Computational Methods in Finance

Transcription

1 Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Computational Methods in Finance AM Hirsa Ltfi) CRC Press VV^ J Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK

2 List of Symbols and Acronyms List of Figures List of Tables xv xvii xxi Preface xxv Acknowledgments xxix I Pricing and Valuation 1 1 Stochastic Processes and Risk-Neutral Pricing Characteristic Function \ ' Cumulative Distribution Function via Characteristic Function Moments of a Random Variable via Characteristic Function Characteristic Function of Demeaned Random Variables Calculating Jensen's Inequality Correction Calculating the Characteristic Function of the Logarithmic of a Martingale Exponential Distribution Gamma Distribution Levy Processes Standard Normal Distribution Normal Distribution Stochastic Models of Asset Prices Geometric Brownian Motion Black-Scholes Stochastic Differential Equation Black-Scholes Partial Differential Equation Characteristic Function of the Log of a Geometric Brownian Motion Local Volatility Models Derman and Kani Stochastic Differential Equation Generalized Black-Scholes Equation Characteristic Function Geometric Brownian Motion with Stochastic Volatility Heston Model Heston Stochastic Volatility Model Stochastic Differential Equation 12 vn

3 viii Heston Model Characteristic Function of the Log Asset Price Mixing Model Stochastic Local Volatility (SLV) Model Geometric Brownian Motion with Mean Reversion Ornstein- Uhlenbeck Process Ornstein-Uhlenbeck Process Stochastic Differential Equation Vasicek Model Cox-Ingersoll-Ross Model Stochastic Differential Equation Characteristic Function oftntegral /T Variance Gamma Model Stochastic Differential Equation Characteristic Function CGMY Model Characteristic Function Normal Inverse Gaussian Model Characteristic Function Variance Gamma with Stochastic Arrival (VGSA) Model Stochastic Differential Equation Characteristic Function Valuing Derivatives under Various Measures ' Pricing under the Risk-Neutral Measure Change of Probability Measure Pricing under Forward Measure Floorlet/Caplet Price Pricing under Swap Measure Types of Derivatives 32 Problems 33 2 Derivatives Pricing via Transform Techniques Derivatives Pricing via the Fast Fourier Transform Call Option Pricing via the Fourier Transform Put Option Pricing via the Fourier Transform Evaluating the Pricing Integral Numerical Integration Fast Fourier Transform.' Implementation of Fast Fourier Transform Damping factor a Fractional Fast Fourier Transform Formation of Fractional FFT Implementation of Fractional FFT.. r Derivatives Pricing via the Fourier-Cosine (COS) Method COS Method Cosine Series Expansion of Arbitrary Functions Cosine Series Coefficients in Terms of Characteristic Function 56

4 ix COS Option Pricing COS Option Pricing for Different Payoffs Vanilla Option Price under the COS Method Digital Option Price under the COS Method...' Truncation Range for the COS method Numerical Results for the COS Method Geometric Brownian Motion (GBM) Heston Stochastic Volatility Model Variance Gamma (VG) Model CGMY Model Cosine Method for Path-Dependent Options Bermudan Options Discretely Monitored Barrier Options Numerical^Results COS versus Monte Carlo Saddlepoint Method Generalized Lugannani-Rice Approximation Option Prices as Tail Probabilities Lugannani-Rice Approximation for Option Pricing Implementation of the Saddlepoint Approximation Numerical Results for Saddlepoint Methods Geometric Brownian Motion (GBM) Heston Stochastic Volatility Model... : Variance Gamma Model \ CGMY Model Power Option Pricing via the Fourier Transform 76 Problems : Introduction to Finite Differences Taylor Expansion Finite Difference Method Explicit Discretization Algorithm for the Explicit Scheme Implicit Discretization Algorithm for the Implicit Scheme Crank-Nicolson Discretization Algorithm for the Crank-Nicolson Scheme Multi-Step Scheme Algorithm for the Multi-Step Scheme Stability Analysis Stability of the Explicit Scheme Stability of the Implicit Scheme Stability of the Crank-Nicolson Scheme Stability of the Multi-Step Scheme Derivative Approximation by Finite Differences: Generic Approach Matrix Equations Solver Tridiagonal Matrix Solver Pentadiagonal Matrix Solver 108

5 x Problems 110 Case Study ' Derivative Pricing via Numerical Solutions of PDEs Option Pricing under the Generalized Black-Scholes PDE Explicit Discretization Implicit Discretization Crank-Nicolson Discretization Boundary Conditions and Critical Points Implementing Boundary Conditions. 121 ' Dirichlet Boundary Conditions Neumann Boundary Conditions Implementing Deterministic Jump Conditions Nonuniform Grid Points..' Coordinate Transformation Black-Scholes PDE after Coordinate Transformation Dimension Reduction Pricing Path-Dependent Options in a Diffusion Framework Bermudan Options American Options Bermudan Approximation Black-Scholes PDE with a Synthetic Dividend Process Brennan-Schwartz Algorithm Barrier Options Single Knock-Out Barrier Options Single Knock-In Barrier Options Double Barrier Options Forward PDEs Vanilla Calls Down-and-Out Calls Up-and-Out Calls Finite Differences in Higher Dimensions Heston Stochastic Volatility Model Options Pricing under the Heston PDE Implementation of the Boundary Conditions Alternative Direction Implicit (ADI) Scheme Derivation of the Craig-Srieyd Scheme for the Heston PDE Heston PDE Numerical Results and Conclusion 161 Problems. 164 Case Studies Derivative Pricing via Numerical Solutions of PIDEs Numerical Solution of PIDEs (a Generic Example) Derivation of the PIDE Discretization 176

6 xi Evaluation of the Integral Term Difference Equation Implementing Neumann Boundary Conditions American Options Heaviside Term - Synthetic Dividend Process Numerical Experiments PIDE Solutions for Levy Processes Forward PIDEs American Options Down-and-Out and Up-and-Out Calls Calculation of g\ and g<i 198 Probfems 199 Case Studies 200 Simulation Methods for Derivatives Pricing Random Number Generation Standard Uniform Distribution Samples from Various Distributions Inverse Transform Method Acceptance-Rejection Method Standard Normal Distribution via Acceptance-Rejection Poisson Distribution via Acceptance-Rejection Gamma Distribution via Acceptance-Rejection Beta Distribution via Acceptance-Rejection Univariate Standard Normal Random Variables Rational Approximation Box-Muller Method Marsaglia's Polar Method...: Multivariate Normal Random Variables Cholesky Factorization Simulating Multivariate Distributions with Specific Correlations Models of Dependence Full Rank Gaussian Copula Model Correlating Gaussian Components in a Variance Gamma Representation Linear Mixtures of Independent Levy Processes Brownian Bridge Monte Carlo Integration Quasi-Monte Carlo Methods Latin Hypercube Sampling Methods Numerical Integration of Stochastic Differential Equations Euler Scheme Milstein Scheme Runge-Kutta Scheme Simulating SDEs under Different Models Geometric Brownian Motion 231

7 xii Ornstein-Uhlenbeck Process CIR Process Heston Stochastic Volatility Model Full Truncation Algorithm Variance Gamma Process Variance Gamma with Stochastic Arrival (VGSA) Process Output/Simulation Analysis Variance Reduction Techniques Control Variate Method Antithetic Variates Method Conditional Monte Carlo Methods 244 ' Algorithm for Conditional Monte Carlo Simulation Importance Sampling Methods Variance Reduction via Importance Sampling Stratified Sampling Methods Findings and Observations ^ Algorithm for Stratified Sampling Methods Common Random Numbers 253 Problems 254 II Calibration and Estimation Model Calibration Calibration Formulation General Formulation Weighted Least-Squares Formulation Regularized Calibration Formulations Calibration of a Single Underlier Model Black-Scholes Model Local Volatility Model Forward Partial Differential Equations for European Options Construction of the Local Volatility Surface Constant Elasticity of Variance (CEV) Model Heston Stochastic Volatility Model Mixing Model Stochastic Local Volatility (SLV) Model Variance Gamma Model CGMY Model Variance Gamma with Stochastic Arrival Model Levy Models Interest Rate Models Short Rate Models Vasicek Model Pricing Swaptions with the Vasicek Model Alternative Vasicek Model Calibration CIR Model Pricing Swaptions with the CIR Model 292

8 Alternative CIR Model Calibration Ho-Lee Model Hull-White (Extended Vasicek) Model Multi-Factor Short Rate Models Multi-Factor Vasicek Model Multi-Factor CIR Model CIR Two-Factor Model Calibration Pricing Swaptions with the CIR Two-Factor Model Alternative CIR Two-Factor Model Calibration Findings... : Affine Term Structure Models ' Forward Rate (HJM) Models Discrete-Time Version of HJM Factor Structure Selection LIBOR Market Models Credit Derivative Models Model Risk Optimization and Optimization Methodology Grid Search Nelder-Mead Simplex Method Genetic Algorithm Davidson, Fletcher, and Powell (DFP) Method Powell Method ' Using Unconstrained Optimization for Linear Constrained Input Trust Region Methods for Constrained Problems Expectation-Maximization (EM) Algorithm Construction of the Discount Curve LIBOR Yield Instruments : Simple Interest Rates to Discount Factors Forward Rates to Discount Factors Swap Rates to Discount Factors Constructing the Yield Curve Construction of the Short End of the Curve Construction of the Long End of the Curve Polynomial Splines for Constructing Discount Curves Hermite Spline Natural Cubic Spline Tension Spline Arbitrage Restrictions on Option Premiums Interest Rate Definitions 331 Problems 333 Case Studies Filtering and Parameter Estimation Filtering Construction of p(xfc zi ;fe ) Likelihood Function 345 xiii

9 xiv 8.3 Kalman Filter Underlying Model Posterior Estimate Covariance under Optimal Kalman Gain and Interpretation of the Optimal Kalman Gain Non-Linear Filters Extended Kalman Filter Unscented Kalman Filter Predict Update Implementation of Unscented Kalman Filter (UKF) Square Root Unscented Kalman Filter (SR_UKF) Particle Filter Sequential Importance Sampling (SIS) Particle Filtering Sampling Importance Resampling (SIR) Particle Filtering Problem of Resampling in Particle Filter and Possible Panaceas Markov Chain Monte Carlo (MCMC) 393 Problems 394 References 395 Index 409