2 l (X,t)s(X,t)]X

121 V s (X,t) 1 V 5 r(t)VXX t2

where subscripts indicate partial derivatives andl(X, t) is referred to as the market price of risk. Theresult is developed in standard texts such as Duffie(1992), Hull (1993) and Shimko (1992). This equa-tion is also derived in Appendix A. Note that unless Xis a traded asset, then the market price of risk cannotbe eliminated. This is especially important for pricinginterest-rate-dependent claims because interest ratesthemselves are not traded assets.

The terms in this pde have a natural interpretation.The left-hand side gives the (risk-adjusted) expectedchange in the security value. The first term is the sen-sitivity of the security’s value to small changes in therandom factor X times the (risk-adjusted) expectedchange in X. The next term is a second-order termsimilar to the convexity term in the deterministicmodel. The third term is the change in the value thatresults from small changes in time. The right-handside is equal to the risk-free rate times the value ofthe security. Hence, the pde simply states that in a


risk-adjusted arbitrage-free stochastic pricing model,the expected return on the security is the instanta-neous risk-free rate.

The solution to the pde depends on the particularsecurity to be valued, since this will determine therelevant boundary conditions. The absence of arbi-trage can be shown to be (essentially) equivalent tovalues being discounted expected values with respectto a risk-neutral probability distribution referred to asan equivalent martingale measure. Shiu (1993, Sec-tion 4) provides a concise discussion of this result forthe finite discrete time case. The continuous timecase is very complex and is covered in Delbaen andSchachermayer (1994). Under these assumptions thevalue of all securities expressed in units of a moneymarket account earning the instantaneous risk freerate is a martingale. The solution to the pde, where itcan be determined analytically, can be shown to takethe form of a (risk adjusted) discounted expectedvalue of the cash flows with respect to the equivalentmartingale measure (Duffie, 1992).

3.2 Interest-Rate-Dependent SecurityPricing

The earliest approach to term structure modeling wasto take the instantaneous spot rate, referred to as the‘‘short rate’’ or ‘‘the’’ spot rate, as the random factorin a continuous time model. This short rate is usuallydefined as the continuous time equivalent of the one-period spot rate in a discrete model. Assume that rfollows the diffusion

dr 5 µ(r,t)dt 1 s(r,t)dZ

The value of an interest-rate-dependent securityP5P(r, t) that is assumed to be a function of the singlefactor r is given by the pde:

12P s(r,t) 1 P µ* 1 P 5 rPrr r t2

The terms in this pde have an intuitive interpretation.This is clearer if it is rewritten in terms of the secu-rity’s effective duration and convexity as follows:

12PCs(r,t) 1 2P(MD)µ* 1 P 5 rPt2

The left-hand side is the (risk-adjusted) expected re-turn on the security that results from an applicationof Ito’s lemma and an arbitrage-free assumption. Thefirst term is a function of the security’s convexity andthe volatility of the short interest rate. The secondterm is the expected change in the security value for

a small change in the ‘‘short’’ interest rate times theexpected change in the short rate, and the third termcaptures the change in value with time. The right-hand side is the risk-free return on the security.Hence the pde states that, on the assumptions that, ifmarkets are complete and arbitrage-free, the shortrate follows a diffusion and the security’s value is afunction of the short rate, then the (risk-adjusted) in-stantaneous return on the bond is equal to the risk-free rate.

The term structure is determined by the zero-cou-pon bond prices since these give the spot rates forvarying maturities. The pde can be solved for zero-coupon bond prices by specifying the process for theshort rate and the boundary conditions for the zero-coupon bond. In the case of the zero-coupon bond,the boundary condition is that the value P(r, T)51 onthe maturity date (T) of the bond. For specific formsof the diffusion for r, an analytical solution for zero-coupon bond prices exists and takes the form:

B (t,T)rP(r,t) 5 A(t,T)e

where A(t, T) and B(t, T) depend only on time andthe maturity of the bond and r is the current shortrate. The exact form of the parameters A and B fordifferent models can be found in the original papersor in Duffie (1992), Hull (1993), and Hull and White(1994).

This zero-coupon bond formula applies for theaffine class of parameter specifications with

µ(r, t) 5 a 1 br

and

=s(r, t) 5 c 1 dr

in the diffusion for the short rate. In this case, A(t,T)and B(t,T) depend only on the time to maturity (T2t).More details are provided in Duffie (1992, pages 133–136), Rebonato (1996, Chapter 15), and Wilmott,Dewynne, and Howison (1993, Chapter 14). Vetzal(1994) provides a comprehensive survey of continu-ous time models. Specific cases of this form of thediffusion are:

d50 gives the Vasicek (1977) modelwhere dr5(a1br)dt1sdZ. In thiscase the volatility s(r, t)5 5s is=cconstant and deterministic and thedrift term allows for mean reversionof the short rate.


c50 gives the Cox, Ingersoll, and Ross(1985) model dr5(a1br)dt1s dZ=rwith s(r, t)5 5s . This is the= =dr rsquare root model with stochasticvolatility depending on r.

b50 and d50 give a Gaussian-path-independentmodel (see Jamshidian, 1991a)since the drift and volatility do notdepend on the short rate r.

All these cases assume no time dependence in theshort-rate parameters. If the volatility is a determin-istic function, then the short rate will have a Gaussiandistribution.

The parameters can be assumed to be time depen-dent so that

µ(r,t) 5 a(t) 1 b(t)r

and

=s(r,t) 5 c(t) 1 d(t)r.

One example is b(t)50, c(t)5c and d(t)50, whichgives the continuous time equivalent of the Ho andLee (1986) model dr5a(t)dt1sdZ. This model has adeterministic and time varying drift and constant vol-atility for the short rate. Another example is d(t)50,which gives the general Gaussian model with time-dependent parameters as analysed by Jamshidian(1990), so that dr5(a(t)1b(t)r)dt1s(t)dZ. In thismodel the forward rate volatilities decline exponen-tially as the forward date increases.

Other cases often do not permit an analytical so-lution, so that numerical techniques are required. Thecontinuous time equivalent of the BDT (1990) modelgiven by

s(t)d log(r) 5 a(t) 1 log(r) dt 1 s(t)dZ@ #s(t)

is an example. In this case the short rate is lognormal.If it is assumed that the volatility s(t) is constant,then s(t)50 and the model becomes a lognormalversion of Ho and Lee.

In the case of lognormal models it is better to modelthe nominal or effective interest rate rather than thecontinuously compounding short rate with a diffusion.The reason for this is that it can be shown that ex-pected accumulation factors over fixed future time in-tervals in a lognormal model for the continuouslycompounded rate are unbounded, as shown by Hoganand Wientraub (1993). This problem does not occurif the interest rate modeled is a nominal rate with a

compounding frequency such as monthly, half-yearlyor annually. Such a model in a discrete implementa-tion is found in Sandmann and Sondermann (1993)and in Sherris (1994).

As the number of parameters in the short-rate pro-cess is increased, either by adding additional param-eters or by making them time-dependent, more inputparameters can be used to specify the models.

The solution to the pde for the value of a zero-cou-pon bond can in general be written in the form of anexpectation. This expectation takes the form:

T

P(r,t,T) 5 E* exp 2* r(s)ds$ %t

where the * denotes an expectation with the driftterm for the diffusion for r adjusted from µ(r, t) toµ(r, t)2l(r, t)s(r, t). This adjusted diffusion is re-ferred to as an equivalent martingale measure. Deri-vation of this result can be found in many papers andtexts. It is the Feynman-Kac solution to the pde. Theinterested reader is referred to Duffie (1992, p. 86).

This result extends to other interest-rate-dependentcash flows such as those for interest rate options. Thevalue of a security with interest-rate-dependent payoffof C(r, T) at time T is given by

T

C(r,t,T) 5 E*[exp 2* r(s)ds C(r,T)]$ %t

It is not correct to simply discount the expected pay-offs from a security at the prevailing spot rate. Therisk-adjusted rate must be used where the expectedgrowth rate in each variable is reduced by the productof the market price of its risk and its volatility. Theform of this expectation is identical to that derivedunder the local expectations hypothesis. The local ex-pectations hypothesis assumes the role of risk-neutralpricing, and the results imply we can obtain a secu-rity’s value by taking an expected value and discount-ing at the risk-adjusted short rate. In more complexcases simulation can be used to obtain this expectedvalue.

If the value of a money market account at time taccumulating at the instantaneous risk-free rate fromtime 0 is denoted by M(t), then

t

M(t) 5 exp * r(s)ds .$ %0

The value of the interest-rate-contingent claim canthen be rewritten as


C(r,t,T) C(r,T)5 E* .@ #M(t) M(T)

So that the expected value of the payoff on the se-curity in units of the money market account is equalto the current value of the security divided by thecurrent value of the money market account. Thismartingale result is fundamental to arbitrage-freepricing and holds if markets are arbitrage-free. If mar-kets are complete, then the adjusted diffusion, re-ferred to as the equivalent martingale measure, isunique.

3.3 Lattice Models

These continuous time models are implemented in adiscrete form in practice by using a lattice of values.Lattices actually arise from finite difference approxi-mations to the fundamental pde for pricing securities.Such discrete models are in fact equivalent to thecontinuous time stochastic approaches. The latticeapproach to the numerical solution to the pde is toproject the values of the factor X on lattices. In thisway the diffusion term in the stochastic equations forthe factors can be represented by the spread of thefactor values on the lattice at each point. Branches ofa tree are not restricted to being binomial: Hull andWhite (1993a) use trinomial trees and Bookstaber, Ja-cob, and Langsam (1986) describe the relationship bi-nomial trees must have with multinomial trees forno-arbitrage relationships to hold. Similarly, anystrategy employed on a discrete lattice can also beemployed in a continuous time framework with theright restrictions and the corresponding algorithm.

A trinomial lattice can be obtained by substitutingappropriate finite difference approximations to thepartial differentials in this pde. Details are providedin Appendix C. The result is a lattice as follows:

P(i+1)a*

P(i)b*

P(i)

c*

time j

P(i–1)

time j+1

where a*, b*, and c* are referred to as risk-neutralprobabilities.

A binomial lattice can also be derived, but this is alittle more tricky, requiring a change of variable in thepde. Once again a full derivation is presented in Ap-pendix C. The binomial lattice takes the form

P(i+1)

P(i)

time j

P(i–1)

time j+1

1–

where the p is the risk-neutral probability of the upjump.

The lattice that results from the discretization ofthe pde is in terms of the bond price, P. In discreteinterest rate models, it is usually the one-period spotinterest rate, or ‘‘short’’ rate, that is modeled on alattice. In fact, it is common to model interest ratesrather than bond prices, since these are often consid-ered to be the natural state variable in an interest-rate-dependent security valuation model. However,since it is assumed that the bond price is a functionof the interest rate, there is a one-to-one correspon-dence between bond prices and this interest rate. Itcould be argued that since bonds trade and prices areobservable, at least for traded maturities, that bondprices are a more appropriate state variable to use.

3.4 (Partial) Equilibrium Models

Various approaches have been taken by researchersin developing term structure models that involvemaking different assumptions about the market priceof risk l(r, t). Equilibrium approaches involve makingsome assumptions about the risk preferences of in-vestors to specify the form for the market price ofrisk. Taking the interest rate as the state variable re-sults in preference-dependent bond prices becauserates of interest are nontraded assets. In contrast, ar-bitrage-free models fit the market price of risk to cur-rent bond prices directly.

Equilibrium models rely on assumptions about thestructure of capital markets and the preferences ofinvestors. If investors’ utility functions are logarith-mic, then the market price of risk is constant (Dothan1978, pp. 61–62). Vasicek (1977) derives bond-pricing


solutions for a constant price of risk in his model.Longstaff (1990), Courtadon (1982), and Dothan(1978) are examples of equilibrium approaches thatassume log utility and specify the market price of riskas a constant. Rendelman and Bartter (1980), in oneof the first discrete models for bond prices, also as-sume logarithmic utility. Cox, Ingersoll, and Ross(1985), in a major contribution to equilibrium assetpricing and term structure modeling, assume the mar-ket price of risk is a function of the interest rate andassume constant relative risk aversion utility func-tions. They also show the importance of consistencybetween assumptions for the utility function and thefunctional form of the market price of risk.

Equilibrium models can incorporate different sto-chastic processes for the instantaneous spot interestrate. These may have drifts that are mean-revertingor linear, and the diffusion may be constant or dependon the level of the factor. With constant prices of risk,these models are able to yield closed-form solutionsthat is, analytical bond pricing formulas for the termstructure. Such formulas allow direct calculation ofbond prices, and prices of bond options, without theuse of a numerical technique.

Multiple factors can be incorporated in equilibriummodels. Brennan and Schwartz (1979) specify sepa-rate correlated processes for the long and short rates.Using arbitrage arguments, Brennan and Schwartz areable to eliminate the market price of risk term for thelong rate, so that in their two-factor model there isonly one market price of risk, which is assumed to beconstant. A similar model is Schaefer and Schwartz(1984), which instead uses the short rate and the dif-ference between the long rate and short rate. Longs-taff and Schwartz (1992) develop a two-factor modelwith stochastic volatility as one of the factors.

Multiple factor models are important for actuarieswho want to value products with multiple random fac-tors affecting their value (for example, mortality, in-terest rates, withdrawals, and so on). One examplewith constant prices of risk for each decrement factorfor pricing life insurance products involving mortalityand withdrawal risk is Manistre (1990).

Equilibrium models are said to endogenize the termstructure. The price of risk must be estimated frombond prices. Rarely will an equilibrium approach gen-erate zero-coupon bond prices that are consistent withthe market prices of all zero-coupon bonds, whateverthe specification of the market price of risk. The fit willusually be only approximate. In contrast, the arbitrage-free approach produces model prices that exactly fitthe observed prices of zero-coupon bonds.

Arbitrage-free approaches have gained favor be-cause in pricing derivative securities on term struc-ture instruments, they ensure that the model at leastprices the underlying term structure correctly. AsHull (1993) notes, traders have little confidence inmodels that generate bond prices dissimilar to the ob-served bond prices in the market. Arbitrage-free ap-proaches use algorithms that choose the parametersof the model such that the prices of the zero-couponbonds that are derived using the algorithm are equalto the market prices of the bonds on the valuationdate.

3.5 Arbitrage-Free Models

Arbitrage-free models are in many ways similar to theequilibrium approaches. In equilibrium approachesthe market price of risk is specified. In an arbitrage-free approach the market price of risk is taken to bethat observed in the market. Like equilibrium models,analytical solutions are available for some models,and both approaches can be implemented by discretemethods or by simulation. Models can have as manyparameters as necessary to fit the observed termstructure and volatilities. Heath, Jarrow, and Morton(1992) have developed a multifactor arbitrage-freemodel that will fit any term and volatility structure.However, despite their current popularity, arbitrage-free approaches are not the panacea for term struc-ture modeling for the following reasons:• Arbitrage-free models involve estimating the param-

eters of prespecified models. They tell us nothingabout the adequacy of the model itself. For example,most Gaussian arbitrage-free approaches permitnegative interest rates. As Marsh (1994) notes, thisis an important consideration in the design phase ofany model, be it equilibrium or arbitrage-free.

• Arbitrage-free models price according to the termstructure. This does not allow us to identify arbi-trage opportunities on the yield curve itself likeyield curve shifts. If the present yield curve has ar-bitrage opportunities, then our model also has thosesame inconsistent arbitrage opportunities. Using anequilibrium model, if the model is right, allows usto trade bonds at a profit if the market prices arenot equal to the model prices. Such arbitrage is notpossible with an arbitrage-free approach.

• The choice of parameters and their subsequent be-havior often restrict the dynamics of the change inthe yield curve through time. For example, the con-stant volatility in the Ho and Lee (1986) model isthe same for both the spot and forward rates, and


the Pederson, Shiu, and Thorlacius (1989) modelassumes that volatility declines with time. These be-haviors of the dynamics of the yield curve may notfit the actual dynamics of the yield curve, and thechoice of model itself is then inappropriate.

• Models are usually built around key rates being ar-bitrage-free (for example, cash, 30-day, 90-day, 1-year, 2-year, 5-year, 10-year yields), so that the in-terim rates may well have arbitrage opportunities.These approaches may be inconsistent in the senseof Bookstaber, Jacob, and Langsam (1986), sincethe full span of rates is not being considered andthe prices obtained on such a model could be un-reliable.

• Most arbitrage-free approaches fit the model param-eters to the zero-coupon bond yield curve. There isno theoretical reason to use the spot rate yieldcurve. The parameters of the model could just aseasily be fitted to other traded instruments. Fittingto one set of securities also does not allow us toidentify arbitrage opportunities with other securi-ties. Some practitioners and researchers use a rangeof instruments including caps and floors to fit themodel parameters. The aim of fitting the model isto determine the state-contingent security prices(Arrow-Debreu securities), since these can be usedto price any security using the model. For this rea-son the model is likely to be more reliable if it usesactively traded interest rate options to fit param-eters, since these are closer in payoff to the state-contingent securities than the zero coupon bonds.

3.6 Black-Scholes Interest Rate Option-Pricing

The pde for equilibrium and arbitrage-free models canbe solved for bond and interest rate option prices. Un-der the assumption that the short rate is Gaussian,bond prices are lognormal and analytical formulas canbe derived. Under this assumption zero-coupon bondand interest rate options are valued using a formulasimilar to the Black-Scholes equity option formulamodified to allow for stochastic interest rates. Thevolatility assumption for equities is replaced with aterm structure of volatilities for forward interest rates.These analytical formulas can also allow for mean re-version in interest rates.

The Black-Scholes formula can be applied to op-tions on zero-coupon bonds. However, the application

to coupon-paying bonds is not as straightforward. Ina single-factor model a form of the Black-Scholesformula can be used for options on coupon-payingbonds. Jamshidian (1991a) demonstrates the tech-nique required and proves the results. Hull and White(1993b) develop results for the case in which short-rate volatilities are deterministic.

The other problem with Black-Scholes is that it cannot be used directly for American options (with earlyexercise). Since many traded options are Americanstyle, this is a significant practical problem. Discretenumerical methods of valuing interest rate optionscan readily allow for early exercise. The Black-Scho-les formula does not allow for path-dependent optionssuch as options on the average of interest rates (Asianoptions).

The Gaussian assumption for interest rates is oftennot satisfactory, since negative spot and forward in-terest rates are possible in such models. More realisticmodels can be used, but this requires the use of dis-crete numerical approaches to determining values.With developments in computational techniques andsignificant increases in computing speed, this is not asignificant problem.

3.7 Discrete Time Methods

The discrete implementation of equilibrium and ar-bitrage-free models is usually done on lattices. Inter-est rates are usually projected on recombining treesfor computational reasons. Tree modeling has its or-igins in the classic stock option-pricing approach ofCox, Ross, and Rubinstein (1979). Jarrow (1996) pro-vides a comprehensive development of discrete timeterm structure modeling and valuation of interest ratederivatives.

Arbitrage-free models require the fitting of themodel parameters to the yield curve before they canbe used for valuation. Valuation of interest-rate-de-pendent securities is carried out by using arbitrage-free paths of the short interest rate. This is done usingforward induction. Valuation (pricing) is then doneusing backward induction. Since the principles arethe same for both equilibrium and arbitrage-free mod-els, this paper concentrates on arbitrage-free models.

In a discrete binomial model the paths of interestrates are determined by three parameters which arefitted to the market data to ensure that the model isarbitrage-free. These are the size of the up jump, thesize of the down jump and the probabilities of the up


jump for each time interval and next period interestrate:

r0

ru

rd

1–

The value of a bond is calculated by backward induc-tion or recursion through the tree. The algorithm canmost generally be specified in terms of Arrow-Debreuprices, also referred to as state-contingent prices,G(n, i, m, j), which are the prices of a security at timen and state i, which pays $1 at time m, m.n, andstate j and $0 elsewhere. The state of the model de-termines the value of the interest rate.

Arrow-Debreu securities are the ‘‘primitive’’ state-contingent securities in the model because the payoffstructure for any state-contingent security can be rep-licated by a portfolio of appropriate Arrow-Debreu se-curities. The sum of the values of the appropriate setof Arrow-Debreu securities is the value of the state-contingent security. These Arrow-Debreu securitiesare the discrete equivalent of Green’s function, whichis the fundamental solution to the pde of any securityvalue (Jamshidian 1991b).

The general backward induction valuation algo-rithm for state contingent security prices can be writ-ten as:

G(n,i,m,j) 5

p(n, i)G(n 1 1, i 1 1, m, j) 1 (1 2 p (n, i))G(n 1 1, i 2 1, m, j)

1 1 r(n, i)

where p(n,i) is the probability of going to state i11from state i at time n, and r(n, i) is the one-periodprevailing interest rate at state i at time n. Note thatthe equation above relates the value of the Arrow-De-breu securities at the two nodes at time n11 to thevalue of the Arrow-Debreu security at the node attime n. The value at time 0 of a security paying C(m,j)51 at time m for all states j, by definition the mmaturity zero-coupon bond, can be written as:

ZCB(0, m) 5 G(0, 0, m, j)C(m, j)Σall states j

Note that the movements in interest rates and theArrow-Debreu prices are path-independent. That is,G(n, i, m, j) has the same value irrespective of the

path taken to state i at time n and the path taken tostate j at time m. Path-dependent valuation isimplemented using path wise valuation in a similarmanner. This valuation formula is also used to deter-mine the value of zero-coupon bonds on future datesfor future states. At each node in the lattice a vectorof zero-coupon bond prices for future maturities isdetermined from this formula.

3.8 Discrete Time Models

In a binomial model there are three degrees of free-dom at each node available to fit three differentparameters. In a single-factor short-rate model of theterm structure, the degrees of freedom can be used tofit the market prices of risk, a volatility structure forthe short rate, and a volatility structure for forwardrates.

Kalotay, Williams, and Fabozzi (1993) develop alognormal model of interest rates in which the spotrates are constrained to be positive. They fit a singledegree of freedom in the down jump and fix the prob-abilities at a half. The KWF approach is fast and com-putationally easy. The model does not fit an explicitvolatility structure.

Methods involving two degrees of freedom can besubdivided into those dealing with the down jump andthe difference between the down and up jump, andthose dealing with the down jump and the probabilityof the up jump. With two parameters, an arbitraryprescribed initial volatility curve can be incorporated.The BDT model (1990) is an implementation involv-ing rd and ru2rd: it has mean-reversion in the shortrate and imposes a volatility structure at each futureepoch. In fact, Jamshidian (1991b) shows that theBDT forecasts an increasing short-rate volatility toaccommodate a flat current volatility curve.

Another model involving two degrees of freedom isHo and Lee (1986), who were the first to propose anarbitrage-free model. Ho and Lee can be implementedby choosing two degrees of freedom, the probabilityp and the down jump rd. As noted earlier, the Ho andLee model has an analytic solution since its contin-uous time limit is an additive normal model for theshort rate of the form dr5a(t)dt1sdZ witha(t)5Ft(0,t)1s2t, where Ft(0, t) is the derivative withrespect to t of the instantaneous forward rate applyingto time t at the initial date of the yield curve used tofit the parameters (time 0). This implies that all spotand forward rates have the same volatility. Since thisis a Gaussian model, negative short rates are possible.


Methods employing three degrees of freedom, p, rd,ru at each time and state, lead to a non-recombiningtree. Such an implementation could be used to valuepath-dependent securities for which the value of thesecurity depends on the past behavior of the shortrate r. The problem with using non-recombining treesis that they are computationally intensive. Such a lat-tice can be used to model a one-factor Heath, Jarrow,and Morton (1992) model. Simulation is often usedinstead of lattice approaches for path-dependent val-uation.

3.9 Model Fitting—Forward Induction

Forward induction is an efficient technique used to fitthe parameters in an arbitrage-free model. Backwardinduction, as previously described, is used to obtainthe value of the cash flows, working backwards intime calculating expected values and discounting. Incontrast, forward induction involves working forwardin time from the current date. The relationship be-tween forward and backward induction is seen in thefundamental pde introduced earlier. In continuoustime, a Green’s function, the fundamental solution ofpde’s, satisfies two fundamental differential equations:the Kolmogorov backward equation and the Kolmo-gorov forward, or the Fokker-Planck, equation (Duffie1992).

The backward equation is the pde introduced ear-lier, which can be rewritten as:

12g 1 g µ(x,t) 1 g s (x,t) 2 r(x,t)g 5 0t x xx2

where µ(x,t) and s2(x,t) are the drift and volatilityterms, respectively, of the process for x. The Fokker-Planck equation is given by:

122g 2 g µ(x,t) 1 g s (x,t) 2 r(x,t)g 5 0t x xx2

In discrete time, the corresponding formulations areforward and backward induction. The forward induc-tion relationship actually arises by substituting back-ward finite differences in the Kolmogorov forwardequation and working through the equations in Ap-pendix C. An insight into the relationships betweenthe two can be considered by a reexamination of thebinomialization of the continuous processes. Forwardinduction gives a relationship between two values ofthe security at t and one value at t11, and backwardinduction gives a relationship between two values ofthe security at t11 and one value at t. This is illus-trated in the following diagram. In the diagram G(i)

represents the value of a primitive security, orGreen’s function, that satisfies the pde.

G(i+1)

G(i)

time tG(i–1)

time t+1

G(i+1)

G(i)

time t time t+1G(i–1)

BackwardInduction

ForwardInduction

1– 1–

Backward induction is used by discrete pricing mod-els to work backwards through a tree to obtain theprice of a security by placing known payoffs at theterminal branches of the tree. To value a zero-couponbond with maturity T, payoffs of $1 are placed at theterminal branches at time T and backward inductionis used to value these back to time 0. In an arbitrage-free discrete model, the parameters at time T are de-termined by making sure the model values the Tmaturity zero correctly. A naive approach to fittingthese parameters will generally involve significantcomputation if backward induction is used to calcu-late model prices for all the zero-coupon bonds inorder to fit the short rate parameters to market prices.

Jamshidian (1991b) uses forward induction to dra-matically cut the computation required to fit the yieldcurve. He uses forward induction to fit the parametersof his model to the yield curve. Consider a model withone parameter, the down jump, with the differencebetween the down and up jump predetermined so thetree recombines. The probabilities of the jumps arefixed at 1/2. The backward equation in terms of statecontingent prices is:

G(n, i, m, j) 51 G(n, i, m, j 2 1) 1 G(n 1 1, i 2 1, m, j)@ #2 1 1 r(n, i)

The corresponding binomial forward equation has itsdiscounting done at the two previous nodes:

1 G(n, i, m, j 2 1)G(n, i, m 1 1, j) 5 @2 1 1 r(m, j 2 1)

G(n, i, m, j 1 1)1 #1 1 r(m, j 1 1)


This formula uses discount factors that push forwardfrom the last nodes of the tree. This gives a techniquefor working prospectively, rather than retrospectively.Having fitted the parameters so that the T maturityzero-coupon bond is priced correctly, forward induc-tion is used to ensure the T11 maturity zero-couponbond is priced correctly without having to work back-wards through the tree.

Sherris (1994) gives a practical example with de-tailed numerical implementation of the principles ofJamshidian’s (1991b) forward induction. A simple ex-ample of this approach is also given in Chapter 9 ofSherris (1996). Both authors consider fitting the driftover time to match the given term structure using log-normal or normal models with initial spot rate vola-tilities defined for each period.

Forward and backward induction can also be usedwhen simulation is used, as discussed in Tilley (1992).Consider the case in which a number of paths of shortrates up to time T have been generated. The T ma-turity zero-coupon bond is valued by discountingalong each path the value of $1 at time T. This isbackward induction. If these paths of interest rates upto time T generate the market value of zero-couponbonds for maturities t51, 2, . . . , T, then the model isarbitrage-free up to time T. Forward induction can beused to fit the parameters for calculating the value ofthe T11 maturity zero once the price of the T ma-turity zero is known. With one degree of freedom (thedrift term), simply alter the model’s drift until theT11 maturity zero price calculated from the modelequals the observed market price.

With two degrees of freedom (the drift and volatilityterms), simply alter the model’s drift and volatility un-til the T11 maturity zero price calculated from themodel equals the observed market price and themodel standard deviation of the T11 maturity zero isequal to the required historical or implied volatility.This algorithm can be used with simulation to estab-lish the probability distribution of paths in the risk-neutral world. Ang (1994) uses the techniques of bothforward and backward induction, along with simula-tion, to value mortgage-backed securities in a modelincorporating variance reduction techniques.

3.10 Forward Yield Curve Models—HJM

Heath, Jarrow, and Morton (1992) developed a gen-eral multifactor framework for term structure mod-eling using the entire forward yield curve and itsarbitrage-free evolution rather than the ‘‘short’’ rate.They recognize the relationship that must hold

between the evolution of the instantaneous returns onbonds, the spot rates, and the forward rates and showthat in an arbitrage-free model the evolution of theyield curve is determined by the initial forward yieldcurve and the forward rate curve volatility structure.Single-factor spot rate models are Markov in the shortrate. These models are special cases of the HJMframework.

In general, the HJM approach leads to path depend-encies in the evolution of the ‘‘short’’ rate over time.Careful development and implementation of HJMmodels can provide Markov non-path-dependent mod-els. Brace and Musiela (1994) develop a model basedon HJM that is Markov in the forward yield curve.

To show the relationship between the evolution ofbond prices, forward rates, and spot rates, considerthe continuous equivalents for the definitions of spotrates, forward rates, bond prices, and yields for thediscrete formulas. These are:

P(t, T) 5 exp[2Y(t, T)(T 2 t)]

] ln P(t, T)f(t, T) 5 2

] T

T

P(t, T) 5 exp 2* f(t, s) ds$ %t

Let P(t, T) denote the price at time t of a T-maturityzero-coupon bond with the instantaneous return onthis bond assumed to follow the stochastic partialdifferential equation (diffusion):

dP(t, T)5 µ (t, T)dt 2 s (t, T)dz(t).P PP(t, T)

The instantaneous forward rates at time t applyingfrom time T to T1dT follow a diffusion of the form

df(t, T) 5 µ (t, T)dt 1 s (t, T)dz(t).f f

Since the forward rates and bond prices are relatedby the definition

] ln P(t, T)f(t, T) 5 2

] T

Ito’s lemma gives the relationships that must hold be-tween the drift and volatility of the instantaneous re-turn on the T maturity bond and the forward rates.These are

]µ (t, T) 5 2 µ (t, T) 1 s (t, T)s (t, T)P f P f]T


and

]s (t, T) 5 s (t, T)P f]T

so that

T

s (t, T) 5 * s (t, s)ds.P ft

The arbitrage-free condition for bond markets canthen be applied to show that

µ (t, T) 5 2 l(t)s (t, T) 1 s (t, T) s (t, T).f f P f

These results are derived in Appendix D. See alsoRitchken and Sankarasubramanian (1995a) and thediscussion by Yong Yao of Sherris (1994).

From these results, it can be seen that the volatilitystructure of the entire forward yield curve and themarket price of risk determine the evolution of the en-tire yield curve through time. This information and theinitial yield curve are all that is required to generatean arbitrage-free interest rate model. Equivalently aterm structure can specify the short-rate process as inthe more traditional models covered earlier.

In general, the evolution of the short rate in theHJM model is non-Markov. The present level of theshort interest rates depends also on the level of inter-est rates in the past. To implement this, a non-recom-bining tree must be used or simulation techniques.Heath, Jarrow and Morton developed the multifactorcase and specify the volatilities of all instantaneousforward rates at all times, referred to as a volatilitystructure. Ritchken and Sankarasubramanian (1995b)show that specification of the volatility structure offorward rates for single-factor models is important invaluing options on interest rates and bonds. Eventhough the model might fit the initial forward rate vol-atilities, the structure of the volatility as a function ofmaturity also determines the pricing performance ofthe model.

3.11 Multiple-Factor Models

Many term structure models are single-factor models.Bond prices are determined by the value of one statevariable or factor. This usually refers to the modelhaving only one source of uncertainty, in which caseyield changes for different maturities are perfectlycorrelated. The single factor can be assumed to bedriven by multiple random shocks, in which case in-stantaneous yield changes for different maturitiesneed not be perfectly correlated.

A single-factor model can be made to fit actual bondprice and volatility data by increasing the number ofparameters fitted. If actual bond prices are influencedby more than one random factor, so that yields fordifferent maturities are not perfectly correlated, thensingle-factor models are unlikely to be adequate forhedging purposes although they might suffice for pric-ing and valuation purposes. This is because they donot capture all the possible future yield curve changesand hence can not hedge against these.

Canabarro (1995) demonstrates how common one-factor models can price bond options and interest ratecaps adequately but misprice interest rate spread op-tions whose values depend on the difference betweentwo interest rates. He also demonstrates that thehedging accuracy of single-factor models is poor.

Sherris (1995) uses Australian 13-week Treasurynote yields and 2-, 5-, and 10-year Treasury bondyields from January 1972 to October 1994 to estimatethe number and importance of the random factorsdriving yield curve changes. Using a factor model andprincipal component analysis, he finds that three fac-tors explain almost all the variability in yield curvechanges over this period.

The three factors can be interpreted as explainingdifferent types of change in the shape of the yieldcurve. The first factor affects yields at all maturitiesby a similar amount and in the same direction. Thisfactor can be interpreted as a parallel shift factor forthis reason. The changes are not exactly parallel,since the effect at the short maturity is less than atthe medium to long maturities. This factor explainsas much as 83% of yield curve changes over the periodof study.

The second factor has an opposite effect on theshort and long yields. This factor can be interpretedas a ‘‘slope’’ factor, since it changes the slope of theyield curve. This second factor explains about 13% ofyield curve changes.

The third factor has a negative effect on mediumyields and a positive effect on short- and long-termyields. For this reason, this factor can be interpretedas a ‘‘curvature’’ factor. The third factor explainsabout 3% of yield curve changes. In total, these threefactors explain more than 99% of yield curve changes.

These results are consistent with overseas studies.However, compared with overseas studies, the slopeand curvature factor appear to be more important inexplaining the variance of yield curve changes in theAustralian bond market. A single-factor model wouldbe expected to explain only about 83% of yield curvechanges. This approach can also be used to estimate


historical interest rate volatilities for use in a HJMmultifactor model.

Heath, Jarrow, and Morton (1990b) use the resultsof a principal components analysis by BARRA to de-termine a volatility structure for a two-factor versionof their model and demonstrate its adequacy for pric-ing a range of interest-rate-dependent securities.Singh (1995) uses principal components analysis andU.S. interest rate data from January 1983 through De-cember 1992 to fit a three-factor Cox, Ingersoll, andRoss square root process model. The parameters ofthe model are quite unstable across time when differ-ent time series data are used.

Other stochastic factors such as a model for theinflation rate or for a stochastic volatility can be in-cluded in the pricing model. Each of these factors canbe driven by a single or multiple random ‘‘shocks.’’These additional factors are state variables for pricingcash flows and the value of the cash flows will be afunction of these state variables.

Models used for short interest rates also assumethat the processes are difference stationary and hencecontain a unit root. Ang and Moore (1994) use theaugmented Dickey Fuller procedure to test for unitroots with Australian interest rate data. They use theofficial cash rate, the 13-week and 26-week Treasurynote yields, and the 2- and 10-year Treasury bondyields from July 1969 to February 1994. They wereunable to reject the hypothesis of a unit root for anyof the series providing empirical support for differ-ence stationary models.

3.12 Equilibrium Relationships

The long-run equilibrium relationship between yieldsof different maturities contains useful information informulating a multifactor model. A number of timeseries are said to be co-integrated when a linear com-bination of them is stationary, even though the indi-vidual series are not stationary. This means that if anumber of variables are co-integrated, then an under-lying equilibrium relationship governs how they movein relation to each other. Thus the individual seriescan be nonstationary, but a difference relationship be-tween the series can have a constant variance.

Co-integration analysis permits equilibrium rela-tionships between economic time series to be exam-ined without requiring the individual series to bedifferenced. For interest rates, if the short-term spotinterest rate and the spreads between longer termspot interest rates and the short-term interest rate areco-integrated, as you might expect, then the term

premiums are determined by long-term equilibriumrelationships.

A co-integrating vector relating interest rates of dif-ferent maturities is consistent with theoretical modelsof the term structure that impose a common stochas-tic process for underlying rates. The significance ofusing co-integration analysis when analyzing time se-ries of economic variables is that forecasts that useinformation about long-term relationships should out-perform conventional time series forecasts that differ-ence the data and remove this information. Theseresults are confirmed by a study by Bradley andLumpkin (1992) on the U.S. Treasury yield curve.

A number of different econometric techniques canbe used to obtain estimates of co-integrating vectors.Ang and Moore (1994) follow Bradley and Lumpkinbut extend the analysis to using techniques that over-come some of the technical shortcomings of the Brad-ley and Lumpkin study. Testing Australian yields, Angand Moore find strong evidence of cointegration ofboth nominal and real yields.

3.13 Pathwise Valuation

A significant issue in implementing path-dependentand multifactor term structure models is the compu-tation time involved. A number of techniques havebeen developed recently to implement these models.These models will prove the most useful for actuarialliabilities, so these recent developments are relevantto actuaries.

In many valuation and hedging problems, thecash flows are path-dependent. Examples are thecash flows on mortgage-backed securities or an op-tion on the average of past interest rates. Standardmethods that rely on backward induction on a lat-tice involve exploding lattices. Amin and Bodurtha(1994) report the implementation of a single-factorHJM model using a binary non-recombining lattice.They use varying time interval lengths, rather thanfixed time intervals, and ten or less steps to obtainaccurate option prices for short-term options. Thisapproach is unlikely to be computationally feasiblefor long-dated interest-sensitive cash flows found inactuarial liabilities.

An alternative equivalent approach to backward in-duction in a lattice is to use pathwise valuation (Tilley1992). Instead of determining expected values anddiscounting, pathwise valuation generates paths of in-terest rates and cash flows and then values the cashflows down each path. An expected value of the path-wise values is then computed to obtain the value.


Such an approach handles path-dependent cash flows.In many cases the interest rate paths are generatedas equally likely paths using simulation. An alterna-tive approach is to weight selected paths according totheir contribution to the expected value, thus reduc-ing the amount of computation. This technique un-derlies the linear path space approach developed byHo (1992). This technique is not dissimilar to strati-fied sampling, in which more observations are takenin the parts of the region which are ‘‘more important.’’A similar idea is importance sampling, which has alsobeen widely studied [see Glynn and Iglehart (1989)].

Pathwise valuation can handle path-dependent cashflows. However, in order to value any set of cash flowsthat involves optimal dynamic decisions, as for Amer-ican-style options, the lattice approach is required. Inhis linear path space Ho (1992) imposes a latticestructure and weights these paths with probabilities,allowing both American-style and path-dependentcash flows to be valued. Pohlman and Wolf (1993)demonstrate how to calculate the weights for Ho’s lin-ear path space. Tilley (1993) demonstrates how sim-ulation can be adapted to value American-styleoptions using another form of path-bundling.

Rose (1994) gives a method based on calculatingexpected cash flows and discounting that allows forboth path-dependent and early-exercise features ininterest-rate-dependent cash flows. The technique al-lows an approach similar to the traditional deter-ministic approach to be applied. The expected cashflows are calculated as effective cash flows that allowfor path dependence and early exercise. The tech-nique uses the concepts of forward and backward in-duction to determine the cash flows and then to valuethem.

3.14 Simulation Techniques

A powerful method of implementing term structuremodels is to use simulation. This technique uses thepathwise approach to valuation. Tilley (1992) pro-vides a comprehensive description of how this can bedone in practice. For most actuarial applications, itwill be essential to use variance reduction to speedthe computation.

Variance reduction techniques (VRTs) improve theefficiency of normal Monte Carlo estimates. In simu-lation problems using arbitrage-free models, generat-ing a value on random paths of interest rates producesrandom outputs. VRTs attempt to reduce the varianceof the output random variable of interest without dis-turbing its expectation. VRTs, if used optimally, will

lead to increases in speeds of computation becausefewer simulations will need to be run. Lewis and Orav(1989) note many limitations of VRTs, especially thatVRT’s may induce non-normality and very wild out-liers. The smaller variance may not tell the entirestory of what is being done by a VRT.

Many of the presentations of VRTs in the literaturerelate to operations problems and not to distributionalproblems of the kind that occur in valuing securities.Because VRTs are very problem specific, this meansit is not easy to just take VRT implementation ‘‘off theshelf’’ from other problems. A good survey of usingVRTs in increasing the speed of algorithms is givenby McGeoch (1992). This paper does not attempt togive a formal presentation of the statistical propertiesof the VRTs; for this the reader is referred to the text-books by Rubinstein (1981), Morgan (1984), and Lawand Kelton (1991). A number of VRTs are readily im-plemented:

Antithetic Variates

First introduced by Hammersley and Morton (1956),this VRT is based on the rationale that if the corre-lation between two random variables, X1 and X2, isvery negative, the average of the negatively correlatedpair is less than the variance of the average of anindependent pair:

X 1 X 11 2var 5 [var (X ) 1 var (X )1 22 4

1 2 cov (X , X )]1 2

If var (X1)5var (X2)5s2, then

X 1 X 11 2 2var 5 s (1 1 r)2 2

where r is the correlation between X1 and X2. If r isnegative, then the variance of the average of the an-tithetic pair is less than that of the independent pair.

Antithetic variates are constructed by taking a setof normal deviates U, and reversing the signs of thedeviates to obtain 2U. For a particular path p gen-erated by random numbers (n1, n2, . . . , nT), an anti-thetic path is generated by the random numbers (2n1,2n2, . . . , 2nT).

Stratified Sampling

This involves breaking the sampling region D into mdisjoint subregions Di, i51, 2, . . . , m. Crude MonteCarlo is used to estimate each interval separately. Theidea of this technique is to take more observations in


the parts of the region D that are more ‘‘important’’;variance reduction is achieved by concentrating ob-servations in the important subsets Di rather than bysampling over the whole distribution. The problemwith stratified sampling is the determination of an op-timal stratification plan. However, the results showthat in practice nonoptimal stratification can yield re-sults far more efficient than crude Monte Carlo.

The practical adaptation of stratified sampling to asimulation model involves dividing the sampling den-sity for the diffusion factor into T strata of equal prob-ability 1/T. A stratified sample can be created bygenerating numbers over each of the T strata or usingthe sample-mean Monte Carlo method, using themean of each stratum for each of the T strata. Therequired random sample of T deviates is obtained byshuffling the ordered stratified sample. An easy wayto do this is to assign each ni, i51 . . . T a randomnumber between 0 and 1. The random numbers arethen sorted into either descending or ascending order,and the sample of T deviates is then shuffled.

Control Variates

Unlike the other VRTs discussed above, control vari-ates operate at the level of the calculation of the ex-pected value, rather than at the level of the generationof the interest rate paths (the random numbers them-selves). Like antithetic variates, they attempt to takeadvantage of correlation between certain random var-iables to obtain a reduction in variance. Unlike theantithetic approach, the random variables do not haveto have a negative correlation. Suppose X is beingused to estimate a parameter q with E(X)5q. Nowconsider another random variable, Y, with known ex-pectation m. Then for any constant c (the control),we can write:

X 5 X 2 c(Y 2 m).c

Xc , like X, is an unbiased estimator of q with;

2Var(X ) 5 Var(X) 1 c Var(Y) 2 2c cov(X, Y).c

The maximum variance reduction is obtained whenc5[cov(X, Y)]/[var(Y)]. Note that Xc is less variablethan X if and only if:

1cov(X, Y) . c Var(Y)

2

However, while cov (X,Y) and possibly var (Y) maynot be known, c can be taken simply as 51, as ap-propriate. Like any ‘‘control’’ problem, there are sta-bility concerns with the choice of covariates and the

value of c. Ideally it is best to use a control variatethat is very similar to the security being valued. Clew-low and Carverhill (1994) develop a control variateapproach based on the risk statistics of the derivative.

Quasi-random or Low-Discrepancy Random Numbers

A recent development in simulation techniques forevaluating interest rate derivatives has been the useof quasi-random numbers. This approach is based onusing deterministic points that are as uniformly dis-tributed as possible instead of the pseudo-randomnumbers used in Monte Carlo simulation. Paskov andTraub (1995) demonstrate that a particular set ofquasi-random numbers called Sobol points producesmore accurate, smoother and faster convergence toderivative values as compared with Monte Carlo sim-ulation. Monte Carlo techniques can be sensitive tothe initial seed and this problem does not occur forquasi-random numbers.

4. CONCLUSION

Actuaries often promote themselves as financial en-gineers. The topics covered in this paper could wellbe classified as ‘‘financial engineering’’ for valuationand risk management of interest-rate-related cashflows. Actuaries are becoming more involved in thisarea as they move into quantitative areas of financeand investment.

It is important to recognize that these techniqueshave practical applications in the traditional actuarialfields of practice. They are of relevance not only toquantitative finance and investment actuaries butmore generally to life insurance, property and casu-alty, and pension actuaries.

The paper includes many references, the majorityof which are dated after 1990. Apart from a numberof early, and significant, contributions to this area, itwas only in the middle and late 1980s that major the-oretical advances were made that laid the foundationfor current research. A major proportion of this cur-rent research is driven by practical application of thetechniques to interest rate risk management. Theamount of research by academics and practitioners inthis area has been nothing more than phenomenal inthe last few years. It continues to be an active re-search area as a better understanding of the basis forinterest rate modeling is gained.

The authors hope that this paper will motivateother actuaries to learn more about this exciting area


and provide a basis for learning how to develop andapply these techniques to actuarial liabilities.

ACKNOWLEDGMENTS

The authors would like to thank Elias Shiu, PrincipalFinancial Group Professor of Actuarial Science, Uni-versity of Iowa, for valuable (e-mail) discussion andcomments on an earlier version of this paper and twoanonymous referees for valuable comments. Any re-maining errors or misconceptions remain the respon-sibility of the authors. Andrew Ang would like toacknowledge the support of AMP Investments duringhis honours year studies, and Michael Sherris wouldlike to acknowledge the financial support of an Aus-tralian Research Council (Small) Grant. An earlierversion of this paper was presented to the 1995 Con-vention of the Institute of Actuaries of Australia, andthe authors thank participants at that convention fortheir useful comments.

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APPENDIX A

Glossary of Key Terms

This glossary provides descriptions of the key con-cepts and terminology used in the paper in alphabet-ical order. It will serve as a useful reference point forreaders who are not familiar with the terminology.

Arbitrage-Free. An interest rate model is arbitrage-free with respect to a set of securities if it is not pos-sible to construct portfolios of these securities withno net future cash flows that require non-zero net ini-tial investment. An arbitrage-free model will producethe market value of these securities when used toprice the securities.

Arrow-Debreu Securities. These are the most basicor ‘‘primitive’’ securities in a pricing model that allowsfor uncertainty from which all other securities can beconstructed. They are also referred to as state-contin-gent securities. In a stochastic interest rate model,Arrow-Debreu securities pay $1 at a specific maturityand for a specified value of the future (uncertain) in-terest rate, and zero for all other maturities and val-ues of future interest rates.

Backward Induction. This is the process used inthe valuation of cash flows that depend on future val-ues of interest rates. Once the cash flows have beendetermined using forward induction, these are valuedworking backwards in time calculating risk-adjustedexpected values and discounting.


Cointegration. A number of time series are said tobe co-integrated when a linear combination of themis stationary, even though the individual series are notstationary. This means that if a number of variablesare co-integrated, then an underlying equilibrium re-lationship governs how they move in relation to eachother.

Convexity. Convexity measures the degree of rela-tive curvature in the price/yield relationship for in-terest-rate-dependent cash flows. It is calculated asthe second derivative of the price with respect to theyield as a proportion of the price. In practice, the sec-ond derivative is usually estimated numerically.

Complete Markets. Markets are said to be completeif there exists a sufficient number of independenttraded securities to dynamically hedge the number ofrandom factors in a security market model. In a com-plete market, interest-rate-dependent cash flows canbe perfectly replicated with portfolios of tradedsecurities.

Delta. The delta of an option is the sensitivity ofthe value of the option to changes in value of the un-derlying asset value. In the case of interest-rate-de-pendent cash flows, the sensitivity of the value of thecash flows to changes in interest rates is often cal-culated as the delta. For interest-rate-dependent cashflows where the cash flows are those of a traded fi-nancial instrument such as an option on a bond, thedelta can be calculated as the sensitivity of the valueof the option to changes in the value of the bond. Thisdefinition of delta is useful for hedging purposeswhere an underlying financial instrument actuallytrades since it gives the hedge ratio.

Diffusion Process. A continuous time Markov pro-cess that assumes small changes in a time series hastwo components, a drift and a diffusion. The drift rep-resents the expected change in the series, and thediffusion provides the random change. The drift anddiffusion components can take a number of differentforms, allowing for mean reversion in the drift and fora variety of distributions in the diffusion.

Duration. This is the weighted average time to re-ceipt of the cash flows, where the weights are thepresent value of the cash flows.

Effective Duration. This is (minus) the sensitivityof the value of the cash flows to a small parallel shiftin interest rates as a proportion of the value of thecash flows. It is also referred to as modified durationfor bond cash flows. It is related to duration but cal-culated as a sensitivity and not as a weighted averagetime to receipt of the cash flows. This difference inthe method of calculating effective duration is important

when considering interest rate derivatives such as fu-tures, swaps, and options.

Expectations Hypothesis. The expectations hypoth-esis has different versions with different assumptions.In one form it states that the expected one-period re-turns on all bonds are equal to the prevailing spot rate,and thus for a given holding period the long-term rateis an unbiased average of the current spot rate andexpected future spot rates (or one-period forwardrates). Under other theories of the term structure suchas the liquidity premium hypothesis and preferred hab-itat theory, risk premiums are assumed for differentmaturities.

Equilibrium Model. Equilibrium interest rate mod-els determine bond prices and the value of other in-terest-rate-contingent claims based on assumptionsabout the risk preferences of investors and the sto-chastic nature of capital markets.

Forward Induction. This is the process of determin-ing future interest-rate-dependent cash flows by work-ing forward from the current time. The process is usedto determine future interest rate paths and then to de-termine future cash flows that depend on these interestrate paths. Forward induction usually refers to the pro-cess of fitting the parameters of an arbitrage-free in-terest rate model to a current yield curve for eachfuture time period by working forward in time.

Forward Interest Rate. A forward interest rate ap-plies to a fixed time period commencing at a pre-spec-ified future (or forward) time. At the current date,forward rates are known and can be derived from theyields to maturity or the spot rates. On future dates,the forward rates that will apply for forward time pe-riods at these dates are uncertain and need to be mod-eled as stochastic variables.

Integration (of a Time Series). If a time series isintegrated of order p, then p-th-differences are re-quired to produce a stationary series. Thus if a seriesis integrated of order 1, then first differences will bestationary. The order of integration of a series has im-portant implications for modeling and inference.

Lattice Model. A discrete model of interest rates,or asset prices, generates future possible values on alattice. The binomial model with two possible valuesconditional on the current value generates a lattice ofvalues with the number of nodes in the lattice eitherincreasing by one or doubling, depending on whethervalues in the lattice recombine.

M2. The weighted ‘‘variance’’ of the time to receipt ofa set of cash flows around the duration of the cash flowswhere the weights are the present values of the cashflows. For asset cash flows, the weights are positive and


M2 is a variance. This will not be the case for net cashflows, equal to asset cash flows minus liability cashflows, since some of the net cash flows and hence someof the weights could be negative.

Market Price of Risk. For a security or portfoliothat is exposed only to a specific risk factor, the mar-ket price of risk is defined as the excess return overthe short rate per unit of volatility. The market priceof interest rate risk is defined as the excess return ona bond over the short rate per unit of volatility of thebond price.

Markov Process. A Markov process has the prop-erty that all future values depend only on the currentvalue of the process. For interest-rate-modeling pur-poses, such processes are computationally simpler toimplement.

Martingale. A martingale is a stochastic processwith the property that its expected value at any futuretime is equal to its current value.

Multiple Factor Model. A model of an interest rateseries, or asset price series, where future values aredetermined by multiple random factors or ‘‘shocks.’’The nature of these random shocks can be estimatedfrom historical data by using principal components orfactor analysis.

Short Rate. In a continuous time model the ‘‘short’’rate is usually defined to be the instantaneous spotinterest rate. In a discrete model the short rate is theone-period spot interest rate. Often the term ‘‘the spotinterest rate’’ is used to refer to the short rate as de-fined here.

Single-factor Model. A model of an interest rate se-ries, or asset price series, that has only one randomfactor determining the future values of the series.

Spot Interest Rates. These are the yields to matur-ity on zero-coupon bonds for varying maturities. Theone-period spot interest rate is the yield to maturityon a zero-coupon bond maturing in one time period.The instantaneous spot interest rate in a continuoustime model is the rate applying from time t to t1dt.

Stationary Series. There are various levels of sta-tionarity. A common definition for a series to be sta-tionary is that the mean and variance are constantand the autocorrelations do not vary through time. Achanging mean could be modeled by using a polyno-mial trend. If after fitting the trend the remaining se-ries has a constant mean, then the series is ‘‘trend’’stationary. Random ‘‘shocks’’ do not have an effect onthe long run level of the series. If it is necessary todifference the series to obtain a stationary mean, thenthe series is said to be ‘‘difference’’ stationary. In thiscase random ‘‘shocks’’ have a permanent influence on

the level of the series, and the trend is stochastic. Itis possible for the variance of the series to be madeconstant by transformation of the series such as a log-arithmic transformation for cases in which the vari-ance increases with the level of the series.

Term Structure of Interest Rates. This is the struc-ture of interest rates as time to maturity varies withthe effect of all other factors such as coupon level,liquidity, and credit risk removed. The term structurereflects only variations in interest rates with respectto term. The spot yield curve for zero-coupon govern-ment bonds is most often used to represent the termstructure.

Variance Reduction Techniques. Used in simula-tion, variance reduction techniques (VRTs) improvethe efficiency of Monte Carlo estimates. VRTs workby reducing the variance of the random variable ofinterest without disturbing its expectation. Somecommon VRTs used in interest rate modeling are an-tithetic variates, stratified sampling, control variates,importance sampling, path-bundling, and the linearpath space.

Volatility. This term has many different meanings.The term is sometimes used to refer to (minus) thedifferential of the value of a security with respect tothe interest rate as a proportion of the value. This ismore often called modified duration or effective du-ration in financial markets. In option pricing the termvolatility refers to the instantaneous standard devia-tion of the rate of return on the underlying asset.

Wiener Process (Brownian Motion). A diffusionprocess with increments that are stationary, indepen-dent, and normally distributed with variance propor-tional to the time interval for the increment.

Yield Curve. The plot of yield against term. Mostoften for the yields to maturity but also for spot andforward interest rates.

APPENDIX B

Derivation of the Standard PDEs forSecurity Valuation

Consider the stochastic process for the factor X:

dX 5 µ(X, t)dt 1 s(X, t)dZ (1)

where dZ is a standard Wiener process.Let H(X, t) and G(X, t) be the values of two secu-

rities that are functions of X and t. These processesmay be described as:

dH5 µ (X, t)dt 1 s (X, t)dZ (2a)H HH


and

dG5 µ (X, t)dt 1 s (X, t)dZ (2b)G GG

Form a portfolio with value P consisting of [sG(X, t)]/Hof the first security H and 2 [sH(X, t)]/G of the secondsecurity G so that:

P 5 s (X, t) 2 s (X, t) (3a)G H

and

dP 5 s (X, t)dH 2 s (X, t)dG (3b)G H

Substitute (2a) and (2b) to obtain:

dP 5 [µ (X, t)s (X, t) 2 µ (X, t)s (X, t)]dt (4)H G G H

Since this is riskless, it must earn the risk-free rater(t):

dP 5 r(t) Pdt (5)

Substituting (2a), (2b), and (3a), (3b) into (5) gives:

µ (X, t)s (X, t) 2 µ (X, t)s (X, t)H G G H (6)

5 r(t)[s (X, t) 2 s (X, t)]G H

which simplifies to:

µ (X, t) 2 r(t) µ (X, t) 2 r(t)H G5 5 l(X, t) (7)s (X, t) s (X, t)H G

where l is the market price of risk. Assuming acomplete market, where it is possible to replicate thepayoffs of a security V(X,t), then

µ (X, t) 2 r(t)V 5 l(X, t),s (X, t)V

which can be written as:

µ (X,t) 2 r(t) 5 l(X,t)s (X,t) (8)V V

Application of lemma gives dV as:Ito’s

12dV 5 V dX 1 V dX 1 V dtX XX t2

5 V [µ(X, t)dt 1 s(X, t)dZ]X

121 V s (X, t)dt 1 V dtXX t2

125 V µ(X, t) 1 V s (X, t) 1 V@ X XX t#2

dt 1 V s(X, t)dZX

and so the parameters for the drift and diffusion fordV are:

12µ (X, t) 5 V µ(X, t) 1 V s (X, t) 1 V (9a)V X XX t2

and

s (X, t) 5 V s(X, t) (9b)V X

Substitute (9a) and (9b) into (8) to obtain the partialdifferential equation, which must be satisfied by V:

V [µ(X, t) 2 l(X, t)s(X, t)]X

121 V s (X, t) 1 V 5 r(t)VXX t2

To obtain a solution for V, impose boundary condi-tions on this pde and solve it analytically ornumerically.

APPENDIX C

The Discretization of Continuous Models

This appendix derives lattice models from finite dif-ference approximations to the fundamental pde forpricing securities. For convenience, it is assumed thatdr5µrdt1srdZ, so that r is a lognormal process,P5P(r,t), and that the bond market is arbitrage-free.The market price of risk is incorporated into the driftterm, so no market price of risk explicitly appears inthe pde. The pde is restated here for convenience,where subscripts denote partial derivatives:

12 2P s r 1 P µr 1 P 5 rP (1)rr r t2

Explicit finite difference approximations to the partialderivatives can be used to derive trinomial or bino-mial lattices. References are He (1990), Hull (1993),Jamshidian (1991b), and Nelson and Ramaswamy(1990).

To obtain a trinomial tree, substitute the followingdifference approximations to the partial derivatives:

P 2 Pi: j11 i: jP 5 (2a)t Dt

where the index i refers to increments in r of size Dr,j refers to increments in t of size Dt.

P 2 Pi11: j11 i21: j11P 5 (2b)r 2Dr

P 1 P 2 2Pi11: j11 i21: j11 i: j11P 5 (2c)rr 2Dr

Equation (2a) is a first difference, (2b) is a centraldifference, and (2c) is a second difference.


Substitute (2a–c) into (1) to get:

P 1 1i: j 2rP 1 5 P iµ 1 s ii: j i11: j11 @ #Dt 2 21 1 1

2 21 P iµ 1 s i 1 P s i 1 (3)i21: j11 @ # i: j11 @ #2 2 Dt

Equation (3) can be written as:

a* P 1 b* P 1 c* Pj i11: j11 j i: j11 j i21: j11P 5 (4a)i: j 1 1 rDt

where

1 12 2a* 5 iµDt 1 s i Dt (4b)j 2 2

2 2b* 5 1 2 s i Dt (4c)j

1 12 2c* 5 2 iµDt 2 s i Dt (4d)j ~ !2 2

a* 1 b* 1 c* 5 1. (4e)j j j

Note that since these sum to 1 they can be interpretedas (pseudo) probabilities.

The result [in 4(a)] states that the PV of a bond atstate i, time t, is a weighted average of the value ofthe bond at time t11 discounted by the one-periodspot rate at state i on a trinomial lattice. Here a*, b*,and c* are the (risk-neutral) probabilities:

P(i+1)a*

P(i)b*

P(i)

c*

time j

P(i–1)

time j+1

A binomial tree can be derived from the pde by mak-ing the change of variable:

dr dr ln rx(r,t) 5 * 5 * 5 (5a)

s(r,t) sr s

1x 5 (5b)r sr

1x 5 2 (5c)rr 2sr

x 5 0 (5d)t

From Lemma:Ito’s

12dx 5 x dr 1 x dr 1 x dt (6)r rr t2

Substituting (5b–d) into (6) yields:

µ sdx 5 bdt 1 dZ, b 5 2 (7)

s 2

The pde in terms of ]r must be transformed intoterms of ]x where r(x,t) is the inverse of x(r,t). Letp(x,t) 5 P(r,t) with:

P 5 p x by the chain ruler x r

pxP 5 (8a)r sr

2P 5 p x 1 p x by the product rule. (8b)rr xx r x rr

Simplifying (8b) gives:

p pxx xP 5 2 (8c)rr 2 2 2s r sr

Substituting (8b–c) into the pde (1) and simplifyinggives:

1p 1 b p 1 p 5 rp (9)xx x t2

Applying finite differences as (2a–c), except now interms of p, gives:

p b 1i: jrp 1 5 p 1i: j i11: j11 ~ !2D t 2Dx 2Dx

1 b p pi: j11 i: j111 p 2 2 1 (10)i21: j11 ~ !2 22Dx 2Dx Dx D t

In continuous time:dx 5 bdt 1 dZ (11a)

2dx 5 dt

and in discrete time the equivalent first-orderapproximations are:

=Dx ' bDt 1 j Dt (11b)2Dx ' Dt

where j;N(0, 1).Simplifying (10) gives:

1 1=p 5 p (b dt 1 1)i: j @ i11: j111 1 rdt 2

1=1 p (1 2 b dt) (12)i21: j11 #2


Or in discrete time:

1p 5 [pp 1 (1 2 p) p ] (13a)i: j i11: j11 i21: j111 1 rDt

where (13b)1

=p 5 [1 1 b Dt]2

Equations (13a) and (13b) can be interpreted as abinomial model, where p and (12p) are the binomialrisk-neutral probabilities:

p(i+1)

p(i)

time j

p(i–1)

time j+1

1–

This appendix has shown that the discrete trinomialand binomial models are in fact discrete approxima-tions to the continuous time approaches. Similarly, adiscrete lattice can always be developed to be consis-tent with the continuous time framework with theright restrictions and corresponding algorithm. Theseresults are then very important in understanding howto select discrete lattice models.

APPENDIX D

The Relationship of the StochasticProcesses of Bond Prices, Forward Rates,and Spot Rates

Let P(t,T) denote the price at time t of a T-maturityzero-coupon bond with instantaneous return given by:

dP(t,T)5 µ (t,T)dt 2 s (t,T)dz(t). (1)P PP(t,T)

The instantaneous forward rates at time t applyingfrom time T to T1dT follow a diffusion of the form

df(t,T) 5 µ (t,T)dt 1 s (t,T)dz(t). (2)f f

Since the forward rates and bond prices are relatedby the definition

] ln P(t,T)f(t,T) 5 2 (3)

]T

From Ito’s lemma:

12d ln [P(t,T)] 5 [µ (t,T) 2 s (t,T)]dtP P2

2 s (t,T)dz(t) (4)P

and

]df(t,T) 5 2 [d ln P(t,T)]

]T

] 125 2 [µ (t,T) 2 s (t,T)]dtP P]T 2

]1 s (t,T)dz(t). (5)P]T

Comparing terms in (2) and (5) gives

] 1 ]2µ (t,T) 5 2 µ (t,T) 1 s (t,T) (6)f P P]T 2 ]T

]s (t,T) 5 s (t,T) (7)f P]T

Rearranging (6), differentiating the squared volatilityterm and substituting (7) gives:

]µ (t,T) 5 2µ (t,T) 1 s (t,T)s (t,T) (8)P f P f]T

Integrating (7) gives:

T

s (t,T) 5 * s (t,s)ds. (9)P t f

The arbitrage-free condition for bond markets is that

µ (t,T) 5 r(t) 1 l(t)s (t,T) (10)P P

Differentiating (10) and using (7) gives

] ]µ (t,T) 5 l(t) s (t,T) 5 l(t)s (t,T) (11)P P f]T ]T

Substitute (11) into (8) and rearrange to get

µ (t,T) 5 2l(t)s (t,T) 1 s (t,T)s (t,T). (12)f f P f

Short-rate dynamics follow from the dynamics of theforward rate curve since r(t)5f(t,t).

Discussions on this paper will be accepted untilOctober 1, 1997. The authors reserve the right to re-ply to any discussion. See the Table of Contents pagefor detailed instructions on the preparation of dis-cussions.

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