Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015

Master’s Thesis

Effective Pricing of Cliquet Options

Peter Warken

December 2015

Supervisor: Prof. Dr. Jorn Saß

University of Kaiserslautern

Department of Mathematics

Financial Mathematics Group

Abstract

This thesis provides cutting-edge conceptions for pricing equity-linked annuities.A semi-closed-form expression of the price of cliquet options is developed ina Black-Scholes market model and compared to a Monte Carlo approach forpricing these path-dependent options. It is presented how the result can beapplied to comparable structured products like sum cap contracts. Finally, it isdemonstrated that the presence of stochastic volatility and stochastic interestrates has a significant impact on the pricing behavior of cliquet options. Severalnumerical experiments are performed to illustrate the influence of market modelsand the associated financial parameters.

Contents

1 Cliquet option market 11.1 Contract specification . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Financial model and pricing concepts . . . . . . . . . . . . . . . . 9

2 A semi-closed-form solution 112.1 Pricing formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Vega of the cliquet option . . . . . . . . . . . . . . . . . . . . . . 172.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Numerical pricing 233.1 Implementing the semi-closed-form solution . . . . . . . . . . . . 233.2 Approximating the Vega of the option price . . . . . . . . . . . . 263.3 Monte Carlo technique I . . . . . . . . . . . . . . . . . . . . . . . 273.4 Comparison of the numerical techniques . . . . . . . . . . . . . . 303.5 On the choice of g and c . . . . . . . . . . . . . . . . . . . . . . . 37

4 The sum cap contract - a similar product 404.1 Pricing formula of sum cap contracts . . . . . . . . . . . . . . . . 404.2 Numerical pricing of sum cap contracts . . . . . . . . . . . . . . 42

5 The influence of stochastic volatility 455.1 The Heston stochastic volatility model . . . . . . . . . . . . . . . 455.2 Stock price simulation in the Heston model . . . . . . . . . . . . 465.3 Monte Carlo technique II . . . . . . . . . . . . . . . . . . . . . . 47

6 The influence of stochastic interest rates 526.1 The Black-Scholes-Hull-White model . . . . . . . . . . . . . . . . 526.2 Stock price simulation in the BSHW model . . . . . . . . . . . . 536.3 Monte Carlo technique III . . . . . . . . . . . . . . . . . . . . . . 536.4 Further short rate models . . . . . . . . . . . . . . . . . . . . . . 57

I

Introduction

Equity-indexed annuities - EIAs - are customized structured products, soldfor instance by insurance companies to provide savings and insurance benefits.These EIAs differ from traditional variable annuities in several significant ways.In particular, the return of an investment in an EIA is guaranteed not to fallbelow a certain minimum level. This feature qualifies these investment productsas insurance products. An EIA provides a fixed return plus the possibility ofan additional return, based on the performance of the underlying. There is awide variety of possible designs. All of them have in common that the guaranteeis financed via a limitation of the returns associated with the underlying (cf.[Pa06], [BB11]).

In the thesis the pricing of cliquet options is discussed. These contracts havea European payoff at a fixed future maturity date. Cliquet options can beinterpreted as a series of forward-starting at-the-money options with a singlepremium determined upfront which locks in any gains on specific reset dates.At these dates the strike price is reset at the current level of the underlyingasset. Thus, any decline in the price of the underlying asset resets the strikeprice to a lower level, while keeping earlier profits. Floors and caps are addedto fix the minimum and maximum returns. By construction, these structuredproducts provide a downside protection yet being affordable priced since thepayoff is capped locally.

In recent years, there has been an increasing interest in such path-dependentoptions. Especially, turmoil in financial markets has led to a demand for invest-ment solutions that reduce downside risk while still offering upside potential(see [BBG11] for further details).

1 Cliquet option market

In this chapter cliquet options are introduced as in [BL13] and the financialmarket model is described.

1.1 Contract specification

Let T , a future point in time, be the maturity date of the contract. The interval[0, T ] is divided into n different periods of length ∆ with T = n∆. For k = 0, ..., nthe dates tk = k∆ are called reset days. The initial investment is denoted byK. By St the price of the underlying asset at time t ∈ [0, T ] is denoted. Furtherspecifications include the guaranteed rate at maturity g as well as a local capc for each reset period tk − tk−1. The payoff of a minimum coupon cliquet isgiven by

XT = K max

(1 + g, 1 +

n∑k=1

max

(0,min

(c,Stk − Stk−1

Stk−1

))).

The payoff of the cliquet option is paid at the fixed maturity date and thus theoption is of European type. The return of each period is furthermore locallycapped at c and the contract also consists of a local floor equal to 0. Theguaranteed rate of return is equal to g. As the contract is linked to periodicalreturns, the notation is simplified by denoting by Rk the return of the underlyingin the kth period for k = 1, ..., n:

Rk :=Stk − Stk−1

Stk−1

.

A modified representation of the payoff of the cliquet option is achieved by

XT = K max

(1 + g, 1 +

n∑k=1

max (0,min (c,Rk))

)

= K (1 + g) +K max

(0,

n∑k=1

Zk

),

where Zk is defined as

Zk := max (0,min (c,Rk))− g

nfor k = 1, ..., n.

If the increments of the underlying asset price process are independent andidentically distributed (i.i.d.), then also the returns of the underlying Rk andthe modified Zk are i.i.d. random variables. Since their distribution does notdepend on k, denote by R and Z a corresponding independent and identicallydistributed random variable.

1

Observe that in a single reset period k the returns are truncated at the floorlevel 0 and capped at c. With the help of figure 1, the payout profile of the kthperiod can thus be interpreted as 1

Stk−1-call spreads, i.e. long calls with strike

Stk−1and short calls with strike Stk−1

(1 + c). It is thus justified to interpret acliquet option as a series of forward-starting at-the-money options.

−0.1 −0.05 0 0.05 0.1−0.1

−0.05

0

0.05

0.1

max

(0,m

in(R

k,c))

Rk

Figure 1: Payout profile in the kth period with c = 5%

2

For illustration purposes a 5-year minimum coupon cliquet with monthly resetdates on the S&P 500 Index is considered. Let the local cap be equal to c = 0.6%and the guaranteed rate is set to g = 16%. Based on a historical data set ofthe closing prices since January 1999 a naive sampling algorithm with A = 105

trials is performed. The figure 2 displays the sampled distribution of the rateof return of the cliquet option at maturity.

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.320

100

200

300

400

500

600

Rate of return at maturity

Fre

quen

cy

Figure 2: Sampled distribution of the rate of return at maturity

3

In the following consider a historical example of a yearly reseted cliquet optionon the S&P 500 Index, that started at the beginning of 2005 and matured atJanuary the first, 2010. The chart below shows the performance of the under-lying during this period.

1998 2000 2002 2004 2006 2008 2010 2012 20141000

1500

2000

2500

3000

3500

Years

Inde

x le

vel a

t the

beg

inni

ng o

f the

yea

r

S&P 500 Index Performance Chart

Figure 3: Performance chart S&P 500 Index

Let the local cap of the contract be equal to c = 8% and the guaranteed rate isset to g = 16%. The yearly returns of the underlying as well as the floored andcapped yearly returns of the cliquet option are illustrated in the following barchart.

4

2005 2006 2007 2008 2009−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Years

Yea

rly r

etur

ns

Yearly returns

yearly returns of the cliquet optionyearly returns of the underlying

Figure 4: Periodical returns

It can easily be recognized that the holder of the cliquet option is insured againstthe large drawdown during the financial crisis in the year 2008. This insuranceis financed via the caps of the yearly returns. Thus, an investor is not able toparticipate in the above 8% returns in the years 2006 and 2009. The final returnat maturity of the cliquet option is equal to 25.2%, whereas a simple buy-and-hold investment in the underlying resulted in a return of 12.5% only. Thus,especially during turmoils in financial markets cliquet options are investmentsolutions that reduce downside risk while still offering upside potential.

To illustrate the in-depth product set-up of the cliquet contract a termsheet ofa cliquet note issued by Vontobel is provided below.Source: https://derinet.vontobel.com/CH/DE/Produkt/CH0100562179

5

Bank Vontobel AG, Gotthardstrasse 43, 8022 Zürich: Financial Products, Tel. +41 (0)58 283 78 88, Fax +41 (0)58 283 57 67

Banque Vontobel Genève SA, Place de l’Université 6, 1205 Genève: Financial Products, Tél. +41 (0)22 809 91 91, Fax +41 (0)22 809 90 99

Internet: http://www.derinet.com

VONTOBEL CLIQUET NOTE ON THE DOW JONES EURO STOXX 50® INDEX

CAPITAL PROTECTION 100% AT EXPIRY

2.00% ANNUAL MINIMUM COUPON

The Vontobel CLIQUET NOTE is characterized by capital protection at maturity and regular coupon payments. The coupon payments

are calculated on the annual coupon fixing dates, based on the sum of the monthly performances of the underlying (performance

component) and the minimum between 0 and the previous year performance component. In this respect, each monthly performance

has a cap. If the coupon calculated in this way is below the annual minimum coupon, then the annual minimum coupon is paid out.

This coupon calculation takes place annually.

PRODUCT INFORMATION

Issuer Vontobel Financial Products Ltd., DIFC Dubai

Lead Manager Bank Vontobel AG, Zurich

Calculation Agent Bank Vontobel AG, Zurich

Guarantor Vontobel Holding AG, Zurich (Standard & Poor's A; Moody's A2)

Underlying value Dow Jones EURO STOXX 50® Index (no di valore svizzero: 846 480)

Issue Price per Note EUR 1000.00

Notional per Note EUR 1000.00

Reference price EUR 2317.36

Initial fixing April 27, 2009

Payment date May 4, 2009

Last trading day April 24, 2014 (12:00 CET)

Final fixing April 28, 2014

Repayment date May 5, 2014

Underlying per Note (Ratio) 0.4315

Swiss Sec. No./ISIN 1005 6217 / CH0100562179

Telekurs Symbol VQDJB

CAPITAL PROTECTION (NOTES)

Capital protection per Note EUR 1000.00 (100% of the issue price)

Net present value EUR 963.47

Taxes From the technical taxation aspect these CLIQUET NOTES are seen as a transparent capital

protected product with a non-predominantly one-off interest payment ("Non-IUP").

Accordingly, the difference between the capital protection and the cash value (EUR 1000.00 –

EUR 963.47 = EUR 36.53) will be subject to income tax for private investors in Switzerland only

at final redemption, the minimum annual coupon (2.00% p.a.) however on its respective due

date. (IRR: 2.79%)

The minimum coupon exceeding disbursements represent tax free capital gain.

TERMSHEET VONTOBEL CLIQUET NOTE

SSPA DESIGNATION: CAPITAL PROTECTION WITH COUPON

+41 (0)58 283 78 88 or www.derinet.ch

Figure 5: Termsheet of a cliquet contract

6




TERMSHEET

SEITE 2

No Swiss withholding tax, no stamp duties at issuance. For Swiss stamp duty purpose, the

product is treated as analogous to a bond. Therefore, secondary market transactions are in

principle subject to Swiss stamp duty (TK22).

For Swiss paying agents this product is subject to the EU taxation of savings income in the form

of interest payments. The guaranteed minimum coupon is liable to tax.

The taxation mentioned above applies on the issue date. The tax legislation and Internal

Revenue Service practice can change at any time.

COUPONS

Coupon frequency Annual

Coupon fixing dates 27.04.2010; 27.04.2011; 27.04.2012; 29.04.2013; 28.04.2014

Ex-dates First trading day following fixing date

Minimum annual coupon 2.00%

Yield cap per month 0.85% (10.20% p.a.)

Calculation of the coupon On each coupon fixing date (initially 27.04.2010) the coupon will be calculated based on the

sum of the monthly performances of the underlying (performance component) and the minimum

between 0 and the previous year performance component (supplementary component). In this

respect, each monthly performance has a cap. If the coupon calculated in this way is below the

annual minimum coupon, then the annual minimum coupon is paid out.

Coupon formula The annual coupon for year t is:

51t,B(t)A(t) , 2.00% MAX C t L=+=

where:

0)0( =A

∑×

+−×= −⎟⎟⎠

⎞⎜⎜⎝

⎛−=

t

ti i

i

I

ICapMintA

12

1)1(12 1

1,)( "Performancekomponente"

( ))1(,0)( −= tAMintB "Supplementary component"

Cap = 0.85%

iI is the closing price of the underlying of the monthly observation date i

51t L= corresponding to 27.04.2010; 27.04.2011; 27.04.2012; 29.04.2013; 28.04.2014

The monthly observation date 0i = corresponds to the initial fixing date and the monthly

observation date 60i = corresponds to the closing fixing date. The monthly observation dates

are on the 27th day of each month, but if a particular day is not a business day, then the

monthly observation date is the first business day following the given date.

FURTHER INFORMATIONS

Reference Currency EUR

Issue Size 50'000 CLIQUET NOTES, the size may be increased any time

Repayment In addition to the last coupon, the investor will receive the capital protection per Note.

Secondary market The secondary market is guaranteed for the entire duration of the product.

The Cliquet Note is traded “flat”, that means accrued interest will be included in the price.

Clearing/Settlement SIX SIS, Euroclear, Clearstream

Sales restrictions USA, US persons, DIFC Dubai and United Kingdom

Listing Will be applied for in the main segment at the SIX Swiss Exchange.

Opportunities / Risks Vontobel CLIQUET NOTES give investors the opportunity to benefit from both capital protection

and participation in the performance of the equity index through regular coupons.

During the term the price can dip below the capital protection.

The value of structured products may depend not only on the development of the underlying

asset, but also on the credit rating of the issuer/guarantor. The investor is exposed to the

default risk of the issuer/guarantor.

Notice The product is not a collective investment within the meaning of the Federal Act on Collective

Investment Schemes (KAG); it is under no approval obligation and is not supervised by the Swiss

Financial Market Supervisory Authority FINMA (FINMA).

"Performance component"

7




TERMSHEET

SEITE 3

Publication of notices All notices to investors regarding products and changes in product conditions (because of

corporate actions, for example) are published at www.derinet.ch; under the rules relating

to IBL (Internet Based Listing), notices concerning products quoted on the SIX Swiss

Exchange are also published at www.six-swiss-exchange.com. Term Sheets are generally not

amended.

The original version of this term sheet is in German; versions in other languages are non-binding translations.

This term sheet does neither constitute a Listing Prospectus in the sense of the Listing Regulation nor an Issuing Prospectus in the sense of Art. 652a and/or 1156 OR.

The alone relevant complete conditions as well as the detailed risk references to this product are contained in the appropriate Listing prospectus.

The Listing prospectus can be ordered free of charge at the Bank Vontobel AG, Financial Products Documentation, Dreikönigstrasse 37, 8022 Zurich (Tel.: 058 283 78 88) or

www.derinet.ch.

We would be glad to answer any questions you may have concerning our products on +41 (0)58 283 78 88 from 08.00-20.00 CET on bank workdays. Please note that all

conversations on this line are recorded. By calling, we assume that you agree to this business practice.

The list and details provided do not represent a recommendation on the specified underlying security; they are for informative purposes only and under no circumstances are

they to be used or considered as an offer to sell or a solicitation of any offer to buy any financial instrument. No responsibility is assumed for the completeness and accuracy

of the information provided herein. The information provided herein is not meant as a substitute for a consultation with your house bank which we consider indispensable

prior to entering any kind of derivatives transaction. Transactions of this nature should only be conducted once investors are fully aware of the risks involved and are in a

position to bear the possible related financial losses. Furthermore, we refer to our brochure «Special Risks in Securities Trading», which we will send you free of charge on

request.

Dow Jones EURO STOXX 50® Index is owned by STOXX LIMITED and is a Service Mark of Dow Jones & Company Inc.

Zurich, April 27, 2009

8

1.2 Financial model and pricing concepts

The following subsection aims to give a short, yet comprehensive overview oncontinuous-time models and option pricing. For proofs and additional details itis recommended to follow [Sh04] or [Jo10].Let (Ω,A,P) be a complete probability space. Suppose that W = (Wt)t∈[0,T ] isam-dimensional Wiener process w.r.t. some filtration F = (Ft)t∈[0,T ], satisfyingthe usual conditions, with trivial F0 (augmented with null sets), and assumethat FT = A. The dynamics of the bond B = (Bt)t∈[0,T ] and stock pricesSi = (Sit)t∈[0,T ] (i = 1, ..., d and d ≤ m) are modeled according to

Bt = B0 exp

(∫ t

0

rsds

)and

Sit = Si0 exp

∫ t

0

µisds+

m∑j=1

(∫ t

0

σijs dW js −

1

2

∫ t

0

(σijs)2

ds

)with constant initial values B0, S

i0 > 0 for i = 1, ..., d.

(rt)t∈[0,T ], (µt)t∈[0,T ] and (σt)t∈[0,T ] are progressively measurable processes sat-isfying

rt ≥ 0

and ∫ T

0

(rt + ‖µt‖+ ‖σt‖2

)dt <∞ a.s.,

where ‖ · ‖ denotes the Euclidean norm. For each t ∈ [0, T ] it should hold thatσtσ>t is non-singular. Thus, the price processes are the unique Ito processes

satisfyingdBt = Btrtdt

and

dSit = Sit

µitdt+

m∑j=1

σijt dW jt

i = 1, ..., d,

associated with the constant initial values. The discount factor at t ∈ [0, T ] isdefined as

βt :=B0

Bt

and the corresponding discounted price process Si = (Sit)t∈[0,T ], i = 1, ..., d, aregiven by

Sit := βtSit .

Suppose that F is the filtration generated by W , augmented by the P-nullsets and that there exists a m-dimensional, progressively measurable processθ = (θt)t∈[0,T ] such that ∫ T

0

‖θt‖2dt <∞ a.s.,

9

σtθt = µt − rt1d for all t ∈ [0, T ] and

H = (Ht)t∈[0,T ] with

Ht = exp

(−∫ t

0

θ>s dWs −1

2

∫ t

0

‖θs‖2ds

)is a martingale under P.

A probability measure Q ∼ P is defined by the Radon Nikodym derivativeHT = dQ

dP . By the Girsanov Theorem

Wt = Wt +

∫ t

0

θsds for t ∈ [0, T ]

defines a m-dimensional Wiener Process W = (Wt)t∈[0,T ] under Q. A probabil-ity measure Q ∼ P under which the discounted price processes are local mar-tingales, is called an equivalent martingale measure or a risk neutral probabilitymeasure. Assuming d = m, the risk neutral measure is uniquely determined.A portfolio process (πB , π) = (πBt , πt)t∈[0,T ], πt = (π1

t , ..., πdt )>, is a (d + 1)-

dimensional, progressively measurable process with∫ T

0

(|πBt + π>t 1d||rt|+ |π>t (µt − rt1d)|+ ‖π>t σt‖2)dt <∞ a.s. .

πit is the amount of money invested in the ith stock and πBt is the amount of

money invested in the bond. The corresponding wealth process XπB ,π is definedby

XπB ,πt := πBt + π>t 1d

and the portfolio process is called self-financing, if for an initial capital x0 ∈ R

XπB ,πt = x0 +

∫ t

0

XπB ,πs rsds+

∫ t

0

π>s dRs.

The portfolio process is then called admissible, if the discounted wealth process

XπB ,π, defined by XπB ,πt := βtX

πB ,πt , satisfies XπB ,π

t ≥ −K for some con-stant K > 0 and all t ∈ [0, T ]. The market model is arbitrage-free, i.e. thereexists no self-financing, admissible portfolio process π for which Xπ

0 ≤ 0 andP(Xπ

T ≥ 0) = 1 and P(XπT > 0) > 0.

A contingent claim CT is an FT -measurable random variable for which βTCT ≥−K for some constant K > 0. An admissible, self-financing portfolio pro-cess π with Xπ

T ≥ CT a.s. is called a superhedging strategy for the contin-gent claim CT and the superreplication price of CT is defined as p(CT ) =infXπ

0 : π superhedging strategy for CT . CT is furthermore called attain-able, if p(CT ) < ∞ and there exists a superhedging strategy π with Xπ

T = CTa.s.. π is then called a replication strategy for CT . The presented market modelis complete, i.e. any claim with p(CT ) <∞ is attainable. In the financial mar-ket model with risk neutral measure Q it holds for any attainable claim thatp(CT ) = EQ[βTCT ] and there exists a martingale generating replication strat-egy π with Xπ

t = β−1t EQ[βTCT |Ft]. The arbitrage-free price of the contingentclaim CT at time t is then equal to

pt(CT ) := β−1t EQ[βTCT |Ft] for t ∈ [0, T ].

10

2 A semi-closed-form solution

In the following d = m = 1 and µ, r and σ > 0 are constant for t ∈ [0, T ], i.e.the complete, standard Black-Scholes market model is considered and the basicconcepts follow [BL13].

2.1 Pricing formula

00.1

0.20.3

0.40.5

0

0.01

0.02

0.031000

1050

1100

1150

1200

1250

1300

Volatility σLocal cap c

Opt

ion

pric

e

Figure 6: Cliquet option price surface w.r.t. σ and c

To derive a pricing formula for the price of the cliquet option, the expectationof the discounted payoff under the risk neutral measure Q has to be calculated.Thus, it is crucial to calculate

E

[max

(0,

n∑k=1

Zk

)],

where in the following the expectation is formed under Q.In this context, characteristic functions provide an useful tool to derive a semi-closed-form expression. The results below are based on [Jo13]. For a Rd-valuedrandom variable X the characteristic function of X is defined as

ϕX(t) := E[eit>X ], t ∈ Rd.

11

For independent, Rd-valued random variables X1, ..., Xn it holds that

ϕ∑nk=1 Xk

(t) =

n∏k=1

ϕXk(t).

Therefore, especially for i.i.d. Rd-valued random variables (Xk)nk=1 and X itholds that ϕ∑n

k=1 Xk(t) = ϕnX(t). A C-valued function f is called µ-integrable,

if Re(f) and Im(f) are µ-integrable. Then∫fdµ :=

∫Re(f)dµ+ i

∫Im(f)dµ

and it suffices to check, that |f | is integrable to verify the µ-integrability of f .

Especially, for f(t) = eit>X integrability w.r.t. a probability measure follows as

|eit>X | = 1.Denote by ϕZ the characteristic function of the random variable Z and letCT = max (0,

∑nk=1 Zk). Given the FT -measurability and boundedness of βT CT

from below, CT forms a contingent claim.

Proposition 2.1. The time-0 price p0 of the contingent claim CT is given by

p0(CT)

=ne−rT

2EZ +

e−rT

π

∫ ∞0

t−2 (1− Re (ϕnZ (t))) dt.

Proof. Let x =∑nk=1 Zk. Then max (0,

∑nk=1 Zk) = max (0, x). A simple case

analysis gives the representation max (0, x) = x+|x|2 . Now a result from [Ha82]

is applied: Note that a substitution of u = |x|t yields∫ ∞0

t−2(1− cos(xt))dt = |x|∫ ∞0

u−2(1− cos(u))du.

Partial integration now implies that∫ ∞0

u−2(1− cos(u))du =

∫ ∞0

sin(u)

udu =

π

2.

Therefore, |x| can be represented by

|x| = 2

π

∫ ∞0

t−2(1− cos(xt))dt.

Let ε be a Rademacher random variable, i.e. P(ε = ±1) = 12 under a probability

measure P. It follows that for a Rademacher random variable it holds that

EP [eixtε] = EP [cos(xtε)] + iEP [sin(xtε)] = cos(xt).

Then

|x| = 2

π

∫ ∞0

t−2(1− EP [eixtε])dt.

12

Thus

ΘZ : = E

[∣∣∣∣∣n∑k=1

Zk

∣∣∣∣∣]

=2

π

∫ ∞0

t−2(

1− EP

[E[ei(∑nk=1 Zk)tε

]])dt

=2

π

∫ ∞0

t−2 (1− EP [ϕnZ(tε)]) dt

=2

π

∫ ∞0

t−2 (1− Re (ϕnZ(t))) dt.

Here the positivity of the integrand as well as the boundedness of eitx enable toexchange the order of integration in the second step (e.g. [Jo13]). Combiningthe earlier steps and using that the Zk are i.i.d. random variables, the resultfollows immediately:

p0(CT ) = e−rTE

[max

(0,

n∑k=1

Zk

)]

=e−rT

2

(E

[n∑k=1

Zk

]+ E

[∣∣∣∣∣n∑k=1

Zk

∣∣∣∣∣])

=ne−rT

2EZ +

e−rT

π

∫ ∞0

t−2 (1− Re (ϕnZ(t))) dt.

The pricing formula for the cliquet option follows directly:

Theorem 2.2. The time-0 price p0 of the cliquet option with payoff

XT = K (1 + g) +K max

(0,

n∑k=1

Zk

)is given by

p0 (XT ) = K (1 + g) e−rT +Ke−rT

2(ΘZ + nEZ) ,

where ΘZ is defined as

ΘZ =2

π

∫ ∞0

t−2 (1− Re (ϕnZ (t))) dt.

Proof. Let XT = K (1 + g) +K max (0,∑nk=1 Zk). Given the FT -measurability

and boundedness of βTXT from below, it forms a contingent claim. Then

p0 (XT ) = e−rT(K (1 + g) +KE

[CT])

= e−rTK (1 + g) +Kp0(CT)

.

The foregoing proposition

p0(CT ) =ne−rT

2EZ +

e−rT

π

∫ ∞0

t−2 (1− Re (ϕnZ (t))) dt,

yields the assertion.

13

An application of the earlier result is possible, if the expression ΘZ is wellunderstood. This one consists of the characteristic function ϕZ . Therefore, ananalysis of the characteristic function is needed.

Proposition 2.3. The characteristic function ϕZ of Z is given by

ϕZ(t) := E[eitZ

]= e−it

gn

(1 + it

∫ c

0

eitxQ (R > x) dx

).

Proof. Let Y be a non-negative random variable with finite expectation. Theuse of Fubini’s theorem yields the following representation: (cf. [Jo13])

ϕY (t) : = E[eitY

]= E

[eit0 + it

∫ Y

0

eitxdx

]

= 1 + it

∫ ∞0

∫ y

0

eitxdxQ(dy)

= 1 + it

∫ ∞0

eitx∫ ∞x

Q(dy)dx

= 1 + it

∫ ∞0

eitxQ (Y > x) dx.

The random variable Z = max (0,min (c,R))− gn is - by construction - bounded

from below by − gn . This is a consequence of the local floor of the cliquet option

at 0. Therefore, the random variable Z + gn is non-negative. A case analysis

argument yields

Q(Z +

g

n> x

)=

0 if x > c

Q (R > x) if x ≤ c.

Furthermore,

ϕZ(t) = E[eitZ

]= e−it

gnE[eit(Z+ g

n )]

= e−itgnϕZ+ g

n(t).

By the use of the expression of the characteristic function of a non-negativerandom variable, in this case Z + g

n , it follows that

ϕZ(t) = e−itgnϕZ+ g

n(t) = e−it

gn

(1 + it

∫ c

0

eitxQ (R > x) dx

).

Denote the density of R under the risk neutral probability Q by fR. Then, amodified expression for the expectation of Z under Q is obtained.

Proposition 2.4. The expectation of Z under the risk neutral measure Q isgiven by

EZ =(c− g

n

)Q (R ≥ c) +

∫ c− gn

− gn

xfR

(x+

g

n

)dx− g

nQ (R < 0) .

14

Proof. A simple case analysis yields the distribution of the random variable Z,which is given by

Q (Z > x) =

0 if x > c− g

n

Q(R− g

n > x)

if − gn ≤ x ≤ c−

gn

1 if x < − gn

.

Thus, Z has a mixed distribution with to mass points at − gn and at c− g

n anda density over

[− gn , c−

gn

]. The expression of the expected value of Z then

follows immediately.

For a further analysis in the Black-Scholes setting, observe that the returns Rkfor k = 1, ..., n are independent and identically distributed random variables(e.g. [Jo10]). Under the risk neutral measure Q the returns can be representedas

Rk = exp

((r − σ2

2

)∆ + σ

(Wtk − Wtk−1

))− 1.

Letξ ∼ N

(mξ, σ

2ξ

)with

mξ =

(r − σ2

2

)∆

andσ2ξ = σ2∆.

ThenRk ∼ eξ − 1.

To price the cliquet options in the Black-Scholes model, the distribution of Zkhas to be calculated. The foregoing representation of the returns Rk = eξ − 1is used and denote by N (·) the cumulative normal distribution function. It isnow possible to calculate the three cases of the mixed distribution of Zk:

Q(Zk = c− g

n

)= Q (Rk ≥ c) = Q (ξ ≥ ln(1 + c)) = N

(mξ − ln(1 + c)

σ√

∆

),

fZ(x) =1

σ(x+ 1 + g

n

)√2π∆

e−(ln(x+1+

gn )−mξ)

2

2σ2∆ if x ∈(− gn, c− g

n

)and

Q(Zk = − g

n

)= Q (Rk ≤ 0) = Q (ξ ≤ 0) = N

(−mξ

σ√

∆

).

These results are now inserted in the expression of the expected value of therandom variable Z under the risk neutral measure Q:

15

EZ =(c− g

n

)Q (R ≥ c) +

∫ c− gn

− gn

xfR

(x+

g

n

)dx− g

nQ (R < 0)

=(c− g

n

)N(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

1

(y − 1− g

n

) 1√2π∆σy

e−(ln(y)−mξ)

2

2σ2∆ dy − g

nN(−mξ

σ√

∆

).

Analogously it holds for the characteristic function of the random variable Zthat

ϕZ(t) = e−itgnE[eit(Z+ g

n )]

= eit(c−gn )N

(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

1

eit(y−1−gn ) 1√

2π∆σye−

(ln(y)−mξ)2

2σ2∆ dy + e−itgnN

(−mξ

σ√

∆

).

All building blocks are now together to give a semi-closed formula for the time-0price p0 of the cliquet option with payoff

XT = K (1 + g) +K max

(0,

n∑k=1

Zk

)in the Black-Scholes model.

Theorem 2.5. In the Black-Scholes model the price of a cliquet option withpayoff

XT = K (1 + g) +K max

(0,

n∑k=1

Zk

)is given by

p0 (XT ) = K (1 + g) e−rT +Ke−rT

2

((2

π

∫ ∞0

t−2 (1− Re (ϕnZ (t))) dt

)+ nEZ

),

where

EZ =(c− g

n

)Q (R ≥ c) +

∫ c− gn

− gn

xfR

(x+

g

n

)dx− g

nQ (R < 0)

=(c− g

n

)N(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

1

(y − 1− g

n

) 1√2π∆σy

e−(ln(y)−mξ)

2

2σ2∆ dy − g

nN(−mξ

σ√

∆

)and

ϕZ(t) = eit(c−gn )N

(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

1


2π∆σye−

(ln(y)−mξ)2


(−mξ

σ√

∆

).

16

2.2 Vega of the cliquet option

0.10.2

0.30.4

0.5

0.01

0.015

0.02−1200

−1000

−800

−600

−400

−200

0


Veg

a

Figure 7: Vega surface of the cliquet option w.r.t. σ and c

To judge the sensitivity of the price of the cliquet option, partial derivatives ofthe option price with respect to various parameters are calculated (cf. [KK01]).Whereas responses to changes in the volatility parameter are well understoodin the case of plain vanilla options, it is first of all not clear how the price ofthe cliquet option is going to behave under such variations. It turns out thatcliquet options are quite sensitive with respect to changes in volatility. It istherefore natural to analyze the Vega of the cliquet option. The Vega of anoption is defined as the partial derivative of the option price with respect tothe volatility parameter σ. As Vega is no Greek letter, it is sometimes calledLambda. Therefore, it is also convenient to use the notation Λ(t) for the Vegaat time t.

Proposition 2.6. The Vega of the cliquet option at time 0 is given by

Λ(0) : =∂

∂σ(p0(XT ))

= Ke−rT

2

(∂

∂σΘZ + n

∂

∂σEZ

)= K

e−rT

2

((− 1

π

∫ ∞0

1

t2∂

∂σ(ϕnZ(t) + ϕnZ(−t)) dt

)+ n

∂

∂σEZ

).

Proof. The first expression follows immediately by differentiating

p0 (XT ) = K (1 + g) e−rT +Ke−rT

2(ΘZ + nEZ) .

17

The partial derivative of ΘZ with respect to the volatility σ is calculated asfollows

∂

∂σΘZ =

∂

∂σ

(2

π

∫ ∞0

t−2 (1− Re (ϕnZ (t))) dt

)=

∂

∂σ

(2

π

∫ ∞0

t−2(

1− ϕnZ(t) + ϕnZ(−t)2

)dt

)= − 1

π

∫ ∞0

1

t2∂

∂σ(ϕnZ(t) + ϕnZ(−t)) dt.

The expression is thus received by rewriting

Re (ϕnZ (t)) =ϕnZ(t) + ϕnZ(−t)

2

and interchanging integration and differentiation in the last step, which is al-lowed due to the positivity of the integrand, cf. [Jo13].

Hence, it is required to calculate the partial derivative of the characteristicfunction ϕZ with respect to the volatility parameter σ to obtain the Vega of thecliquet option. In the Black-Scholes model the partial derivative is calculatedas follows.

Proposition 2.7. The partial derivative of the characteristic function ϕZ withrespect to the financial parameter σ is given by

∂ (ϕZ(t))

∂σ= ϕσZ1

(t) + ϕσZ2(t) + ϕσZ3

(t),

where

ϕσZ1(t) =

ln(1 + c)−mξ − σ2∆

σ2√

2πTeit(y−1−

gn )e−

(mξ−ln(1+c))2

2σ2∆ ,

ϕσZ2(t) =

mξ + σ2∆

σ2√

2πTe−it

gn e−

m2ξ

2σ2∆

and

ϕσZ3

(t) =

∫ 1+c

1

1

σy√

2π∆eit(y−1− g

n

) − 1 + (ln(y) −mξ)

σ+

(ln(y) −mξ

)2σ3∆

e−

(ln(y)−mξ

)22σ2∆ dy.

Proof. The foregoing expression of the characteristic function is used to calculatethe partial derivative ∂

∂σϕZ :

ϕZ(t) = eit(c−gn )N

(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

1


2π∆σye−

(ln(y)−mξ)2


(−mξ

σ√

∆

).

18

2.3 Approximation

In the following, an approximation scheme is provided that enables to calculatethe integral in ΘZ . Such a method is needed, as an evaluation of the character-istic function of the truncated lognormal distribution of Z is not possible in anexplicit way, see [Le91]. However, in the cliquet contract the periodical returnsare capped at c. Thus, the support of each characteristic function is bounded.The two approximation steps are carried out below. The first step consists oftruncating the integral in ΘZ . In a second step the exponential function in thecharacteristic function is approximated by a finite Taylor series.

Approximation. Define the truncated version of ΘZ as

ΘZ(U) :=2

π

∫ U

0

t−2 (1− Re (ϕnZ (t))) dt.

A finite Taylor series approximation of ϕZ yields

ϕZ,u(t) := e−itgn

(1 + it

∫ c

0

u∑k=0

(itx)k

k!Q (R > x) dx

).

Then, define the approximation as

ΘZ(U, u) :=2

π

∫ U

0

t−2(1− Re

(ϕnZ,u (t)

))dt.

It is now possible to show that the convergence of the foregoing approximationto ΘZ is guaranteed, if u and U are chosen suitably large.

Proposition 2.8. For any positive integer U it holds, that

|ΘZ −ΘZ(U)| ≤ 4

Uπ.

For a fixed U , the convergence of the approximation follows:

limu→∞

|ΘZ(U)−ΘZ(U, u)| = 0.

Proof. The first result follows from the following inequality:

|ΘZ −ΘZ(U)| ≤∣∣∣∣ 2π∫ ∞U

t−2 (1− Re (ϕnZ (t))) dt

∣∣∣∣≤ 2

π

∫ ∞U

2

t2dt

≤ 4

Uπ.

According to Proposition 2.3, ϕZ can be expressed as

ϕZ(t) = e−itgn

(1 + it

∫ c

0

eitxQ (R > x) dx

).

By setting ϕZ,u equal to

ϕZ,u(t) = e−itgn

(1 + it

∫ c

0

u∑k=0

(itx)k

k!Q (R > x) dx

),

19

the following estimation is obtained:

|ϕZ(t)− ϕZ,u(t)| ≤ t∫ c

0

∣∣∣∣∣eitx −u∑k=0

(itx)k

k!

∣∣∣∣∣Q (R > x) dx

≤ 2tu+2

(u+ 1)!

∫ c

0

xu+1Q (R > x) dx

≤ 2tu+2

(u+ 2)!cu+2.

Hence,limu→∞

|ΘZ(U)−ΘZ(U, u)| = 0

follows by using the inequality hereafter and the fact that U and n are fixed.

|ΘZ(U)−ΘZ(U, u)| ≤ 2

π

∫ U

0

t−2∣∣Re (ϕnZ(t))− Re

(ϕnZ,u (t)

)∣∣dt≤ 2

π

∫ U

0

t−2∣∣ϕnZ (t)− ϕnZ,u (t)

∣∣dt≤ 2

π

∫ U

0

t−2 |ϕZ (t)− ϕZ,u (t)|

∣∣∣∣∣n−1∑k=0

ϕn−kZ (t)ϕkZ,u (t)

∣∣∣∣∣ dt≤ 2

π

∫ U

0

t−2 |ϕZ (t)− ϕZ,u (t)|n−1∑k=0

|ϕZ,u (t) |kdt

≤ 2

π

∫ U

0

t−2 |ϕZ (t)− ϕZ,u (t)|n−1∑k=0

(1 + |ϕZ (t)− ϕZ,u (t) |)k dt

≤ 2

π

∫ U

0

(2

tu

(u+ 2)!cu+2

) n−1∑k=0

(1 +

(2

tu+2

(u+ 2)!cu+2

))kdt.

A further investigation of the approximation is now carried out in the Black-Scholes setting. As shown before, the periodical returns Rk are lognormallydistributed in this framework, i.e.

Rk = eξ − 1

withξ ∼ N (mξ, σξ) .

For a local cap c and k > 0 define

µk := µ(c)k =

∫ c

0

kxk−1N(mξ − ln(1 + x)

σξ

)and assume, that

µ(c)0 = 1.

20

The approximated characteristic function of Z can now be rewritten as

ϕZ,u(t) : = e−itgn

(1 + it

∫ c

0

u∑k=0

(itx)k

k!Q (R > x) dx

)

= e−itgn

(1 + it

∫ c

0

u∑k=0

(itx)k

k!N(mξ − ln(1 + x)

σξ

)dx

)

= e−itgn

(1 +

∫ c

0

u∑k=0

(it)k+1

(k + 1)!(k + 1)xkN

(mξ − ln(1 + x)

σξ

)dx

)

= e−itgn

u+1∑k=0

(it)k

k!µk.

Hence,

ϕnZ,u(t) = e−itg

(u+1∑k=0

(it)k

k!µk

)n= e−itg

n(u+1)∑l=0

αl(it)l,

where

αl :=∑

(j1,...,jn)|∑ns=1 js=l

(n∏k=1

µjkjk!

).

The real part of ϕnZ,u is now expressed as

Re(ϕnZ,u(t)

)= cos(gt)

bn(u+1)2 c∑l=0

α2l(−1)lt2l + sin(gt)

bn(u+1)−12 c∑l=0

α2l+1(−1)lt2l+1.

Finally, it is possible to explicitly calculate the approximation

ΘZ(U, u) =2

π

∫ U

0

t−2(1− Re

(ϕnZ,u (t)

))dt.

Calculating the terms up to α2, the approximation is given as follows:

ΘZ(U, u) = ΘU,uZ,0 + ΘU,u

Z,1 + ΘU,uZ,2 + ΘU,u

Z,sin + ΘU,uZ,cos,

where

ΘU,uZ,0 =

2

π

∫ U

0

t−2 (1− cos(gt)) dt,

ΘU,uZ,1 = − 2

π

∫ U

0

sin(gt)α1

tdt,

ΘU,uZ,2 =

2

π

∫ U

0

cos(gt)α2dt,

21

ΘU,uZ,sin =

2

π

bn(u+1)−12 c∑l=1

∫ U

0

sin(gt)α2l+1(−1)l+1t2l−1dt

and

ΘU,uZ,cos =

2

π

bn(u+1)2 c∑l=1

∫ U

0

cos(gt)α2l(−1)l+1t2l−2dt.

Furthermore, the following simplifications and asymptotics for U →∞ are valid:

ΘU,uZ,0 =

2

π

∫ U

0

t−2 (1− cos(gt)) dt

= g − 2

π

∫ ∞U

t−2 (1− cos(gt)) dt

∼ g,

ΘU,uZ,1 = − 2

π

∫ U

0

sin(gt)α1

tdt

= −α1 +2

π

∫ ∞U

sin(gt)α1

tdt

∼ −α1

and

ΘU,uZ,2 =

2

π

∫ U

0

cos(gt)α2dt

=2

π

sin(gU)α2

g.

22

3 Numerical pricing

In the following chapter, the numerical pricing of a cliquet contract is analyzed.The contract and the parameters are specified as follows:

T 5n 60K 1,000g 0.30c 0.01r 0.02

These specification are set in such a way to represent a cliquet contract, heavilytraded in financial markets. To obtain the numerical result the pricing methodsare implemented in Matlab.

3.1 Implementing the semi-closed-form solution

The price of the cliquet option in the Black-Scholes model is given by Theorem2.5. Therefore, a numerical result is obtained via the implementation of thispricing formula. In the code, the procedure ”integral(·)” is used. This proce-dure approximates the integrals in the pricing formula by using global adaptivequadrature and default error tolerances.The integral ΘZ is approximated by ΘZ(1, 000), i.e. by setting the upperbound of integration equal to 1, 000, whereas the theoretical result demandsfor an unbounded integration. A numerical evaluation of the integral ΘZ(U)for varying U shows that the choice of U = 1, 000 is satisfactory. The followingchart illustrates the level of accuracy w.r.t. the upper bound of the integralΘZ(U). Choosing U larger than 1, 000 leads to an increased computationaleffort, which is not justified by the additionally achieved level of correctness.

23

0 400 800 1200 1600 20000.028

0.029

0.03

0.031

0.032

0.033

0.034

0.035

0.036

0.037

0.038

Upper bound U

ΘZ(U

)

Figure 8: Influence of the upper bound in the approximation ΘZ(U)

To verify the earlier statement on an additional computational effort for largerupper bounds, the running time of the integration algorithm has to be checked.By performing the evaluation for U ranging from 1, 000 to 2, 000, where eachcalculation is done 1, 000 times, the average time needed serves as an indicator ofthe computational effort. The experiment proves that the average time neededto calculate ΘZ(U) increases sharply w.r.t. U and also the time fluctuationsescalate for larger upper bounds. This result is presented in the chart below.

24

1000 1200 1400 1600 1800 20000.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Upper bound U

Ave

rage

tim

e to

cal

cula

te Θ

Z(U

)

Figure 9: Average time needed to calculate ΘZ(U)

Another crucial component of the formula is the normal distribution. In thecode, the procedure ”randn(·)” is used to generate normally distributed pseu-dorandom numbers.

25

In figure 10, the price of the cliquet contract is reported as function of thevolatility σ.

0 0.1 0.2 0.3 0.4 0.51015

1020

1025

1030

1035

Volatility σ

Tim

e−0

Pric

e of

the

Cliq

uet

Figure 10: Price of the cliquet contract w.r.t. σ

In this example, one recognizes, that the time-0 price is generally above theinitial investment of K = 1, 000. Furthermore, the graph displays a concavestructure. In particular, the price of the cliquet option is strongly increasing inσ for small values of the volatility. But the slope of the function changes sign atσ ≈ 0.30. Thus, a non-monotonicity of the price with respect to the volatilityparameter σ and a high sensitivity with respect to changes in this parametercan be observed. Therefore, the corresponding Vega has to be studied in moredetail.

3.2 Approximating the Vega of the option price

An advantage of using the semi-closed-form solution of the price of the cliquetoption is the fact, that the Vega of the option price is approximated quite easily.Recall, the Vega of the cliquet option is defined as the partial derivative of theoption price with respect to the volatility σ. Assuming that the option pricep0(XT ) is a function of σ (denoted by p0(XT )(σ)), the sensitivity with respectto the volatility can be approximated via finite differences (cf. [Ja02],[Gl03]).Therefore, set

Λ(0) ≈ p0(XT )(σ + κσ)− p0(XT )(σ)

κσ,

26

where κ is chosen sufficiently small - in this case κ = 0.01. Having calculatedthe price of the cliquet option with respect to the volatility σ in the first step,the Vega is thus obtained by just computing this fraction.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

0

100

200

300

400

500

600

700

Volatility σ

Tim

e−0

Veg

a of

the

Cliq

uet

Figure 11: Vega of the cliquet option w.r.t σ

Observe that the figure 11 fits the interpretation from the foregoing subchapter:the Vega of the cliquet option as function of σ is strictly decreasing, convex andchanges the sign for larger volatilities.

3.3 Monte Carlo technique I

In the following the price of the cliquet option is determined via Monte Carlosimulation, see [Ja02] and [Gl03]. This numerical technique consists of two

steps. First of all, A independent realizations(XiT

)Ai=1

of the final payoff XT

are simulated. In the second step

e−rT

(1

A

A∑i=1

XiT

)is chosen as an approximation for the time-0 price of the contract. It is clear,that the foundation of Monte Carlo simulation is the strong law of large numbers,which guarantees the convergence of the approximation to p0(XT ), the fair priceof the cliquet option. As the final payoff is a functional of the price process S, itis necessary to simulate the path (St)t∈[0,T ] under the risk neutral measure Q tosimulate XT . In the case of cliquet options, only the returns of the underlyingprice process contribute to the final payoff, thus the following approximationprocedure can be used for the purpose of simulating XT . First, n independent,

27

standard normally distributed random variables (Yk)nk=1 are generated. Thereturns (Rk)nk=1 are then simulated by

Rk = exp

((r − 1

2σ2

)T

n+ σ

√T

nYk

)− 1.

This procedure is now independently repeated for i = 1, ..., A to generate thereturns Rik for k = 1, ..., n and i = 1, ..., A in order to simulate the payoffs(XiT

)Ai=1

via

XiT = K max

(1 + g, 1 +

n∑k=1

max(0,min

(c,Rik

))).

The implementation of the Monte Carlo technique is thus straightforward. Therequired independent, normally distributed random variables are given by theMatlab procedure randn(·), which generates normally distributed pseudoran-dom numbers.In the examined pricing example A is chosen to be equal to 105. During theperformance of the Monte Carlo technique for the given contract specificationsand parameters, the standard deviation s of the realized final payoffs is approx-imately equal to

s :=

√√√√ 1

A− 1

A∑i=1

(XiT −

1

A

A∑i=1

XiT

)2

≈ 91.94.

28

In figure 12 the simulated price of the cliquet option is plotted as function of σ.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51154

1155

1156

1157

1158

1159

1160

1161

1162

1163

Cliq

uet O

ptio

n P

rice

Volatility σ


The basic principle for approximating the Vega presented in the foregoing chap-ter can also easily be applied in the concept of Monte Carlo simulation. Recallthat the Vega is approximated by

Λ(0) ≈ p0(XT )(σ + κσ)− p0(XT )(σ)

κσ.

It is therefore crucial to calculate p0(XT )(σ+ κσ) and p0(XT )(σ) in the MonteCarlo setting. To obtain the approximated Vega, the standard technique, calledpath recycling, is used. This method computes the prices of the cliquet optionat σ + κσ and σ by using the same random numbers.

29

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

0

100

200

300

400

500

600

700

800

Volatility σ

Tim

e−0

Veg

a of

the

Cliq

uet

Figure 13: Vega of the cliquet option w.r.t. σ

3.4 Comparison of the numerical techniques

A comparison of the results presented earlier is provided in the two figureshereafter. Observe that both pricing techniques lead to similiar prices and thesame particular behavior with respect to the volatility parameter can be seen.Whereas the price of the cliquet option is rather smooth in the semi-closed-formsolution approach, the Monte Carlo technique also shows some smaller devia-tions from a fully smooth function. Moreover, there is a tendency to slightlylower prices in the semi-closed-form solution.

30

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51154

1155

1156

1157

1158

1159

1160

1161

1162

1163C

lique

t Opt

ion

Pric

e

Volatility σ

Semi−Closed−Form SolutionMonte Carlo Technique


The same behavior can also be observed by investigating the Vega. The Vegaas function of the volatility σ is smooth in both cases. The Monto Carlo tech-niques leads to slightly lower Vegas, but the differences seem to be negligible.

31

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

0

100

200

300

400

500

600

700

Volatility σ

Tim

e−0

Veg

a of

the

Cliq

uet

Semi−Closed−Form SolutionMonte Carlo Technique

Figure 15: Vega of the cliquet option w.r.t. σ

32

In terms of a comparison of different pricing techniques another important per-formance measure is the computational effort needed to price the contracts.Therefore, in the following experiment the running time of the implementedsemi-closed-form solution and the Monte Carlo technique (with σ = 0.20 andA = 105) is measured 100 times each for the exemplary cliquet option contract.On average it takes 0.8156 min to price the contracts with a standard devia-tion of 0.0097 using the semi-closed-form solution. The pricing procedure usingthe Monte Carlo algorithm lasts slightly longer with 0.8725 min and a largerstandard deviation of 0.0209.

0 20 40 60 80 1000.75

0.80

0.85

0.90

Iteration

Tim

e to

cal

cula

te o

ptio

n pr

ice

Figure 16: Time needed using the semi-closed-form solution

The two charts enable to verify the running time differences in more detail.Besides the in general higher level of running time especially the higher fluctu-ations, when using the Monte Carlo technique are easily observed.

33

0 20 40 60 80 1000.75

0.8

0.85

0.9

0.95

1

Iteration

Tim

e to

cal

cula

te o

ptio

n pr

ice

Figure 17: Time needed using the Monte Carlo technique

It should be clear from the earlier considerations on cliquet options, that thenumber of periods n, specified in the contract, has an enormous impact on therunning time of the algorithms. Interestingly, both methods behave differently,when the number of periods is changed. In the experiment the pricing routinehas been run for n ranging from 1 to 60, where each contract is priced 100 times.In the following charts the average time to perform one pricing routine for thecliquet contract is compared.

34

0 20 40 600.5

1

1.5

2

2.5

3

3.5

Number of periods n

Ave

rage

tim

e to

cal

cula

te th

e op

tion

pric

e

Figure 18: Average time needed using the semi-closed-form solution

Surprisingly, an increase in the number of periods n leads to a drop in thetime needed to price the cliquet options. To price the contract with a singleperiod takes 4 times longer than pricing the monthly reseted cliquet option. Theopposite is true when using the Monte Carlo pricer. The running time increasesalmost linear from approximately 20 sec to more than 85 sec by increasing thenumber of periods. Thus, it seems to be reasonable to use the Monte Carlotechnique in the case of only few periods, whereas the semi-closed-form solutionis superior for a larger amount of periods.

35

0 20 40 600

0.25

0.5

0.75

1

Number of periods n

Ave

rage

tim

e to

cal

cula

te th

e op

tion

pric

e

Figure 19: Average time needed using the Monte Carlo technique

36

3.5 On the choice of g and c

In terms of product origination, it is crucial to understand the linkage betweenthe choice of g and c. By the construction of cliquet options an increase in theguaranteed rate g, ceteris paribus, shifts the distribution of possible outcomesupwards. On the other hand, if the local cap is decreased, then each periodi-cal outcome is more limited and thus also the final payoff is influenced. Thisparameter dependence is illustrated in the following two figures, where in eachcase all other factors are held constant.

00.1

0.20.3

0.40.5

0

0.01

0.02

0.031000

1050

1100

1150


Opt

ion

pric

e

Figure 20: Option price surface w.r.t. c and σ

37

00.1

0.20.3

0.40.5

0

0.05

0.11000

1020

1040

1060

1080

Volatility σGuaranteed rate g

Opt

ion

pric

e

Figure 21: Option price surface w.r.t. g and σ

A product issuer might now be interested in the contract design of a cliquetoption with a fixed initial price p0(XT ), where the issuer is allowed to set theparameters g and c. If the originator decides to offer a higher guaranteed rate,this has to be financed via a limitation on the periodical returns, i.e. c has tobe chosen sufficiently small. Vice versa, if the issuer is interested in originatinga cliquet contract with a higher local cap c, the guaranteed rate g has to bereduced in order to offer the same fixed initial price p0(XT ). As the parameterdependence is slightly non-linear, the figure 22 shows the linkage between thetwo components.

38

00.05

0.10.15

0.20.25

0.30.35

0.40.45

0.5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

Guaranteed rate gLocal cap c

Opt

ion

pric

e

Figure 22: Option price surface w.r.t. g and c

39

4 The sum cap contract - a similar product

The sum cap contract is a typical example of a globally floored and locallycapped contract. It thus appears to be a quite similar product compared tothe cliquet contract presented earlier. The only difference is the local floor inthe cliquet contract. This chapter provides a short overview on the semi-closed-form solution for the price of the the sum cap contract. The proofs of the resultsfollow the same pattern as in the chapter on cliquet options and are thus sup-pressed. The interested reader is advised to transfer the proofs presented in theearlier chapter to this product or follow [BL13].

4.1 Pricing formula of sum cap contracts

The payoff of the sum cap contract is given by

SCT = K max

(1 + g, 1 +

n∑k=1

min (c,Rk)

)

= K (1 + g) +K max

(0,

n∑k=1

Lk

),

where Lk is defined as

Lk := min (c,Rk)− g

nfor k = 1, ..., n.

As the increments of the underlying asset price process are i.i.d., the modifiedLk are i.i.d. random variables. Since their distribution does not depend on k,denote by L a corresponding i.i.d. random variable.

For illustration purposes a 5-year monthly sum cap contract on the S&P 500Index is considered - a similar experiment has been performed in the first chapteron cliquet options. Let the local cap be equal to c = 8.5% and the guaranteedrate is set to g = 10%. Based on a historical data set of the closing prices sinceJanuary 1999 a naive sampling algorithm with A = 105 trials is performed. Thefigure 23 displays the sampled distribution of the rate of return of the cliquetoption at maturity.

40

0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000

1500

2000

2500

3000

3500

4000

Rate of return at maturity

Fre

quen

cy

Figure 23: Sampled distribution of the rate of return at maturity

It is possible to derive a semi-closed-form solution for the price of a sum capcontract as in the case of cliquet options. Here, the time-0 price p0 of the sumcap contract is given by

p0 (SCT ) = K (1 + g) e−rT +Ke−rT

2(ΘL + nEL) ,

where ΘL is defined as

ΘL =2

π

∫ ∞0

t−2 (1− Re (ϕnL (t))) dt.

The characteristic function ϕL of L is given by

ϕL(t) := E[eitLk

]= e−it(1+

gn )(

1 + it

∫ 1+c

0

eitxQ (R ≥ x− 1) dx

).

The expectation of L under the risk neutral measure Q is equal to

EL =(c− g

n

)Q (R ≥ c) +

∫ c− gn

−1− gn

xfR

(x+

g

n

)dx.

In the Black-Scholes model the price of a sum cap contract with payoff

SCT = K (1 + g) +K max

(0,

n∑k=1

Lk

)

41

is given by

K (1 + g) e−rT +Ke−rT

2

((2

π

∫ ∞0

t−2 (1− Re (ϕnL (t))) dt

)+ nEL

),

where

EL =(c− g

n

)N(mξ − ln(1 + c)

σ√

∆

)+

∫ 1+c

0

(y − 1− g

n

) 1√2π∆σy

e−(ln(y)−mξ)

2

2σ2∆ dy

and

ϕL(t) = e−it(1+gn )(

1 + it

∫ 1+c

0

eitxN(mξ − ln(x)

σ√

∆

)dx

).

4.2 Numerical pricing of sum cap contracts

The aim is now to display differences between the particular behavior of theprices of sum cap and cliquet contracts. Therefore a numerical example isinvestigated. In the following a monthly sum cap contract specified by thefollowing parameters is considered.

T 5n 60K 1,000g 0.10c 0.085r 0.02

Having introduced the numerical pricing of cliquet options, the implementationof the semi-closed-form solution and also the Vega calculation of the sum capcontract are now straightforward. In figure 24, the price of the sum cap contractis reported as a function of the volatility parameter σ.

42

0 0.1 0.2 0.3 0.4 0.5980

1000

1020

1040

1060

1080

1100

Volatility σ

Tim

e−0

Pric

e of

the

sum

cap

con

trac

t

Figure 24: Price of the sum cap contract w.r.t. σ

Observe that the price has a very particular behavior with respect to the volatil-ity parameter σ. Interesting facts include the non-monotonicity and the changein the curvature of the price of the sum cap contract. Although the sum capand the cliquet contract seem to be quite similar at a first glance, it becomesobvious that the non-existence of a local floor in the sum cap contract, changesthe outcome dramatically.It is moreover interesting to examine the Vega of the contract. Of course, bythe foregoing figure the Vega changes the sign at σ ≈ 16.5%. But the shape ofthe curve - see figure 25 - shows an intriguing behavior. The curvature of theVega of the sum cap contract with respect to σ changes several times.

43

0 0.1 0.2 0.3 0.4 0.5−600

−400

−200

0

200

400

600

800

1000

Volatility σ

Tim

e−0

Veg

a of

the

sum

cap

con

trac

t

Figure 25: Vega of the sum cap contract w.r.t. σ

44

5 The influence of stochastic volatility

As presented in the analysis of the Vega of the cliquet option, the price of thecontract reacts quite sensitively to small changes in the volatility parameter.Although it is appealing that a semi-closed form solution can be derived in theBlack-Scholes market model, many assumption like constant volatility do notfind justifications in financial markets. Relaxing the assumption of constantvolatility might therefore be important to receive market-consistent prices andto account for the sensitivity of the option price to changes in volatility.

5.1 The Heston stochastic volatility model

The Heston model (introduced as in [Ha10], cf. [Ga06]) belongs to the class ofmodels, relaxing the constant volatility assumption by making volatility stochas-tic and thereby incorporating phenomena like volatility clustering. Among thestochastic volatility models this model stands out as a closed-form solution forEuropean call options is provided [He93].The dynamics of the stock price in the Heston model (St)t∈[0,T ] under the riskneutral measure Q are given by

dSt = St(rtdt+

√νtdW

St

)with constant initial level S0 ≥ 0. The stochastic variance process (νt)t∈[0,T ]

with constant initial value ν0 ≥ 0 is specified by

dνt = λ (ς − νt) dt+ γ√νtdW

νt .

In this setting (WS ,W ν) is a two-dimensional Wiener process under Q withinstantaneous correlation ρ, i.e.

dWSt dW ν

t = ρdt.

The model thus consists of the following parameters: the initial stock priceS0, the initial variance ν0, the long run variance ς ≥ 0, the mean reversionrate λ ≥ 0, the volatility of the variance γ ≥ 0 and the correlation, calledleverage parameter, ρS,ν with |ρS,ν | ≤ 1. The denomination of ρS,ν is justifiedas typically −1.0 < ρS,ν < −0.6, which implies that a decline in the stock pricecorrelates with a rise in the volatility, a phenomenon called leverage effect inthe traditional finance literature. For simplicity, it assumed that interest ratesare constant in the following, i.e. rt ≡ r.The figure below illustrates a possible evolution of the stock price and varianceprocess in the Heston model for 200 trading days. The parameters are chosenas follows:

S0 10v0 0.20r 0.02λ 0.50γ 0.15ς 0.20ρS,ν -0.70

45

0 20 40 60 80 100 120 140 160 180 200

10

12

14

Days

Sto

ck p

rice

S

Dynamics in the Heston stochastic volatility model

0 20 40 60 80 100 120 140 160 180 2000,15

0,20

0,25

Days

Var

ianc

e ν

Figure 26: Sample paths in the Heston model

The mean reverting property of the variance process as well as the leverage ef-fect can easily be noticed in this particular sample path.The variance process is well understood and several important results have beenshown (e.g. [BM01], [KT81]). For instance, the Feller condition guarantees thatthe process is strictly positive, if 2λς > γ2. On the other hand, if 2λς ≤ γ2, theorigin is accessible and strongly reflecting, i.e. the process will not stay at zero.Furthermore, the conditional distribution of the variance process is known tobe proportional to a non-central chi-squared distribution.

5.2 Stock price simulation in the Heston model

A naive implementation of the variance dynamics in order to simulate the stockprice in the Heston model might lead to difficulties. Using a Euler scheme forthe discretization of the dynamics might lead to negative values for the varianceprocess and one should be aware how to overcome these difficulties in a propersimulation scheme.For t > s ≥ 0 a naive Euler discretization of variance process leads to

νt = νs + λ(t− s) (ς − νs) + γ√νs(t− s)Yν ,

where Yν is a standard normally distributed random variable. Notice, that theprobability of νt becoming negative decreases in the time step t − s, but it isstrictly positive for every step size, unless γ = 0. In principal, there exist twobest practices: making zero an absorbing or reflecting boundary for the varianceprocess (cf. [Ha10]). The proposed, almost bias-free discretization scheme forthe variance, called full truncation, is specified by:

νt = νs + λ(t− s)(ς − ν+s

)+ γ

√ν+s (t− s)Yν .

Provided with the scheme for the variance process, the stock price process hasto be approximated. An application of the Ito formula yields the exact solution

46

of the stock price dynamics, which is given by

St = Ss exp

(∫ t

s

(r − 1

2νu

)du+

∫ t

s

√νudWS

u

).

Therefore, the following log-Euler scheme for the asset price process is obtained:

log (St) = log (Ss) +

(r − 1

2ν+s

)(t− s) +

√ν+s (t− s)YS ,

where YS is a normally distributed random variable with correlation ρS,ν to Yν .In an implementation the correlated random variables Yν and YS are generated

by setting Yν = Y1 and YS = ρS,νYν +√

1− ρ2S,νY2, where Y1 and Y2 are two

independent random samples of a standard normal distribution. In Matlab themultivariate normal random numbers generator mvnrnd(·, ·) is used for thispurpose. Thus, a simple and computing time efficient discretization scheme isimplemented.

5.3 Monte Carlo technique II

Having introduced the Monte Carlo technique to price cliquet options in theBlack-Scholes model, an application to the Heston model is now straightfor-ward. The simulation of the stock price is used to calculate the periodicalreturns, the further steps are identical as only the dynamics of the underlyingasset price process change.Consider a 5-year cliquet option with monthly resets. The local cap c is equalto 1% and the guaranteed rate g is set to 30%. In the Heston model, the meanreversion rate λ is set to 0.5 and the correlation ρ is given by −0.7. Moreover,suppose that ν0 = ς. In the experiment, the influence of ς and γ on the price ofthe cliquet option is analyzed. Therefore, the Monte Carlo technique is appliedfor ς varying from 0.01 to 0.20 and the values for γ range from 0 to 0.5. Theassociated option price surface is displayed below.

47

0.00.1

0.20.3

0.40.5

0.0

0.1

0.21176

1178

1180

1182

1184

Volatility of the variance γ

Option price surface

Long run variance ς

Pric

e of

the

cliq

uet o

ptio

n

Figure 27: Option price surface w.r.t. ς and γ

Under the assumption that ν0 = ς, the dynamics of the asset price processes inthe Black-Scholes and Heston model coincide for γ = 0 and σ =

√ν0. Therefore,

the price of the cliquet option for this special case is the same in both marketmodels. But, it is not clear how the price of the option is influenced by an in-crease in the volatility of the variance (leaving all other parameters unchanged).The simulation supports the intuition, that an increase in γ should result in alower fair value of the cliquet contract. In this example ν0 is chosen to be equalto 0.20.

48

0 0.1 0.2 0.3 0.4 0.51177

1178

1179

1180

1181

1182

1183

1184Comparision of the price of the cliquet option


Pric

e of

the

cliq

uet o

ptio

n

Price in the Heston modelPrice in the Black Scholes model

Figure 28: Option price in the Black-Scholes and Heston model w.r.t. γ

In the Monte Carlo approach the final payoffs of the contract are simulated. Foreach fixed value of γ the standard deviation of 105 simulated payoffs is calcu-lated. Interestingly, the same qualitative behavior as for the price of the optionis recognized for the standard deviation.

49

0 0.1 0.2 0.3 0.4 0.54

6

8

10

12

14

16


Sta

ndar

d de

viat

ion

Figure 29: Standard deviation of the simulated payoffs w.r.t. γ

Suppose now, in the Heston model it holds that γ = 0.25 for the variance pro-cess. Based on the earlier numerical example one might expect the price in theHeston model to be lower than the price in the Black-Scholes market. A varyinglong-term variance ς has a significant effect on this relationship. Therefore thedifference in the prices of the contracts in the both models w.r.t. ς = ν0 = σ2 isshown in the following figure. The graph has a humped shape with a maximumat ς ≈ 0.05.

50

0 0.1 0.2 0.3 0.4 0.5−1

0

1

2

3

4

5

6Price difference in the market models w.r.t. ς

Long run variance ς

Bla

ck−

Sch

oles

pric

e −

Hes

ton

pric

e

Figure 30: Difference of the option price in the market models w.r.t. ς

51

6 The influence of stochastic interest rates

The theory for pricing equity derivatives in the Black-Scholes model is based onthe assumption of deterministic interest rates. From a practitioner’s point ofview this simplification might be harmless in most situations as the variabilityof interest rates is negligible compared to the volatility in the equity markets.Nevertheless, when pricing long-dated options the fluctuations in interest rateshave a stronger impact on the fair price of a contingent claim. It is therefore ad-visable to relax the assumption of deterministic or even constant interest rates.

6.1 The Black-Scholes-Hull-White model

The Black-Scholes-Hull-White (BSHW) model (introduced as in [Ha10]) com-bines the Black-Scholes model for the dynamics of the asset price process andthe Hull-White model for the dynamics of the short rate. Under the risk neutralmeasure Q (using the bank account as numeraire), the dynamics of the stockprice in the BSHW model (St)t∈[0,T ] are given by

dSt = St(rtdt+ σdWS

t

)with constant initial level S0 ≥ 0. The short rate (rt)t∈[0,T ] follows an Ornstein-Uhlenbeck process with constant initial value r0, i.e.

drt = (τt − brt) dt+ ψdW rt .

In this setting (WS ,W r) is a two-dimensional Wiener process under Q withinstantaneous correlation ρ, i.e.

dWSt dW r

t = ρdt.

τt is a time-dependent, deterministic function, that describes the long-termmean level of the instantaneous interest rate and is in general chosen in such away, that it exactly fits the currently observed term structure of interest rates.The parameter b ≥ 0 is the speed of reversion and characterizes the velocity ofregrouping at τt. ψ is interpreted as the instantaneous volatility. Thus, the shortrate follows a mean-reverting, stationary, Gaussian and Markovian process (see[BM01]). For simplicity, it assumed that the long-term mean level is constantin the following, i.e. τt ≡ τ (the Vasıcek model for the dynamics of the shortrate).The figure below illustrates a possible evolution of the stock price and interestrate process in the BSHW model for 200 trading days. The parameters arechosen as follows:

S0 10r0 0.02τ 0.04b 2ψ 0.10σ 0.20ρS,r 0.00

52

0 20 40 60 80 100 120 140 160 180 2008

10

12

14

Days

Sto

ck p

rice

S

Dynamics in the BSHW model

0 20 40 60 80 100 120 140 160 180 200−0.1

0

0.1

Days

Sho

rt r

ate

r

Figure 31: Sample paths in the BSHW model

6.2 Stock price simulation in the BSHW model

The simulation of the processes in the BSHW model is straightforward. Fort > s ≥ 0 a naive Euler discretization of short rate process leads to

rt = rs + (τ − brs)(t− s) + ψ√

(t− s)Yr,

where Yr is a standard normally distributed random variable. The log-Eulerscheme then yields the discretization of the asset price process:

log (St) = log (Ss) +

(rs −

1

2σ2

)(t− s) + σ

√(t− s)YS ,

where YS is a normally distributed random variable with correlation ρS,r to Yr.

6.3 Monte Carlo technique III

After revisiting the Monte Carlo technique in the Heston model, the basic idea isobvious: the simulation of the stock price in the BSHW model is used to calcu-late the periodical returns to simulate the final payoffs of the cliquet contract.But in comparison to the Black-Scholes and the Heston model, not only thedynamics of the asset price change, also the discount factor has to be adapted.Remember, that for constant rates r, discounting is achieved by multiplying by

exp(−rT ). This corresponds to exp(−∫ T0rtdt

)for time-dependent rates. Hav-

ing simulated a sample path for the short rate, the discount factor is obtainedvia exp (−∆

∑nk=1 rtk). The resulting sample distribution of the discount factor

is displayed in the histogram below for A = 105 trials.

53

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

1000

2000

3000

4000

5000

6000

7000

8000Sample distribution of the discount factor

Discount factor

Fre

quen

cy

Figure 32: Sample distribution of the discount factor

The sample distribution of the non-discounted final payoffs of the cliquet con-tract can be found in the figure below.

1200 1250 1300 1350 1400 14500

1000

2000

3000

4000

5000

6000

7000

8000

9000Sample distribution of the final payoff

Final payoff

Fre

quen

cy

Figure 33: Sample distribution of the final payoff

The associated distribution of the time-0 price of the contract is displayed here-after.

54

600 800 1000 1200 1400 1600 1800 20000

1000

2000

3000

4000

5000

6000

7000

8000Sample distribution of the price of the cliquet option

Price of the cliquet option

Fre

quen

cy

Figure 34: Sample distribution of the price of the cliquet option

In the following, the sensitivity of the price of the cliquet option w.r.t. theinstantaneous volatility ψ is investigated.

0 0.1 0.2 0.3 0.4 0.51140

1160

1180

1200

1220

1240

1260

1280

1300

1320Price of the cliquet option

Instantaneous volatility ψ

Pric

e of

the

cliq

uet o

ptio

n

Figure 35: Price of the cliquet option w.r.t. ψ

55

In this example, the price of the cliquet contract is strictly increasing and convexin ψ. This qualitative behavior is unchanged for different correlations betweenthe stock price and short rate process. Nevertheless, there exist an influence ofthe correlation of the Wiener processes, which is displayed in the chart hereafter.

0 0.1 0.2 0.3 0.4 0.5−20

−15

−10

−5

0

5

10

15


Diff

eren

ce to

non

−co

rrel

ated

pro

cess

es

perfectly positive correlatedperfectly negative correlated

Figure 36: Difference of the option price due to correlation

If the processes are perfectly positive correlated the price difference to the non-correlated option price is positive and shows an upward-sloping trend, whereasin the negatively correlated case the opposite is true.In the following consider the model specified by the parameters hereafter.

S0 10r0 0.03τ 0.03b 2ψ 0.10σ 0.20ρS,r 0.00

The long-term mean level of the instantaneous interest rate is thus given byr = τ

b = 0.015. Therefore, it is interesting to check how the feature of astochastic interest rate changes the option price compared to pricing with theconstant rate r.

56

Interest rate model Option price Standard deviation

Vasıcek 1,211.4 15.09Constant rates 1,212.2 14.61

In this case, the price of the cliquet option is slightly lower in the Vasıcek model,whereas the standard deviation increases by 3.3%.

6.4 Further short rate models

Interest rate modeling was traditionally based on assumptions on the dynamicsof the short rate r. Besides the earlier considered time-homogeneous Vasıcekmodel, there is a wide variety of possible designs (cf. [BM01] for the resultshereafter). In this subchapter the influence of the specific model on the resultingoption price is analyzed. Recall that the dynamics of the short rate in theVasıcek model are given by

drt = (τ − brt) dt+ ψdW rt .

This model allows for non-positive interest rates and the short rate is normallydistributed. Therefore, it is interesting to check the pricing results in modelswith different properties. In the Cox-Ingersoll-Ross model the short rate isspecified as follows

drt = (τ − brt) dt+ ψ√rtdW

rt .

Here, the short rate is characterized by a non-central chi-squared distribution.For parameters ranging in a reasonable region the model admits positive ratesonly. In the Black-Karasinski model, the short rate is lognormally distributedand does not allow for non-positive interest rates. In this case, the dynamicsare given by

drt = rt (o− h ln rt) dt+ ψrtdWrt .

The figure below illustrates a possible evolution of the interest rate processesin the short rate model for 200 trading days. The parameters are chosen asfollows:

r0 0.02τ 0.03o -6b 2h 1.5ψ 0.10

57

0 50 100 150 200−0.1

−0.05

0

0.05

0.1V

asic

ekSample paths of the dynamics in the short rate models

0 50 100 150 2000

0.01

0.02

0.03

Cox

−In

gers

oll−

Ros

s

0 50 100 150 2000.01

0.02

0.03

Days

Bla

ck−

Kar

asin

ski

Figure 37: Sample paths in the short rate models

The assumptions on the short rate in the models lead to different pricing results.For the exemplary cliquet option contract the Monte Carlo approach yields thefollowing prices.

Interest rate model Option price Standard deviation

Vasıcek 1,214.4 12.63Cox-Ingersoll-Ross 1,209.5 14.06Black-Karasinski 1,191.6 14.09

Interestingly, there is an approximately 2% price difference between the Vasıcekand the Black-Karasinski model. Furthermore, the standard deviation in thepricing procedure increases slightly when the Vasıcek model is not used.Recall, an foregoing example showed that the price of the cliquet option isincreasing and convex in the instantaneous volatility ψ.

58

0 0.1 0.2 0.3 0.4 0.51200

1250

1300

1350

1400


Pric

e of

the

cliq

uet o

ptio

nPrice in the Vasicek model

Figure 38: Price w.r.t. ψ in the Vasıcek model

Observe that in this case the prices of the contract range from slightly above1, 200 to almost 1, 400. In the other models, the same linkage w.r.t. the param-eter ψ is illustrated in the two charts below.

59

0 0.1 0.2 0.3 0.4 0.51208

1210

1212

1214

1216


Pric

e of

the

cliq

uet o

ptio

nPrice in the Cox−Ingersoll−Ross model

Figure 39: Price w.r.t. ψ in the Cox-Ingersoll-Ross model

But in both cases the price range is tighter. Pricing the option in the Cox-Ingersoll-Ross model yields a 6 units wide price range (from 1, 209 to 1, 215). Inthe Black-Karasinki model, the lowest price is 1, 191 and reaches a maximum of1, 195.

60

0 0.1 0.2 0.3 0.4 0.51190

1192

1194

1196


Pric

e of

the

cliq

uet o

ptio

nPrice in the Black−Karasinski model

Figure 40: Price w.r.t. ψ in the Black-Karasinski model

The qualitative behavior w.r.t. the parameter ψ is thus unchanged in the dif-ferent short rate models, but the detailed analysis prevails the influences anddifferences of the models.

61

Conclusion

In a publication from 2002, P. Wilmott has already described cliquet optionsas ”the height of fashion in the world of equity derivatives” [Wi05]. As thesecontracts provide a downside protection while simultaneously offering an enor-mous upside potential, they might be a perfect fit during turmoils in financialmarkets. In the aftermaths of the subprime crisis, these contracts were rarelytraded and cliquet options are drawing attention more heavily only since 2010.Nevertheless, research literature on this topic is not widespread, neither in thefields of finance nor in mathematical areas, as the pricing concepts are ratherproprietarily developed by investment banks.This master’s thesis tries to capture recent research highlights on the pricingconcepts used for these type of contracts. The presented ansatz of developinga semi-closed-form solution for the option price might serve as starting pointfor further applications on various equity-linked derivatives. Besides the com-putational tractability, the formula enables to calculate hedging parameters likethe Vega of the option price easily. From an investor’s perspective as well asproduct issuer the choice of the local cap and the guaranteed rate is crucialand changes the pricing results vehemently. Moreover, the numerical examplesshowed the tremendous influence of the pricing parameters. These illustrationsprevail that exotic traders at investment banks face challenges when trying toprice these path-dependent structures. A reinforcement of these issues is showedin financial market models allowing for stochastic volatility and stochastic in-terest rates. The main difficulty in pricing and hedging the cliquet options ina non-Monte Carlo fashion is the design of a expression for the expectation aswell as for the characteristic function of the truncated returns of the underlyingasset.Cliquet options as equity-linked annuities provide interesting opportunities forinvestors - also in terms of insurance products. But as many aspects regardingthe pricing of these structured products are not clarified yet, further researchshould focus on these investment vehicles to offer an in-depth understanding ofthe contracts.

References

[BBG11] C. Bernard, P. P. Boyle, W. Gornall (2011): Locally-capped investmentproducts and the retail investor, The Journal of Derivatives, Summer2011, Vol. 18 (4): pp. 72-88

[BB11] C. Bernard, P. P. Boyle (2011): A natural hedge for equity indexedannuities, Annals of Actuarial Science, September 2011, Vol. 5 (2): pp.211-230

[BL13] C. Bernard, W. V. Li (2013): Pricing and hedging of cliquet contractsand locally-capped contracts, SIAM Journal on Financial Mathematics,April 2013, Vol. 4 (1): pp. 353-371

[BM01] D. Brigo, F. Mercurio (2001): Interest Rate Models - Theory and Prac-tice, Springer Finance

[Ga06] J. Gatheral (2006): The Volatility Surface - A Practitioner’s Guide,Wiley Finance

[Gl03] P. Glasserman (2003): Monte Carlo Methods in Financial Engineering,Springer Finance

[Ha82] U. Haagerup (1982): The best constants in the Khintchine inequality,Studia Mathematica, 1981, Vol. 70 (3): pp. 231-283

[Ha10] A. v. Haastrecht (2010): Pricing Long-Term Options with StochasticVolatility and Stochastic Interest Rate, PhD Thesis

[He93] S. L. Heston (1993): A closed-form solution for options with stochasticvolatility with applications to bond and currency options, The Review ofFinancial Studies, 1993, Vol. 6 (2): pp. 327-343

[Ja02] P. Jaeckel (2002): Monto Carlo Methods in Finance, Wiley Finance

[KT81] S. Karlin, H. M. Taylor (1981): A Second Course in Stochastic Pro-cesses, Gulf Professional Publishing

[KK01] E. Korn, R. Korn (2001): Option Pricing and Portfolio Optimization:Modern Methods of Financial Mathematics, Oxford University Press

[Le91] R. B. Leipnik (1991): On lognormal random variables: I-the charac-teristic function, The Journal of the Australian Mathematical Society,1991, Vol. 32 (Series B): pp. 327-347

[Pa06] B. A. Palmer (2006): Equity-Indexed Annuities: Fundamental Conceptsand Issues, Insurance Information Institute

[Wi05] P. Wilmott (2005): The Best of Wilmott 1: Incorporating the Quanti-tative Finance Review, Wiley Finance

[Jo10] J. Sass (2010): Financial Mathematics 1, Lecture Notes

[Jo13] J. Sass (2013): Probability Theory, Lecture Notes

[Sh04] S. E. Shreve (2004): Stochastic Calculus for Finance II, Springer Fi-nance

Statutory Declaration

Peter Warken

I hereby declare, that I have authored the thesis

Effective Pricing of Cliquet Options

independently and that I have not used other than the declared sources/resources.

Frankfurt am Main, December 6, 2015

Best regards,

Peter Warken

Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015

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