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1 Computational Finance Using C and C# Derivatives and Valuation SECOND EDITION George Levy ELSEVIER AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an Imprint of Elsevier

2 Contents Preface 1 Overview of Financial Derivatives 2 Introduction to Stochastic Processes 2.1 Brownian Motion A Brownian Model of Asset Price Movements Ito's Formula (or Lemma) Girsanov's Theorem Ito's Lemma for Multi-Asset GBM Ito Product and Quotient Rules in Two Dimensions Ito Product Rule Ito Quotient Rule Ito Product in n Dimensions The Brownian Bridge Time Transformed Brownian Motion Scaled Brownian Motion Mean Reverting Process Ornstein Uhlenbeck Process The Ornstein Uhlenbeck Bridge Other Useful Results Fubini's Theorem Ito's Isometry Expectation of a Stochastic Integral Selected Exercises 31 3 Generation of Random Variates 3.1 Introduction Pseudo-Random and Quasi-Random Sequences Generation of Multivariate Distributions: Independent Variates Normal Distribution Lognormal Distribution Student's (-Distribution Generation of Multivariate Distributions: Correlated Variates Estimation of Correlation and Covariance Repairing Correlation and Covariance Matrices Normal Distribution Lognormal Distribution Selected Exercises 56 4 European Options 4.1 Introduction Pricing Derivatives using A Martingale Measure Put Call Parity Discrete Dividends 58 xvii vii

3 viii Contents Continuous Dividends Vanilla Options and the Black-Scholes Model The Option Pricing Partial Differential Equation The Multi-asset Option Pricing Partial Differential Equation The Black-Scholes Formula Historical and Implied Volatility Pricing Options with Microsoft Excel Barrien Options Introduction Analytic Pricing of Down and Out Call Options Analytic Pricing of Up and Out Call Options Monte Carlo Pricing of Down and Out Options Selected Exercises 90 5 Single Asset American Options 5.1 Introduction Approximations for Vanilla American Options American Call Options with Cash Dividends The Macmillan, Barone-Adesi, and Whaley Method Lattice Methods for Vanilla Options Binomial Lattice Constructing and using the Binomial Lattice Binomial Lattice with a Control Variate The Binomial Lattice with BBS and BBSR Grid Methods for Vanilla Options Introduction Uniform Grids Nonuniform Grids The Log Transformation and Uniform Grids The Log Transformation and Nonuniform Grids The Double Knockout Call Option Pricing American Options using a Stochastic Lattice Selected Exercises Multi-asset Options 6.1 Introduction The Multi-asset Black-Scholes Equation Multidimensional Monte Carlo Methods Introduction to Multidimensional Lattice Methods Two-asset Options European Exchange Options European Options on the Maximum or Minimum American Options Three-asset Options 193

4 Contents ix 6.7 Four-asset Optio ns Selected Exercises Other Financial Derivatives 7.1 Introduction Interest Rate Derivatives Forward Rate Agreement Interest Rate Swap Timing Adjustment Interest Rate Quantos Foreign Exchange Derivatives FX Forward European FX Option Credit Derivatives Defaultable Bond Credit Default Swap Total Return Swap Equity Derivatives TRS Equity Quantos Selected Exercises C# Portfolio Pricing Application 8.1 Introduction Storing and Retrieving the Market Data Equity Deal Classes Single Equity Option Option on Two Equities Generic Equity Basket Option Equity Barrier Option FX Deal Classes FX Forward Single FX Option FX Barrier Option Selected Exercises A Brief History of Finance 9.1 Introduction Early History The Sumerians Biblical Times The Greeks Medieval Europe Early Stock Exchanges The Anwterp Exchange 280

5 X A B C D E F Contents Amsterdam Stock Exchange Other Early Financial Centres Tulip Mania Early Use of Derivatives in the USA Securitisation and Structured Products Collateralised Debt Obligations The 2008 Financial Crisis The Collapse of AIC 297 The Greeks for Vanilla European Options A.1 Introduction 301 A.2 Gamma 302 A.3 Delta 303 A.4 Theta 303 A.5 Rho 304 A.6 Vega 305 Barrier Option Integrals B.1 The Down and Out Call 307 B.2 The Up and Out Call 310 Standard Statistical Results C.1 The Law of Large Numbers 315 C.2 The Central Limit Theorem 315 C.3 The Variance and Covariance of Random Variables 31 7 C.3.1 Variance 317 C.3.2 Covariance 319 C.3.3 Covariance Matrix 321 C.4 Conditional Mean and Covariance of Normal Distributions 321 C.5 Moment Generating Functions 323 Statistical Distribution Functions D.1 The Normal (Gaussian) Distribution 325 D.2 The Lognormal Distribution 327 D.3 The Student's t Distribution 328 D.4 The General Error Distribution 330 D.4.1 Value of A for Variance /z, 330 D.4.2 The Kurtosis 331 D.4.3 The Distribution for Shape Parameter, a 332 Mathematical Reference E.1 Standard Integrals 333 E.2 Gamma Function 333 E.3 The Cumulative Normal Distribution Function 334 E.4 Arithmetic and Geometrie Progressions 335 Black-Scholes Finite-Difference Schemes

6 Contents xi G H F.1 The General Case 337 F.2 The Log Transformation and a Uniform Grid 337 The Brownian Bridge: Alternative Derivation Brownian Motion: More Results H.1 Some Results Concerning Brownian Motion 345 H.2 Proof of Equation (H.1.2) 346 H.3 Proof of Equation (H.1.4) 347 H.4 Proof of Equation (H.1.5) 347 H.5 Proof of Equation (H.1.6) 347 H.6 Proof of Equation (H.1.7) 349 H.7 Proof of Equation (H.1.8) 349 H.8 Proof of Equation (H.1.9) 350 H.9 Proof of Equation (H.1.10) 350 I Feynman-Kac Formula I.1 Some Results 353 Glossary 355 Bibliography 357 Further Reading 359 Index 363

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