Stochastic simulation and applications in finance with MATLAB programs / Huu Tue Huynh, Van Son Lai and Issouf Soumaré.

Accompanied by: 1 computer disc (CD-ROM)

Includes bibliographical references (pages 327-338) and index.

1. Introduction to Probability -- 1.1. Intuitive Explanation -- 1.1.1. Frequencies -- 1.1.2. Number of Favorable Cases Over The Total Number of Cases -- 1.2. Axiomatic Definition -- 1.2.1. Random Experiment -- 1.2.2. Event -- 1.2.3. Algebra of Events -- 1.2.4. Probability Axioms -- 1.2.5. Conditional Probabilities -- 1.2.6. Independent Events -- 2. Introduction to Random Variables -- 2.1. Random Variables -- 2.1.1. Cumulative Distribution Function -- 2.1.2. Probability Density Function -- 2.1.3. Mean, Variance and Higher Moments of a Random Variable -- 2.1.4. Characteristic Function of a Random Variable -- 2.2. Random vectors -- 2.2.1. Cumulative Distribution Function of a Random Vector -- 2.2.2. Probability Density Function of a Random Vector -- 2.2.3. Marginal Distribution of a Random Vector -- 2.2.4. Conditional Distribution of a Random Vector -- 2.2.5. Mean, Variance and Higher Moments of a Random Vector -- 2.2.6. Characteristic Function of a Random Vector -- 2.3. Transformation of Random Variables -- 2.4. Transformation of Random Vectors -- 2.5. Approximation of the Standard Normal Cumulative Distribution Function -- 3. Random Sequences -- 3.1. Sum of Independent Random Variables -- 3.2. Law of Large Numbers -- 3.3. Central Limit Theorem -- 3.4. Convergence of Sequences of Random Variables -- 3.4.1. Sure Convergence -- 3.4.2. Almost Sure Convergence -- 3.4.3. Convergence in Probability -- 3.4.4. Convergence in Quadratic Mean -- 4. Introduction to Computer Simulation of Random Variables -- 4.1. Uniform Random Variable Generator -- 4.2. Generating Discrete Random Variables -- 4.2.1. Finite Discrete Random Variables -- 4.2.2. Infinite Discrete Random Variables: Poisson Distribution -- 4.3. Simulation of Continuous Random Variables -- 4.3.1. Cauchy Distribution -- 4.3.2. Exponential Law -- 4.3.3. Rayleigh Random Variable -- 4.3.4. Gaussian Distribution -- 4.4. Simulation of Random Vectors -- 4.4.1. Case of a Two-Dimensional Random Vector -- 4.4.2. Cholesky Decomposition of the Variance-Covariance Matrix -- 4.4.3. Eigenvalue Decomposition of the Variance-Covariance Matrix -- 4.4.4. Simulation of a Gaussian Random Vector with MATLAB -- 4.5. Acceptance-Rejection Method -- 4.6. Markov Chain Monte Carlo Method (MCMC) -- 4.6.1. Definition of a Markov Process -- 4.6.2. Description of the MCMC Technique -- 5. Foundations of Monte Carlo Simulations -- 5.1. Basic Idea -- 5.2. Introduction to the Concept of Precision -- 5.3. Quality of Monte Carlo Simulations Results -- 5.4. Improvement of the Quality of Monte Carlo Simulations or Variance Reduction Techniques -- 5.4.1. Quadratic Resampling -- 5.4.2. Reduction of the Number of Simulations Using Antithetic Variables -- 5.4.3. Reduction of the Number of Simulations Using Control Variates -- 5.4.4. Importance Sampling -- 5.5. Application Cases of Random Variables Simulations -- 5.5.1. Application Case: Generation of Random Variables as a Function of the Number of Simulations -- 5.5.2. Application Case: Simulations and Improvement of the Simulations' Quality --

6. Fundamentals of Quasi Monte Carlo (QMC) Simulations -- 6.1. Van Der Corput Sequence (Basic Sequence) -- 6.2. Halton Sequence -- 6.3. Faure Sequence -- 6.4. Sobol Sequence -- 6.5. Latin Hypercube Sampling -- 6.6. Comparison of the Different Sequences -- 7. Introduction to Random Processes -- 7.1. Characterization -- 7.1.1. Statistics -- 7.1.2. Stationarity -- 7.1.3. Ergodicity -- 7.2. Notion of Continuity, Differentiability and Integrability -- 7.2.1. Continuity -- 7.2.2. Differentiability -- 7.2.3. Integrability -- 7.3. Examples of Random Processes -- 7.3.1. Gaussian Process -- 7.3.2. Random Walk -- 7.3.3. Wiener Process -- 7.3.4. Brownian Bridge -- 7.3.5. Fourier Transform of a Brownian Bridge -- 7.3.6. Example of a Brownian Bridge -- 8. Solution of Stochastic Differential Equations -- 8.1. Introduction to Stochastic Calculus -- 8.2. Introduction to Stochastic Differential Equations -- 8.2.1. Ito's Integral -- 8.2.2. Ito's Lemma -- 8.2.3. Ito's Lemma in the Multi-Dimensional Case -- 8.2.4. Solutions of Some Stochastic Differential Equations -- 8.3. Introduction to Stochastic Processes with Jumps -- 8.4. Numerical Solutions of some Stochastic Differential Equations (SDE) -- 8.4.1. Ordinary Differential Equations -- 8.4.2. Stochastic Differential Equations -- 8.5. Application Case: Generation of a Stochastic Differential Equation using the Euler and Milstein Schemes -- 8.5.1. Sensitivity with Respect to the Number of Simulated Series -- 8.5.2. Sensitivity with Respect to the Confidence Interval -- 8.5.3. Sensitivity with Respect to the Number of Simulations -- 8.5.4. Sensitivity with Respect to the Time Step -- 8.6. Application Case: Simulation of a Stochastic Differential Equation with Control and Antithetic Variables -- 8.6.1. Simple Simulations -- 8.6.2. Simulations with Control Variables -- 8.6.3. Simulations with Antithetic Variables -- 8.7. Application Case: Generation of a Stochastic Differential Equation with Jumps -- 9. General Approach to the Valuation of Contingent Claims -- 9.1. The Cox, Ross and Rubinstein (1979) Binomial Model of Option Pricing -- 9.1.1. Assumptions -- 9.1.2. Price of a Call Option -- 9.1.3. Extension To N Periods -- 9.2. Black and Scholes (1973) and Merton (1973) Option Pricing Model -- 9.2.1. Fundamental Equation for the Valuation of Contingent Claims -- 9.2.2. Exact Analytical Value of European Call and Put Options -- 9.2.3. Hedging Ratios and the Sensitivity Coefficients -- 9.3. Derivation of the Black-Scholes Formula using the Risk-Neutral Valuation Principle -- 9.3.1. The Girsanov Theorem and the Risk-Neutral Probability -- 9.3.2. Derivation of the Black and Scholes Formula Under The Risk Neutralized or Equivalent Martingale Principle -- 10. Pricing Options using Monte Carlo Simulations -- 10.1. Plain Vanilla Options: European put and Call -- 10.1.1. Simple Simulations -- 10.1.2. Simulations with Antithetic Variables -- 10.1.3. Simulations with Control Variates -- 10.1.4. Simulations with Stochastic Interest Rate -- 10.1.5. Simulations with Stochastic Interest Rate and Stochastic Volatility -- 10.2. American options -- 10.2.1. Simulations Using The Least-Squares Method of Longstaff and Schwartz (2001) -- 10.2.2. Simulations Using The Dynamic Programming Technique of Barraquand and Martineau (1995) -- 10.3. Asian options -- 10.3.1. Asian Options on Arithmetic Mean -- 10.3.2. Asian Options on Geometric Mean -- 10.4. Barrier options -- 10.5. Estimation Methods for the Sensitivity Coefficients or Greeks -- 10.5.1. Pathwise Derivative Estimates -- 10.5.2. Likelihood Ratio Method -- 10.5.3. Retrieval of Volatility Method -- 11. Term Structure of Interest Rates and Interest Rate Derivatives --

11.1. General Approach and the Vasicek (1977) Model -- 11.1.1. General Formulation -- 11.1.2. Risk Neutral Approach -- 11.1.3. Particular Case: One Factor Vasicek Model -- 11.2. The General Equilibrium Approach: The Cox, Ingersoll and Ross (CIR, 1985) model -- 11.3. The Affine Model of the Term Structure -- 11.4. Market Models -- 11.4.1. The Heath, Jarrow and Morton (HJM, 1992) Model -- 11.4.2. The Brace, Gatarek and Musiela (BGM, 1997) Model -- 12. Credit Risk and the Valuation of Corporate Securities -- 12.1. Valuation of Corporate Risky Debts: The Merton (1974) Model -- 12.1.1. The Black and Scholes (1973) Model Revisited -- 12.1.2. Application of the Model to the Valuation of a Risky Debt -- 12.1.3. Analysis of the Debt Risk -- 12.1.4. Relation Between The Firm's Asset Volatility and its Equity Volatility -- 12.2. Insuring Debt Against Default Risk -- 12.2.1. Isomorphism Between a Put Option and a Financial Guarantee -- 12.2.2. Insuring The Default Risk of a Risky Debt -- 12.2.3. Establishing a Lower Bound for the Price of the Insurance Strategy -- 12.3. Valuation of a Risky Debt: The Reduced-Form Approach -- 12.3.1. The Discrete Case with a Zero-Coupon Bond -- 12.3.2. General Case in Continuous Time -- 13. Valuation of Portfolios of Financial Guarantees -- 13.1. Valuation of a Portfolio of Loan Guarantees -- 13.1.1. Firms' and Guarantor's Dynamics -- 13.1.2. Value of Loss Per Unit of Debt -- 13.1.3. Value of Guarantee Per Unit of Debt -- 13.2. Valuation of Credit Insurance Portfolios using Monte Carlo Simulations -- 13.2.1. Stochastic Processes -- 13.2.2. Expected Shortfall and Credit Insurance Valuation -- 13.2.3. MATLAB Program -- 14. Risk Management and Value at Risk (VaR) -- 14.1. Types of Financial Risks -- 14.1.1. Market Risk -- 14.1.2. Liquidity Risk -- 14.1.3. Credit Risk -- 14.1.4. Operational Risk -- 14.2. Definition of the Value at Risk (VaR) -- 14.3. The Regulatory Environment of Basle -- 14.3.1. Stress Testing -- 14.3.2. Back Testing -- 14.4. Approaches to compute VaR -- 14.4.1. Non-Parametric Approach: Historical Simulations -- 14.4.2. Parametric Approaches -- 14.5. Computing VaR by Monte Carlo Simulations -- 14.5.1. Description of the Procedure -- 14.5.2. Application: VaR of a Simple Bank Account -- 14.5.3. Application: VaR of a Portfolio Composed of One Domestic Stock and One Foreign Stock -- 15. Value at Risk (VaR) and Principal Components Analysis (PCA) -- 15.1. Introduction to the Principal Components Analysis -- 15.1.1. Graphical Illustration -- 15.1.2. Analytical Illustration -- 15.1.3. Illustrative Example of the PCA -- 15.2. Computing the VaR of a Bond Portfolio -- 15.2.1. Sample Description and Methodology -- 15.2.2. Principal Components Analysis (PCA) -- 15.2.3. Linear Interpolation or Bootstrapping for the Intermediate Spot Rates -- 15.2.4. Computing VaR by MC and QMC Simulations -- Appendix A. Review of Mathematics -- Appendix A.1. Matrices -- Appendix A.1. 1 Elementary Operations on Matrices -- Appendix A.1. 2 Vectors -- Appendix A.1. 3 Properties -- Appendix A.1. 4 Determinants of Matrices -- Appendix A.2. Solution of a System of Linear Equations -- Appendix A.3. Matrix Decomposition -- Appendix A.4. Polynomial and Linear Approximation -- Appendix A.5. Eigenvectors and Eigenvalues of a Matrix -- Appendix B. MATLAB Functions.

"Stochastic Simulation and Applications in Finance with MATLAB Programs explains the fundamentals of Monte Carlo simulation techniques, their use in the numerical resolution of stochastic differential equations and their current applications in finance. Building on an integrated approach, it provides a pedagogical treatment of the need-to-know materials in risk management and financial engineering. The book takes readers through the basic concepts, covering the most recent research and problems in the area, including: the quadratic re-sampling technique, the Least Squared Method, the dynamic programming and Stratified State Aggregation technique to price American options, the extreme value simulation technique to price exotic options and the retrieval of volatility method to estimate Greeks. The authors also present modern term structure of interest rate models and pricing swaptions with the BGM market model, and give a full explanation of corporate securities valuation and credit risk based on the structural approach of Merton. Case studies on financial guarantees illustrate how to implement the simulation techniques in pricing and hedging. The book also includes an accompanying CD-ROM which provides MATLAB programs for the practical examples and case studies, which will give the reader confidence in using and adapting specific ways to solve problems involving stochastic processes in finance ."--Publisher's website.

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