Implementing Models in Quantitative Finance: Methods and Cases

Transcription

1 Gianluca Fusai Andrea Roncoroni Implementing Models in Quantitative Finance: Methods and Cases vl Springer

2 Contents Introduction xv Parti Methods 1 Static Monte Carlo Motivation and Issues Issue 1: Monte Carlo Estimation Issue 2: Efficiency and Sample Size Issue 3: How to Simulate Samples Issue 4: How to Evaluate Financial Derivatives The Monte Carlo Simulation Algorithm Simulation of Random Variables Uniform Numbers Generation Transformation Methods Acceptance-Rejection Methods Hazard Rate Function Method Special Methods Variance Reduction Antithetic Variables Control Variables Importance Sampling Comments 39 2 Dynamic Monte Carlo Main Issues Continuous Diffusions Method I: Exact Transition Method II: Exact Solution Method III: Approximate Dynamics 46

3 viii Example: Option Valuation under Alternative Simulation Schemes Jump Processes Compound Jump Processes Modelling via Jump Intensity Simulation with Constant Intensity Simulation with Deterministic Intensity Mixed-Jump Diffusions Statement of the Problem Method I: Transition Probability Method II: Exact Solution Method III.A: Approximate Dynamics with Deterministic Intensity Method III.B: Approximate Dynamics with Random Intensity Gaussian Processes Comments 66 3 Dynamic Programming for Stochastic Optimization Controlled Dynamical Systems The Optimal Control Problem The Bellman Principle of Optimality Dynamic Programming Stochastic Dynamic Programming Applications American Option Pricing Optimal Investment Problem Comments 81 4 Finite Difference Methods Introduction Security Pricing and Partial Differential Equations Classification of PDEs From Black-Scholes to the Heat Equation Changing the Time Origin Undiscounted Prices From Prices to Returns Heat Equation Extending Transformations to Other Processes Discretization Setting Finite-Difference Approximations Grid Explicit Scheme Implicit Scheme Crank-Nicolson Scheme Computing the Greeks 109

4 ix 4.4 Consistency, Convergence and Stability General Linear Parabolic PDEs Explicit Scheme Implicit Scheme Crank-Nicolson Scheme A VBA Code for Solving General Linear Parabolic PDEs Comments Numerical Solution of Linear Systems Direct Methods: The LU Decomposition Iterative Methods Jacobi Iteration: Simultaneous Displacements Gauss-Seidel Iteration (Successive Displacements) SOR (Successive Over-Relaxation Method) Conjugate Gradient Method (CGM) Convergence of Iterative Methods Code for the Solution of Linear Systems VBA Code MATLABCode Illustrative Examples Pricing a Plain Vanilla Call in the Black-Scholes Model (VBA) Pricing a Plain Vanilla Call in the Square-Root Model (VBA) Pricing American Options with the CN Scheme (VBA) Pricing a Double Barrier Call in the BS Model (MATLAB and VBA) Pricing an Option on a Coupon Bond in the Cox-Ingersoll- Ross Model (MATLAB) Comments Quadrature Methods Quadrature Rules Newton-Cotes Formulae Composite Newton-Cotes Formula Gaussian Quadrature Formulae Matlab Code Trapezoidal Rule Simpson Rule Romberg Extrapolation VBA Code Adaptive Quadrature Examples Vanilla Options in the Black-Scholes Model Vanilla Options in the Square-Root Model Bond Options in the Cox-Ingersoll-Ross Model 190

5 X Discretely Monitored Barrier Options Pricing Using Characteristic Functions MATLAB and VBA Algorithms Options Pricing with Levy Processes Comments The Laplace Transform Definition and Properties Numerical Inversion The Fourier Series Method Applications to Quantitative Finance Example Example Comments Structuring Dependence using Copula Functions Copula Functions Concordance and Dependence Frechet-Hoeffding Bounds Measures of Concordance ' Measures of Dependence Comparison with the Linear Correlation Other Notions of Dependence Elliptical Copula Functions Archimedean Copulas Statistical Inference for Copulas Exact Maximum Likelihood Inference Functions for Margins Kernel-based Nonparametric Estimation Monte Carlo Simulation Distributional Method Conditional Sampling Compound Copula Simulation Comments 265 Part II Problems Portfolio Management and Trading Portfolio Selection: "Optimizing" an Error Problem Statement Model and Solution Methodology Implementation and Algorithm Results and Comments In-sample Analysis 281

6 xi Out-of-sample Simulation Alpha, Beta and Beyond Problem Statement Solution Methodology Constant Beta: OLS Estimation Constant Beta: Robust Estimation Constant Beta: Shrinkage Estimation Constant Beta: Bayesian Estimation Time-Varying Beta: Exponential Smoothing Time-Varying Beta: The Kalman Filter Comparing the models Implementation and Algorithm Results and Comments Automatic Trading: Winning or Losing in a kbit Problem Statement Model and Solution Methodology Measuring Trading System Performance Statistical Testing Code Results and Comments 322 Vanilla Options Estimating the Risk-Neutral Density Problem Statement Solution Methodology Implementation and Algorithm Results and Comments An "American" Monte Carlo Problem Statement Model and Solution Methodology Implementation and Algorithm Results and Comments Fixing Volatile Volatility Problem Statement Model and Solution Methodology Analytical Transforms Model Calibration Implementation and Algorithm Code Description Results and Comments 362

7 xii Exotic Derivatives An Average Problem Problem Statement Model and Solution Methodology Moment Matching Upper and Lower Price Bounds Numerical Solution of the Pricing PDE Transform Approach Implementation and Algorithm Results and Comments Quasi-Monte Carlo: An Asian Bet Problem Statement Solution Metodology Stratification and Latin Hypercube Sampling Low Discrepancy Sequences Digital Nets The Sobol' Sequence Scrambling Techniques Implementation and Algorithm Results and Comments Lookback Options: A Discrete Problem Problem Statement Model and Solution Methodology Analytical Approach Finite Difference Method Monte Carlo Simulation Continuous Monitoring Formula Implementation and Algorithm Results and Comments Electrifying the Price of Power Problem Statement The Demand Side The Bid Side The Bid Cost Function The Bid Strategy A Multi-Period Extension Solution Methodology Implementation and Experimental Results A Sparkling Option Problem Statement Model and Solution Methodology 444

8 II xiii 19.3 Implementation and Algorithm Results and Comments Swinging on a Tree Problem Statement Model and Solution Methodology Implementation and Algorithm Gas Price Tree Backward Recursion Code Results and Comments 464 Interest-Rate and Credit Derivatives Floating Mortgages Problem Statement and Solution Method Fixed-Rate Mortgage Flexible-Rate Mortgage Implementation and Algorithm Markov Control Policies Dynamic Programming Algorithm Transaction Costs Code Results and Comments Basket Default Swaps Problem Statement Models and Solution Methodologies Pricing nth-to-default Homogeneous Basket Swaps Modelling Default Times Monte Carlo Method A One-Factor Gaussian Model Convolutions, Characteristic Functions and Fourier Transforms The Hull and White Recursion Implementation and Algorithm Monte Carlo Method Fast Fourier Transform Hull-White Recursion Code Results and Comments Scenario Simulation Using Principal Components Problem Statement and Solution Methodology Implementation and Algorithm Principal Components Analysis 508

9 xiv Code Results and Comments 511 Financial Econometrics Parametric Estimation of Jump-Diffusions Problem Statement Solution Methodology Implementation and Algorithm The Continuous Square-Root Model The Mixed-Jump Square-Root Model Results and Comments Estimating a Continuous Square-Root Model Estimating a Mixed-Jump Square-Root Model Nonparametric Estimation of Jump-Diffusions Problem Statement Solution Methodology Implementation and Algorithm Results and Comments A Smiling GARCH Problem Statement Model and Solution Methodology Implementation and Algorithm Code Description Results and Comments 554 A Appendix: Proof of the Thinning Algorithm 557 B Appendix: Sample Problems for Monte Carlo 559 C Appendix: The Matlab Solver 563 D Appendix: Optimal Control 569 D. 1 Setting up the Optimal Stopping Problem 569 D.2 Proof of the Bellman Principle of Optimality 570 D.3 Proof of the Dynamic Programming Algorithm 570 Bibliography 573 Index 599