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Pricing FX Quanto Options underStochastic VolatilityA dissertation submitted to theWARWICK BUSINESS SCHOOLUNIVERSITY OF WARWICKin partial fulﬁllment of the requirements for the degree ofMSc in Financial Mathematicssubmitted byTANMOY NEOG (0850324)supervised byProf. Dr. Nick Webber7 September, 2009

All the work contained is my own unaided eﬀort and conforms to the University guidelines onplagiarism

AcknowledgmentI would like to begin by thanking my supervisor Prof. Nick Webber. I thank him for his patientguidance and his enthusiasm to answer my questions. This dissertation introduced me to theintricacies of derivative pricing. It was the enthusiasm of my supervisor that gave me the impetusto try to improve my results. Hopefully I did not do very badly. I also thank the WBS authoritieswho ensured that we had the adequate facilities to work eﬃciently.I thank all the faculty members involved with the Financial Mathematics course. I could learn alot from the rigour involved in this course. I thank my batchmates Vineet Thakkar, Zenon, RaviGanesan and Piyush Singh for several discussions related to Financial Mathematics. Comingfrom a background where I had no knowledge of Computational Finance; these individualshelped me to learn a lot of things in lesser time than I would have taken. Last but not the leastspecial thanks to my dear friend Vallu with whom there was never a dull moment.ii

ContentsList of FiguresviList of TablesviiiAbstractx1 Literature Review41.1Bennett and Kennedy’s methodology for pricing the FX Quanto Option . . . . .41.2Pricing the FX quanto option under diﬀerent frameworks . . . . . . . . . . . . .72 Pricing FX Quanto Options in the Black Scholes Framework2.114The standard market practice in pricing FX Quanto Options . . . . . . . . . . .142.1.115Pricing using the Black Scholes Model . . . . . . . . . . . . . . . . . . . .3 Pricing of FX Quanto under the Heston Model173.1Revisiting the Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173.2Option Pricing under the Heston model . . . . . . . . . . . . . . . . . . . . . . .183.2.1The Heston Pricer using Fast Fourier Transform . . . . . . . . . . . . . .193.2.2The Heston Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213.3The Heston model with jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . .iii23

3.4Pricing the FX Quanto option . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Pricing of FX Quanto under the GARCH Option Pricing Model24264.1Duan’s GARCH Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . .264.2Analytical Approximation of the GARCH Option Pricing Model . . . . . . . . .285 Modelling the Dependence Structure Using a Copula5.15.25.331Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335.1.1Archimedean Copulas and their Measures of Dependence . . . . . . . . .335.1.2Identifying the Right Copula from the Archimedean Family to Model theDependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35The T Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.2.1Calibration of the T Copula . . . . . . . . . . . . . . . . . . . . . . . . . .38Pricing using the copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .396 Data416.1Discount Factors, Implied Volatility quotes . . . . . . . . . . . . . . . . . . . . .416.2Recovering Strikes and Prices of FX vanilla options . . . . . . . . . . . . . . . . .427 Implementation of Pricing Methods7.145Implementation of the Heston Stochastic Volatility Model with Jumps . . . . . .457.1.1Calibration of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .457.1.2Finding Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . .49Implementation of the GARCH Option Pricing Model . . . . . . . . . . . . . . .507.2.1Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .537.3Choosing a model to price the Quanto option . . . . . . . . . . . . . . . . . . . .567.4Results of pricing the quanto option . . . . . . . . . . . . . . . . . . . . . . . . .567.2iv

7.57.6Implementing a copula to price the FX quanto option . . . . . . . . . . . . . . .587.5.1Parameter estimation for the T Copula . . . . . . . . . . . . . . . . . . .597.5.2Parameter estimation for the Frank Copula . . . . . . . . . . . . . . . . .607.5.3Calibration of marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . .607.5.4Numerical Results for pricing using copula . . . . . . . . . . . . . . . . . .62Monte Carlo Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .648 Conclusion69Bibliography71v

List of Figures5.1Frank Copula with 𝛼 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.2Clayton Copula with 𝛼 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375.3Gumbel Copula with 𝛼 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377.1Time series observations of USD/JPY spot exchange rate . . . . . . . . . . . . .467.2Market and model volatility(in %) for USD/YEN call options with maturity 1months7.350. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51Market and model volatility(in %) for USD/YEN call options with maturity 12months7.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Market and model volatility(in %) for USD/YEN call options with maturity 6months7.550Market and model volatility(in %) for USD/YEN call options with maturity 3months7.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51Market and model volatility(in %) for USD/YEN call options with maturity 24months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .527.7Market Implied Volatility Surface . . . . . . . . . . . . . . . . . . . . . . . . . . .537.8Heston with Jumps Implied Volatility Surface . . . . . . . . . . . . . . . . . . . .537.9Canonical Log likelihood function (Mashal and Zeevi) . . . . . . . . . . . . . . .607.10 Market and model volatility (in %) for EUR/USD call options with maturity 1months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi62

7.11 Market and model volatility(in %) for EUR/USD call options with maturity 6months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .627.12 Market and model volatility(in %) for EUR/USD call options with maturity 12months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .637.13 Market and model volatility(in %) for EUR/USD call options with maturity 2years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .637.14 Market and model volatility(in %) for EUR/JPY call options with maturity 1months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647.15 Market and model volatility(in %) for EUR/JPY call options with maturity 3months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647.16 Market and model volatility(in %) for EUR/JPY call options with maturity 6months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .657.17 Market and model volatility(in %) for EUR/JPY call options with maturity 2 years 657.18 Frank Copula with 𝛼 0.0075 . . . . . . . . . . . . . . . . . . . . . . . . . . . .667.19 T Copula with 𝜈 14 and 𝜌 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . .66vii

List of Tables5.1Archimedean Copulas and their Generators . . . . . . . . . . . . . . . . . . . . .335.2Archimedean Copulas and Measures of Dependence . . . . . . . . . . . . . . . . .346.1𝑖Discount factors 𝐷0,𝑇corresponding to currency i and maturity T . . . . . . . .426.2EUR/USD implied volatility quotes of call options corresponding to standardvalues of Black Delta and maturity T6.342EUR/JPY implied volatility quotes of call options corresponding to standardvalues of Black Delta and maturity T6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43USD/JPY implied volatility quotes of call options corresponding to standard. . . . . . . . . . . . . . . . . . . . . . . .437.1Parameters of Heston with jumps calibration on USD/JPY rates . . . . . . . . .487.2Parameters of GARCH calibration on USD/JPY rates . . . . . . . . . . . . . . .537.3Implied volatilities for maturity of 1 month . . . . . . . . . . . . . . . . . . . . .547.4Implied volatilities for maturity of 3 months . . . . . . . . . . . . . . . . . . . . .547.5Implied volatilities for maturity of 6 months . . . . . . . . . . . . . . . . . . . . .557.6Implied volatilities for maturity of 1 year. . . . . . . . . . . . . . . . . . . . . .557.7Implied volatilities for maturity of 2 years . . . . . . . . . . . . . . . . . . . . . .557.8Root mean squared errors for the implied volatility ﬁts on USD/JPY FX rate . .56values of Black Delta and maturity Tviii

7.9Quanto prices for maturity of 1 month. Spot price is 0.0083 dollars. All the pricesare in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . . . . .577.10 Quanto prices for maturity of 3 months.Spot price is 0.0083 dollars. All the pricesare in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . . . . .577.11 Quanto prices for maturity of 6 months. Spot price is 0.0083 dollars. All theprices are in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . .587.12 Quanto prices for maturity of 1 year. Spot price is 0.0083 dollars. All the callprices are in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . .587.13 Quanto prices for maturity of 2 year. Spot price is 0.0083 dollars. All the pricesare in the Euro currency.Strike prices in dollars. . . . . . . . . . . . . . . . . . . .597.14 Parameters for the T copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .597.15 Parameters for the Frank copula . . . . . . . . . . . . . . . . . . . . . . . . . . .607.16 Parameters for EUR/USD and EUR/JPY after calibration . . . . . . . . . . . .617.17 Root mean squared errors for the implied volatility ﬁts on EUR/USD and EUR/JPYFX rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .617.18 Relative Diﬀerence (RD)with Black Scholes for the models for maturity of 1month. Values have been rounded oﬀ. . . . . . . . . . . . . . . . . . . . . . . . .667.19 Relative Diﬀerence(RD)with Black Scholes for the models for maturity of 3 months.Values have been rounded oﬀ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .677.20 Relative Diﬀerence(RD)with Black Scholes for the models for maturity of 6 months.Values have been rounded oﬀ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .677.21 Relative Diﬀerence(RD)with Black Scholes for the models for maturity of 1 year.Values have been rounded oﬀ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .687.22 Average Standard Monte Carlo errors.Each error is an average of 25 simulationsfor 25 diﬀerent quanto options. . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix68

AbstractThis dissertation looks at the pricing of FX quanto options using stochastic volatility models.There are a lot of stochastic volatility models available. However we look only at the applicabilityof the Heston model with jumps and the GARCH Option Pricing Model. By a no arbitragecondition on FX rates we can view the FX quanto as a multi currency option. In a Black typeof model the pricing of the FX quanto depends on implied volatilities of three exchange rates.This encourages the use of a copula function to model the dependency. We do so by pricingthe quanto with a T Copula; member of the elliptic family and the Frank Copula; a member ofthe Archimedean Copula family with the marginals as Heston with jumps process. We observethat under stochastic volatility there is a considerable diﬀerence in option prices from the BlackScholes model. This is more so for options which have a low delta. Further given our data setthe Heston model with jumps ﬁts the market behaviour for plain vanilla FX options better thanthe GARCH Option Pricing model.x

IntroductionIn this dissertation we take up the problem of pricing a European style FX quanto optionunder stochastic volatility. An FX quanto option has as its underlying an exchange rate with adomestic and foreign currency. The payoﬀ at maturity is converted into a third currency. Thisthird currency is called the quanto currency. An individual would buy a quanto call option if hebelieves that the exchange rate would appreciate. Further he would expect the quanto currencyto appreciate more than the foreign currency for that particular exchange rate.In the past various authors have priced these options in a non stochastic volatility framework.The FX quanto is viewed as a multi asset option as volatility of the underlying exchange rateis dependent on the volatilities of the quanto domestic and quanto foreign rates. We have aclosed form solution under the Black Scholes model for pricing FX quanto options. There havebeen authors who have taken a diﬀerent approach such as employing a copula to model thedependence structure between exchange rates.(please refer to the literature review) The jointdistribution of asset prices is extracted from market implied volatilities. The price of the quantooption is then evaluated as an integral involving the joint density of asset prices.There have been expressions for quanto options under discrete time GARCH models which we1

consider as well. While pricing a multi asset product like a quanto option one needs to takeinto account multivariate models which can handle the co-movement of the underlying priceprocesses. The multivariate normal distribution is a very easy way of analyzing returns onmultiple assets. In a multivariate normal distribution the dependency between the margins ismeasured by linear correlation. The actual association between the diﬀerent assets may notbe so. The use of a copula is alternative measure of association between assets. The basicidea of copulas is to separate the dependence structure between variables from their marginaldistributions.Through the choice of copula, we can inﬂuence control on the association of certain parts of thedistribution.For instance at the tails. To give an example it may so happen that the returns oftwo stocks might be correlated in the extreme tails, but not elsewhere in the distributions, andthere are copulas which can model this behaviour.The aim of this dissertation would be to investigate as to how diﬀerent quanto prices are undera stochastic volatility framework. At ﬁrst we do not incorporate a dependency between thequanto, foreign and domestic rates. After this we do incorporate dependency through a copula.It is unlikely that single copula family model can take care of the asymmetry of the underlyingasset process. If not then the answer could lie in perturbation of the copula family as used byBennett and Kennedy (2003) or mixture copulas.I extend the standard FX quanto option pricing in two ways. First the marginals are assumed tofollow a Heston wih jumps process and the dependence is modelled using a copula. We choose at-copula from the elliptic family and the Frank copula from the Archimedean family. The choiceof a Frank copula is based on best ﬁt to historical data.2

When we model the individual FX rates as Geometric Brownian Motion as in the Black’sformulae the log returns follow a Normal distribution. From an empirical point of view weobserve that the log returns of exchange rates we consider in this paper have a high excesskurtosis of around 9. We need to replace GBM dynamics. Hence I take the marginals as nonlinear GARCH processes. The GARCH process allows fatter tails. We use the Duan’s GARCHOption Pricing Model for our purpose.When using a pricing model it is important for the model to ﬁt the market implied volatilityquotes for plain vanilla options. We observe that given our dataset the Heston with jumps modelgives us a much better ﬁt to market quotes as compared to the GARCH model.3

Chapter 1Literature ReviewOver the years valuation of contingent claims has been an area of extensive research. Theseminal work done by Black and Scholes (1973) and Cox et al. (1979) introduced us to riskneutral pricing models. However there is a shortcoming of these models. The assumption ofa ﬁxed volatility is violated in real time markets. The problem of pricing FX quanto optionsin a non stochastic volatility model has been elaborately explained by Bennett and Kennedy(2003). This paper is like a starting point for our dissertation. We give a brief review of themethodology adopted by the authors in pricing the FX quanto options.1.1Bennett and Kennedy’s methodology for pricing the FXQuanto OptionBenett and Kennedy express the payoﬀ of a plain vanilla FX option in terms of two otherexchange rates. This is obtained from the triangular no arbitrage condition for currencies. The4

price of this European multi asset option is written as an integral involving the joint densityof the asset prices at expiry. Hence the calculation of the price of the option involves theintegration of a joint density function of the two exchange rates. The authors then express thejoint density as a product of the marginal density functions of the individual exchange ratesand a copula function. This is done by using Sklar’s Theorem. The copula function determinesa joint density function of the two exchange rates. The use of the copula allows the authorsto separate the modelling of the marginal distributions from the modelling of the dependencestructure. The marginal densities are taken as mixture of lognormal distributions. The initialcalibration of marginal distributions to implied volatility quotes is done by a weighted nonlinear least squares optimisation. This process of using option implied densities is similar toDupire (1994). A Gaussian copula is then used which is perturbed by a cubic spline to get adependence structure between the three currency pairs.The modiﬁcation of the upper and lowertail dependence characteristics through this perturbation is to allow calibration to a smile inimplied volatilities of the FX rates.At the outset Bennett and Kennedy have considered a Gaussian copula. However to calibratethe joint distribution to the implied volatility smile on the FX rates the dependence structureassociated with the Normal copula is perturbed. With this perturbation the tail dependencecharacteristics are modiﬁed. The authors have used the following result which is due to Genest(2000):Theorem 1.1. Let 𝜑 : [0, 1] [0, 1] be a continuous, twice diﬀerentiable concave functionsuch that 𝜑(0) 0 and 𝜑(1) 1. Then𝐶𝜑 (𝑢, 𝑣) 𝜑 1 (𝐶(𝜑(𝑢), 𝜑(𝑣)))5(1.1)

is a copula if C(.,.) is a copula.For Benett and Kennedy the starting point is a bivariate Normal distribution with correlationparameter 𝜌. A transformation function is then used to modify tail dependence. I will give areview of how the authors obtained 𝜑 later. I proceed by giving a mathematical form of thetransformation function.The Normal Copula has the density function as𝐶Normal (𝑢, 𝑣 𝜌) 𝑁𝜌 (𝑁 1 (𝑢), 𝑁 1 (𝑣))(1.2)In equation (1.2) 𝑁𝜌 denotes the standard bivariate normal distribution with correlation 𝜌. 𝑁denotes the standard univariate Normal distribution function. To obtain the density of thetransformed copula 𝐶𝜑 the partial derivatives of equation (1.1) with respect to both arguementsare taken. Bennett and Kennedy obtain 𝑐𝜑 (𝑢, 𝑣), the density of the transformed copula as𝑐𝜑 (𝑢, 𝑣) 𝜑′′ (𝐶𝜑 (𝑢, 𝑣)) 𝑑𝐶 𝑑𝐶 )𝜑′ (𝑢)𝜑′ (𝑣) (𝑐(𝜑(𝑢),𝜑(𝑣)) 𝜑′ (𝐶𝜑 (𝑢, 𝑣)[𝜑′ (𝐶𝜑 (𝑢, 𝑣)]2 𝑑𝑢 𝑑𝑣(1.3)The ﬁnal step which we can observe from equation (1.3) is t

In this dissertation we take up the problem of pricing a European style FX quanto option under stochastic volatility. An FX quanto option has as its underlying an exchange rate with a domestic and foreign currency. The payoﬁ at maturity is converted into a third currency. This third currency is called the quanto currency.