ADVANCED MONTE CARLO TECHNIQUES: AN APPROACH FOR FOREIGN EXCHANGE DERIVATIVE PRICING A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2007 Xiao Xiao School of Mathematics Contents Abstract 11 Declaration 12 Copyright 13 Acknowledgements 14 1 Option Pricing Theory 15 1.1 History of Derivative Securities . 16 1.2 Introduction to Options . 20 1.3 Option Pricing Fundamentals . 22 1.3.1 Arbitrage Pricing Method . 22 1.3.2 Equilibrium Pricing Method . 23 1.4 Black-Scholes-Merton Theory . 24 1.5 Option Pricing Implementation . 25 1.5.1 Analytical Solutions . 26 1.5.2 Numerical Techniques . 26 1.6 layout of the thesis . 29 2 Mathematical Preliminaries 31 2.1 Probability Space . 32 2.2 Random Variables . 32 2.2.1 Measurability . 32 2 2.2.2 Conditional Expectation . 33 2.2.3 Stopping Time . 33 2.3 Stochastic Processes . 34 2.3.1 Brownian Motions . 35 2.3.2 Poisson Processes . 37 2.3.3 Markov Processes . 37 2.3.4 Martingales . 38 2.4 Convergence and the Central Limit Theorem . 39 2.4.1 Convergence . 39 2.4.2 The Law of Large Numbers . 40 2.4.3 Central Limit Theorem . 41 2.5 Stochastic Integration . 41 2.5.1 It^o'sIntegral . 41 2.5.2 It^o'sIsometry . 42 2.6 Stochastic Di®erential Equations . 43 2.6.1 Geometric Brownian Motion . 43 2.6.2 Ornstein-Uhlenbeck Process . 44 2.6.3 Square-root Process . 44 2.7 It^o'sLemma . 45 2.7.1 One-dimensional Case . 45 2.7.2 Multi-dimensional Case . 46 2.8 Change of Measure . 47 2.8.1 Radon-Nikod¶ymDerivative . 47 2.8.2 Girsanov's Formula . 48 2.8.3 Equivalent Martingale Measure . 48 2.9 Summary . 49 3 Introduction to Foreign Exchange Markets 50 3.1 Overview of the Foreign Exchange Market . 50 3.1.1 Foreign Exchange Market Structure . 51 3 3.1.2 Participants in Foreign Exchange Market . 51 3.2 Introduction of Currency Options . 52 3.3 The Structure of Currency Option Models . 54 3.3.1 Exchange Rate Models . 54 3.3.2 Volatility Models . 55 3.3.3 Interest Rate Models . 56 3.4 Literature Review . 63 3.4.1 Review of Currency Option Modelling . 63 3.4.2 Review of Numerical Techniques . 72 4 Advanced Monte Carlo Methods 77 4.1 Introduction . 77 4.2 Monte Carlo Integration . 78 4.3 Basic Monte Carlo Simulation . 80 4.4 Variance Reduction Techniques . 82 4.4.1 Control Variates . 82 4.4.2 Antithetic Variates . 83 4.4.3 Importance Sampling . 84 4.4.4 Strati¯ed Sampling . 85 4.5 The Multi-dimensional Simulation . 86 4.6 Least-squares Monte Carlo Method . 89 4.6.1 Least-squares Fitting . 89 4.6.2 LSM Approach for American/Bermudan Options . 91 5 American Currency-Options 95 5.1 Introduction . 95 5.2 The Amin and Bodurtha Model . 97 5.2.1 Assumptions . 97 5.2.2 Numerical Scheme . 99 5.2.3 Numerical Results . 104 4 5.3 Improved Interest-rate Modelling . 107 5.3.1 Assumptions . 107 5.3.2 Numerical Scheme . 108 5.3.3 Numerical Results . 109 5.4 Further Improved Stochastic Volatility Modelling . 111 5.4.1 Assumptions . 111 5.4.2 Numerical Scheme . 113 5.4.3 Numerical Results . 113 5.5 Summary . 116 6 Discrete Barrier Currency-Options 120 6.1 Introduction . 121 6.2 European Up-and-Out Call Option . 122 6.2.1 Hedging Error . 123 6.2.2 Numerical Scheme . 124 6.2.3 Analysis of Error . 125 6.3 Discretely Monitored Up-and-out Call . 126 6.3.1 Hedging Error . 127 6.3.2 Numerical Scheme . 128 6.3.3 Analysis of Error . 129 6.4 Summary . 131 7 Quantile Parisian and ParAsian Options 133 7.1 Introduction . 133 7.2 Parisian and ParAsian Options . 136 7.2.1 Model Setup . 136 7.2.2 Assumptions . 138 7.2.3 Numerical Scheme . 139 7.2.4 Results and Analysis . 140 7.3 Quantile Barriers | A New Feature . 143 5 7.3.1 De¯nition . 145 7.3.2 Numerical Scheme . 146 7.3.3 Numerical Results . 146 7.3.4 Quantile Parisian and Quantile ParAsian . 148 7.4 Application to Currency Options . 154 7.5 Summary . 156 8 Conclusions 160 8.1 Summaries . 160 8.2 Future Research . 162 References 164 A Matlab \randn" Test 181 B Exchange Rate Process 183 Word count 42666 6 List of Tables 4.1 Comparison of Basic Monte Carlo method and Antithetic Monte Carlo method. 84 5.1 American/Bermudan currency-option valuation parameters I . 104 5.2 European put prices (comparison with the analytical solution) . 105 5.3 Comparison of tree method and enhanced LSM method for the Amin and Bodurtha Model . 107 5.4 American/Bermudan currency-option valuation parameters II . 109 5.5 American/Bermudan currency-option valuation parameters III . 114 6.1 Testing parameters . 126 7.1 Parisian and ParAsian options valuation parameters . 140 7.2 Permutations of the di®erent types of Parisian option . 145 7.3 Quantile level of a down-and-out option . 147 7.4 Quantile Parisian and ParAsian currency-option valuation parameters 155 A.1 Random number generator testing . 182 7 List of Figures 5.1 Amin and Bodurtha model for an at-the-money American put (K = x(0)) with 4000, 8000, 16000, 32000 sample paths . 105 5.2 Amin and Bodurtha model for an out-of-the-money American put (K = 0:95x(0)) with 4000, 8000, 16000, 32000 sample paths . 106 5.3 Extended model for an in-the-money American put (K = 1:05x(0)) with 4000, 8000, 16000, 32000 sample paths . 110 5.4 Extended model for an at-the-money American put (K = x(0)) with 4000, 8000, 16000, 32000 sample paths . 111 5.5 Extended model for an out-of-the-money American put (K = 0:95x(0)) with 4000, 8000, 16000, 32000 sample paths . 112 5.6 Stochastic volatility model for in-the-money American put (K = 1:05x(0)) with 4000, 8000, 16000, 32000 sample paths . 115 5.7 Stochastic volatility model for at-the-money American put (K = x(0)) with 4000, 8000, 16000, 32000 sample paths . 116 5.8 Stochastic volatility model for out-of-the-money American put (K = 0:95x(0)) with 4000, 8000, 16000, 32000 sample paths . 117 5.9 Influence of correlation parameters for an in-the-money American put 118 5.10 Influence of correlation parameters for an at-the-money American put 118 5.11 Influence of correlation parameters for an out-of-the-money American put . 119 6.1 Pro¯t and loss function for an up-and-out call option . 123 8 6.2 Mis-hedging error with one million units of domestic currency (U.S. dollar). ..