Quantitative Finance

List of Figures xv List of Tables xvii Part I Stochastic Processes and Finance 1 1 Stochastic Processes 3 1.1 Introduction 3 1.2 General Characteristics of Stochastic Processes 4 1.2.1 The Index Set I 4 1.2.2 The State Space S 4 1.2.3 Adaptiveness, Filtration, and Standard Filtration 5 1.2.4 Pathwise Realizations 7 1.2.5 The Finite Dimensional Distribution of Stochastic Processes 8 1.2.6 Independent Components 9 1.2.7 Stationary Process 9 1.2.8 Stationary and Independent Increments 10 1.3 Variation and Quadratic Variation of Stochastic Processes 11 1.4 Other More Specific Properties 13 1.5 Examples of Stochastic Processes 14 1.5.1 The Bernoulli Process (Simple Random Walk) 14 1.5.2 The Brownian Motion (Wiener Process) 17 1.6 Borel--Cantelli Lemmas 19 1.7 Central Limit Theorem 20 1.8 Stochastic Differential Equation 20 1.9 Stochastic Integral 21 1.9.1 Properties of the Stochastic Integral 22 1.10 Maximization and Parameter Calibration of Stochastic Processes 22 1.10.1 Approximation of the Likelihood Function (Pseudo Maximum Likelihood Estimation) 24 1.10.2 Ozaki Method 24 1.10.3 Shoji-Ozaki Method 25 1.10.4 Kessler Method 25 1.11 Quadrature Methods 26 1.11.1 Rectangle Rule: (n = 1) (Darboux Sums) 27 1.11.2 Midpoint Rule 28 1.11.3 Trapezoid Rule 28 1.11.4 Simpson's Rule 28 1.12 Problems 29 2 Basics of Finance 33 2.1 Introduction 33 2.2 Arbitrage 33 2.3 Options 35 2.3.1 Vanilla Options 35 2.3.2 Put-Call Parity 36 2.4 Hedging 39 2.5 Modeling Return of Stocks 40 2.6 Continuous Time Model 41 2.6.1 Itô's Lemma 42 2.7 Problems 45 Part II Quantitative Finance in Practice 47 3 Some Models Used in Quantitative Finance 49 3.1 Introduction 49 3.2 Assumptions for the Black-Scholes-Merton Derivation 49 3.3 The B-S Model 50 3.4 Some Remarks on the B-S Model 58 3.4.1 Remark 1 58 3.4.2 Remark 2 58 3.5 Heston Model 60 3.5.1 Heston PDE Derivation 61 3.6 The Cox-Ingersoll-Ross (CIR) Model 63 3.7 Stochastic alpha, beta, rho (SABR) Model 64 3.7.1 SABR Implied Volatility 64 3.8 Methods for Finding Roots of Functions: Implied Volatility 65 3.8.1 Introduction 65 3.8.2 The Bisection Method 65 3.8.3 The Newton's Method 66 3.8.4 Secant Method 67 3.8.5 Computation of Implied Volatility Using the Newton's Method 68 3.9 Some Remarks of Implied Volatility (Put-Call Parity) 69 3.10 Hedging Using Volatility 70 3.11 Functional Approximation Methods 73 3.11.1 Local Volatility Model 74 3.11.2 Dupire's Equation 74 3.11.3 Spline Approximation 77 3.11.4 Numerical Solution Techniques 78 3.11.5 Pricing Surface 79 3.12 Problems 79 4 Solving Partial Differential Equations 83 4.1 Introduction 83 4.2 Useful Definitions and Types of PDEs 83 4.2.1 Types of PDEs (2-D) 83 4.2.2 Boundary Conditions (BC) for PDEs 84 4.3 Functional Spaces Useful for PDEs 85 4.4 Separation of Variables 88 4.5 Moment-Generating Laplace Transform 91 4.5.1 Numeric Inversion for Laplace Transform 92 4.5.2 Fourier Series Approximation Method 93 4.6 Application of the Laplace Transform to the Black-Scholes PDE 96 4.7 Problems 99 5 Wavelets and Fourier Transforms 101 5.1 Introduction 101 5.2 Dynamic Fourier Analysis 101 5.2.1 Tapering 102 5.2.2 Estimation of Spectral Density with Daniell Kernel 103 5.2.3 Discrete Fourier Transform 104 5.2.4 The Fast Fourier Transform (FFT) Method 106 5.3 Wavelets Theory 109 5.3.1 Definition 109 5.3.2 Wavelets and Time Series 110 5.4 Examples of Discrete Wavelets Transforms (DWT) 112 5.4.1 Haar Wavelets 112 5.4.2 Daubechies Wavelets 115 5.5 Application of Wavelets Transform 116 5.5.1 Finance 116 5.5.2 Modeling and Forecasting 117 5.5.3 Image Compression 117 5.5.4 Seismic Signals 117 5.5.5 Damage Detection in Frame Structures 118 5.6 Problems 118 6 Tree Methods 121 6.1 Introduction 121 6.2 Tree Methods: the Binomial Tree 122 6.2.1 One-Step Binomial Tree 122 6.2.2 Using the Tree to Price a European Option 125 6.2.3 Using the Tree to Price an American Option 126 6.2.4 Using the Tree to Price Any Path-Dependent Option 127 6.2.5 Using the Tree for Computing Hedge Sensitivities: the Greeks 128 6.2.6 Further Discussion on the American Option Pricing 128 6.2.7 A Parenthesis: the Brownian Motion as a Limit of Simple Random Walk 132 6.3 Tree Methods for Dividend-Paying Assets 135 6.3.1 Options on Assets Paying a Continuous Dividend 135 6.3.2 Options on Assets Paying a Known Discrete Proportional Dividend 136 6.3.3 Options on Assets Paying a Known Discrete Cash Dividend 136 6.3.4 Tree for Known (Deterministic) Time-Varying Volatility 137 6.4 Pricing Path-Dependent Options: Barrier Options 139 6.5 Trinomial Tree Method and Other Considerations 140 6.6 Markov Process 143 6.6.1 Transition Function 143 6.7 Basic Elements of Operators and Semigroup Theory 146 6.7.1 Infinitesimal Operator of Semigroup 150 6.7.2 Feller Semigroup 151 6.8 General Diffusion Process 152 6.8.1 Example: Derivation of Option Pricing PDE 155 6.9 A General Diffusion Approximation Method 156 6.10 Particle Filter Construction 159 6.11 Quadrinomial Tree Approximation 163 6.11.1 Construction of the One-Period Model 164 6.11.2 Construction of the Multiperiod Model: Option Valuation 170 6.12 Problems 173 7 Approximating PDEs 177 7.1 Introduction 177 7.2 The Explicit Finite Difference Method 179 7.2.1 Stability and Convergence 180 7.3 The Implicit Finite Difference Method 180 7.3.1 Stability and Convergence 182 7.4 The Crank-Nicolson Finite Difference Method 183 7.4.1 Stability and Convergence 183 7.5 A Discussion About the Necessary Number of Nodes in the Schemes 184 7.5.1 Explicit Finite Difference Method 184 7.5.2 Implicit Finite Difference Method 185 7.5.3 Crank-Nicolson Finite Difference Method 185 7.6 Solution of a Tridiagonal System 186 7.6.1 Inverting the Tridiagonal Matrix 186 7.6.2 Algorithm for Solving a Tridiagonal System 187 7.7 Heston PDE 188 7.7.1 Boundary Conditions 189 7.7.2 Derivative Approximation for Nonuniform Grid 190 7.8 Methods for Free Boundary Problems 191 7.8.1 American Option Valuations 192 7.8.2 Free Boundary Problem 192 7.8.3 Linear Complementarity Problem (LCP) 193 7.8.4 The Obstacle Problem 196 7.9 Methods for Pricing American Options 199 7.10 Problems 201 8 Approximating Stochastic Processes 203 8.1 Introduction 203 8.2 Plain Vanilla Monte Carlo Method 203 8.3 Approximation of Integrals Using the Monte Carlo Method 205 8.4 Variance Reduction 205 8.4.1 Antithetic Variates 205 8.4.2 Control Variates 206 8.5 American Option Pricing with Monte Carlo Simulation 208 8.5.1 Introduction 209 8.5.2 Martingale Optimization 210 8.5.3 Least Squares Monte Carlo (LSM) 210 8.6 Nonstandard Monte Carlo Methods 216 8.6.1 Sequential Monte Carlo (SMC) Method 216 8.6.2 Markov Chain Monte Carlo (MCMC) Method 217 8.7 Generating One-Dimensional Random Variables by Inverting the cdf 218 8.8 Generating One-Dimensional Normal Random Variables 220 8.8.1 The Box-Muller Method 221 8.8.2 The Polar Rejection Method 222 8.9 Generating Random Variables: Rejection Sampling Method 224 8.9.1 Marsaglia's Ziggurat Method 226 8.10 Generating Random Variables: Importance Sampling 236 8.10.1 Sampling Importance Resampling 240 8.10.2 Adaptive Importance Sampling 241 8.11 Problems 242 9 Stochastic Differential Equations 245 9.1 Introduction 245 9.2 The Construction of the Stochastic Integral 246 9.2.1 Itô Integral Construction 249 9.2.2 An Illustrative Example 251 9.3 Properties of the Stochastic Integral 253 9.4 Itô Lemma 254 9.5 Stochastic Differential Equations (SDEs) 257 9.5.1 Solution Methods for SDEs 259 9.6 Examples of Stochastic Differential Equations 260 9.6.1 An Analysis of Cox-Ingersoll-Ross (CIR)-Type Models 263 9.6.2 Moments Calculation for the CIR Model 265 9.6.3 Interpretation of the Formulas for Moments 267 9.6.4 Parameter Estimation for the CIR Model 267 9.7 Linear Systems of SDEs 268 9.8 Some Relationship Between SDEs and Partial Differential Equations (PDEs) 271 9.9 Euler Method for Approximating SDEs 273 9.10 Random Vectors: Moments and Distributions 277 9.10.1 The Dirichlet Distribution 279 9.10.2 Multivariate Normal Distribution 280 9.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 281 9.11.1 Generating Gaussian Vectors 281 9.12 Problems 283 Part III Advanced Models for Underlying Assets 287 10 Stochastic Volatility Models 289 10.1 Introduction 289 10.2 Stochastic Volatility 289 10.3 Types of Continuous Time SV Models 290 10.3.1 Constant Elasticity of Variance (CEV) Models 291 10.3.2 Hull-White Model 292 10.3.3 The Stochastic Alpha Beta Rho (SABR) Model 293 10.3.4 Scott Model 294 10.3.5 Stein and Stein Model 295 10.3.6 Heston Model 295 10.4 Derivation of Formulae Used: Mean-Reverting Processes 296 10.4.1 Moment Analysis for CIR Type Processes 299 10.5 Problems 301 11 Jump Diffusion Models 303 11.1 Introduction 303 11.2 The Poisson Process (Jumps) 303 11.3 The Compound Poisson Process 304 11.4 The Black-Scholes Models with Jumps 305 11.5 Solutions to Partial-Integral Differential Systems 310 11.5.1 Suitability of the Stochastic Model Postulated 311 11.5.2 Regime-Switching Jump Diffusion Model 312 11.5.3 The Option Pricing Problem 313 11.5.4 The General PIDE System 314 11.6 Problems 322 12 General Lévy Processes 325 12.1 Introduction and Definitions 325 12.2 Lévy Processes 325 12.3 Examples of Lévy Processes 329 12.3.1 The Gamma Process 329 12.3.2 Inverse Gaussian Process 330 12.3.3 Exponential Lévy Models 330 12.4 Subordination of Lévy Processes 331 12.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 332 12.5.1 Rescaled Range Analysis (Hurst Analysis) 332 12.5.2 Detrended Fluctuation Analysis 334 12.5.3 Stationarity and Unit Root Test 335 12.6 Problems 336 13 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 337 13.1 Introduction 337 13.1.1 Stable Distributions 337 13.2 The Lévy Flight Models 339 13.2.1 Background 339 13.2.2 Kurtosis 343 13.2.3 Self-Similarity 345 13.2.4 The H - alpha Relationship for the Truncated Lévy Flight 346 13.3 Sum of Lévy Stochastic Variables with Different Parameters 347 13.3.1 Sum of Exponential Random Variables with Different Parameters 348 13.3.2 Sum of Lévy Random Variables with Different Parameters 351 13.4 Examples and Applications 352 13.4.1 Truncated Lévy Models Applied to Financial Indices 352 13.4.2 Detrended Fluctuation Analysis (DFA) and Rescaled Range Analysis Applied to Financial Indices 357 13.5 Problems 362 14 Approximating General Derivative Prices 365 14.1 Introduction 365 14.2 Statement of the Problem 368 14.3 A General Parabolic Integro-Differential Problem 370 14.3.1 Schaefer's Fixed Point Theorem 371 14.4 Solutions in Bounded Domains 372 14.5 Construction of the Solution in the Whole Domain 385 14.6 Problems 386 15 Solutions to Complex Models Arising in the Pricing of Financial Options 389 15.1 Introduction 389 15.2 Option Pricing with Transaction Costs and Stochastic Volatility 389 15.3 Option Price Valuation in the Geometric Brownian Motion Case with Transaction Costs 390 15.4 Stochastic Volatility Model with Transaction Costs 392 15.5 The PDE Derivation When the Volatility is a Traded Asset 393 15.5.1 The Nonlinear PDE 395 15.5.2 Derivation of the Option Value PDEs in Arbitrage Free and Complete Markets 397 15.6 Problems 400 16 Factor and Copulas Models 403 16.1 Introduction 403 16.2 Factor Models 403 16.2.1 Cross-Sectional Regression 404 16.2.2 Expected Return 406 16.2.3 Macroeconomic Factor Models 407 16.2.4 Fundamental Factor Models 408 16.2.5 Statistical Factor Models 408 16.3 Copula Models 409 16.3.1 Families of Copulas 411 16.4 Problems 412 Part IV Fixed Income Securities and Derivatives 413 17 Models for the Bond Market 415 17.1 Introduction and Notations 415 17.2 Notations 415 17.3 Caps and Swaps 417 17.4 Valuation of Basic Instruments: Zero Coupon and Vanilla Options on Zero Coupon 419 17.4.1 Black Model 419 17.4.2 Short Rate Models 420 17.5 Term Structure Consistent Models 422 17.6 Inverting the Yield Curve 426 17.6.1 Affine Term Structure 427 17.7 Problems 428 18 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization 431 18.1 Introduction 431 18.2 Exchange Traded Funds (ETFs) 431 18.2.1 Index ETFs 432 18.2.2 Stock ETFs 433 18.2.3 Bond ETFs 433 18.2.4 Commodity ETFs 433 18.2.5 Currency ETFs 434 18.2.6 Inverse ETFs 435 18.2.7 Leverage ETFs 435 18.3 Credit Default Swap (CDS) 436 18.3.1 Example of Credit Default Swap 437 18.3.2 Valuation 437 18.3.3 Recovery Rate Estimates 439 18.3.4 Binary Credit Default Swaps 439 18.3.5 Basket Credit Default Swaps 439 18.4 Mortgage Backed Securities (MBS) 440 18.5 Collateralized Debt Obligation (CDO) 441 18.5.1 Collateralized Mortgage Obligations (CMO) 441 18.5.2 Collateralized Loan Obligations (CLO) 442 18.5.3 Collateralized Bond Obligations (CBO) 442 18.6 Problems 443 Bibliography 445 Index 459