Pricing of path-dependent basket options using a copula approach Christ Church University of Oxford A thesis submitted in partial fulfillment of the requirements for the MSc in Mathematical Finance i Abstract The pricing of basket options is usually a difficult task as assets of a basket usually show significant dependence structures which have to be incorporated appropriately in mathe- matical models. This becomes especially important if a derivative depends on the whole path of an option. In general pricing approaches linear correlation between the different assets are used to describe the dependence structure between them. This does not take into account that empirical multivariate distributions tend to show fat tails. One tool to construct multivariate distributions to impose a nonlinear dependence structure is the use of copula functions. In the thesis the general framework of the use of copulas and pricing of basket options using Monte Carlo simulation is presented. On the base of the general framework an algorithm for the pricing of path-dependent basket options with copulas is developed and implemented. This algorithm conducts the calibration of the model to market data and performs a simulation and estimates the fair price of a basket option. In order to investigate the impact of the use of different copulas and marginals the algorithm is applied to a selection of basket options. It is analyzed how the proposed alternative approach affects the fair price of the option. In particular, a comparison to standard approaches assuming multivariate normal distributions is made. The results show that the use and the choice of copulas and especially the choice of alternative marginals can have a significant impact on the price of the options. Contents 1 Introduction 1 2 Basket Options 2 2.1 Definition .................................... 2 2.2 Examples .................................... 3 2.3 Valuing ..................................... 5 2.3.1 Black-Scholes .............................. 6 2.3.2 FiniteDifferences ............................ 6 2.3.3 MonteCarlo............................... 6 2.4 Hedging ..................................... 7 3 Standard Pricing using Monte Carlo 9 4 Copulas 12 4.1 DefinitionsandBasicProperties . 13 4.2 Sklar’sTheorem................................. 15 4.3 MeasuresofAssociation ............................ 16 4.4 CopulaFamilies................................. 18 4.4.1 EllipticalCopulas............................ 18 4.4.2 ArchimedeanCopulas . 21 4.5 Estimation and Calibration from Market Data . ..... 27 4.6 SimulationMethodsforCopulas. 29 4.6.1 EllipticalCopulas............................ 29 4.6.2 ArchimedeanCopulas . 30 5 Monte Carlo Simulations with Copulas 33 6 Numerical Experiments 37 6.1 ExaminedOptions ............................... 37 6.2 Results...................................... 44 6.2.1 EstimationofParameters . 44 6.2.2 PricingoftheOptions . 45 7 Conclusions 53 A Student’s t-distribution 55 ii CONTENTS iii B Maximum Likelihood Method 56 List of Figures 4.1 Gaussiancopula................................. 19 4.2 RandomdrawsfromaGaussiancopula . 19 4.3 Student’st-copula............................... 20 4.4 RandomdrawsfromaStudent’st-copula . 21 4.5 Gumbelcopula ................................. 23 4.6 RandomdrawsfromaGumbelcopula. 24 4.7 Claytoncopula ................................. 24 4.8 Random draws from a Clayton copula . 25 4.9 Frankcopula .................................. 26 4.10 RandomdrawsfromaFrankcopula. 26 6.1 Scatterplot of the returns in the observed time series . ........ 41 6.2 Histograms of the observed returns for the underlying assets . ....... 42 6.3 Q-Q plots of the observed returns against the normal distribution . 43 iv List of Tables 4.1 Selected conditional transforms for copula generation . ........... 32 6.1 Statistics on historical returns of the Bayer/BASF basket . ........ 38 6.2 Statistics on historical returns of the BMW/VW basket . ..... 39 6.3 Parametersofexaminedoptions . 40 ˆ marginals ˆ marginal ˆ marginal 6.4 Parameters Φ of the underlyings Bayer(Φ1 )/BASF (Φ2 )(daily monitoring) ................................... 44 ˆ marginals ˆ marginal ˆ marginal 6.5 Parameters Φ of the underlyings Bayer (Φ1 )/BASF (Φ2 ) (weeklymonitoring)............................... 44 ˆ marginals ˆ marginal ˆ marginal 6.6 Parameters Φ of the underlyings BMW (Φ1 )/VW (Φ2 ) (weeklymonitoring)............................... 45 ˆ marginals ˆ marginal ˆ marginal 6.7 Parameters Φ of the underlyings BMW (Φ1 ) /VW (Φ2 )(dailymonitoring)............................... 45 6.8 Parameters Φcopula of the basket with underlyings BMW/VW (weekly mon- itoring)...................................... 46 6.9 Parameters Φcopula of the basket with underlyings BMW/VW (daily mon- itoring)...................................... 46 6.10 Parameters Φcopula of the basket with underlyings Bayer/BASF (weekly monitoring) ................................... 46 6.11 Parameters Φcopula of the basket with underlyings Bayer/BASF (daily mon- itoring)...................................... 47 6.12 Prices of basket options for underlyings Bayer/BASF in Euro (daily mon- itoring)...................................... 48 6.13 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings Bayer/BASF in Euro (daily monitoring) . ...... 49 6.14 Prices of basket options for underlyings BMW/VW in Euro (daily moni- toring)...................................... 49 6.15 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings BMW/VW in Euro (daily monitoring) . 50 6.16 Prices of basket options for underlyings BMW/VW in Euro (weekly mon- itoring)...................................... 50 6.17 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings BMW/VW in Euro (weekly monitoring) . 51 6.18 Prices of basket options for underlyings Bayer/BASF in Euro (daily mon- itoring)...................................... 51 v LIST OF TABLES vi 6.19 Differences of the prices in % compared to the Gaussian/Gaussian approach for underlyings Bayer/BASF in Euro (weekly monitoring) . ...... 52 Chapter 1 Introduction Many models in finance assume that asset returns are normally distributed. Usually, lin- ear correlation is the chosen measure of dependence between risky assets. The problem with linear correlation is that equities’ prices exhibit a greater tendency to crash together than to boom simultaneously. A number of empirical papers have shown that Gaussian distributions do not fit return data well (Mashal and Zeevi [2002], Dobric and Schmid [2005]). This implies that the standard models face some problems when used to calculate fair prices, sensitivities, hedge ratios, etc. for financial derivatives. Copulas are a proposed framework to model dependence between random variables (RVs), which are able to capture different properties of dependence structures. A copula gener- alizes linear correlation as a measure of dependence. If returns are normally distributed, then variance of the returns is a commonly used measure of risk, and linear correlation describes dependence. Copulas allow the construction of joint distributions which specify the distributions of individual returns separately from each other and separate from the dependence structure. This increases the flexibility in specifying distributions of multiple random variables. The aim of this thesis is to show how copulas can be incorporated in Monte Carlo simula- tions and to study the impact of this amendment in comparison to the standard models. To study the impact path-dependent basket options are used. Therefore, the second Chap- ter is devoted to basket options, as it gives definitions and outlines the standard methods for dealing with theses kind of options. The standard pricing approach with Monte Carlo simulations is described in more depth in the third Chapter. The fourth Chapter is a general introduction to the theory behind copulas. The term copula is defined and basic properties of copulas are described. Some examples of common copulas are given and mathematical methods used for copulas are outlined. On the basis of the general frame- work a modification of the standard Monte Carlo simulation which enables to use copulas to model different dependence structures of the returns is proposed in the fifth Chapter. The developed model is applied to different basket options in the sixth Chapter and the impact on the fair price of the option when pricing path-dependent basket options on the price of the option is investigated. In the seventh Chapter the results are summarized. 1 Chapter 2 Basket Options 2.1 Definition Definitions and classifications of basket options overlap one another and in the literature numerous definitions and classifications of basket options can be found. They often overlap other options such as Mountain Range options and Rainbow options because of their multi-asset characteristic. This thesis builds on the following general definition: Definition 2.1 (Basket Option) A Basket Option is an option whose payoff depends on the value of a portfolio (or basket) of assets. In general, the corresponding assets are related. The payoff p of a path-dependent basket options depends on the underlying assets at specified points in time tj. Therefore, the payoff depends on time t and on the values of the underlying assets Si, i = 1,...,n at the monitored points in time tj, j = 1,...,m. th Let S~i = (Si(t0),Si(t1) ...,Si(tm)) denote the set of prices of the i underlying at the monitored points in