Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Transcription

1 Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer

2 Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic Processes Discrete-Time Stochastic Processes Conditional Expectations and Filtrations Discrete-Time Markov Chains An Introductory Example Definitions and Examples Chapman-Kolmogorov Equations Long-Range Behavior Discrete-Time Martingales Definitions and Examples Stopping Times and Optional Stopping Theorem Doob's Decomposition 21 2 Some Classes of Continuous-Time Stochastic Processes Continuous-Time Stochastic Processes Generalities Continuous-Time Martingales The Poisson Process and Continuous-Time Markov Chains The Poisson Process Two-State Continuous Time Markov Chains Birth-and-Death Processes ' Brownian Motion Definition and Basic Properties Random Walk Approximation. 35' Second Order Properties Markov Properties First Passage Times of a Standard Brownian Motion Martingales Associated with Brownian Motion First Passage Times of a Drifted Brownian Motion Geometric Brownian Motion 43

3 xiv Contents 3 Elements of Stochastic Analysis Stochastic Integration Integration with Respect to a Symmetric Random Walk The Ito Stochastic Integral for Simple Processes The General Ito Stochastic Integral Stochastic Integral with Respect to a Poisson Process Semimartingale Integration Theory (*) Ito Formula Introduction Ito Formulas for Continuous Processes Ito Formulas for Processes with Jumps (*) Brackets (*) Stochastic Differential Equations (SDEs) Introduction Diffusions Jump-Diffusions (*) Girsanov Transformations Girsanov Transformation for Gaussian Distributions Girsanov Transformation for Poisson Distributions Abstract Bayes Formula Feynman-Kac Formulas (*) Linear Case Backward Stochastic Differential Equations (BSDEs) Nonlinear Feynman-Kac Formula Optimal Stopping 78 Part II Pricing Equations 4 Martingale Modeling General Setup Pricing by Arbitrage Hedging Markovian Setup Factor Processes Markovian Reflected BSDEs and Obstacles PIDE Problems Hedging Schemes Extensions More General Numeraires Defaultable Derivatives Ill Intermittent Call Protection...' From Theory to Practice Model Calibration Hedging Benchmark Models Black-Scholes and Beyond 123

4 xv Black-Scholes Basics Heston Model Merton Model Bates Model Log-Spot Characteristic Functions in Affine Models Libor Market Model of Interest-Rate Derivatives Black Formula Libor Market Model Caps and Floors Adding Correlation Swaptions Model Simulation One-Factor Gaussian Copula Model of Portfolio Credit Risk Credit Derivatives Gaussian Copula Model Benchmark Models in Practice Implied Parameters Implied Delta-Hedging Vanilla Options Fourier Transform Pricing Formulas Fourier Calculus Black-Scholes Type Pricing Formula Carr-Madan Formula 153 Part III Numerical Solutions 6 Monte Carlo Methods Uniform Numbers Pseudo-Random Generators Low-Discrepancy Sequences Non-uniform Numbers Inverse Method Gaussian Pairs Gaussian Vectors Principles of Monte Carlo Simulation Law of Large Numbers and Central Limit Theorem Standard Monte Carlo Estimator and Confidence Interval Variance Reduction Antithetic Variables Control Variates Importance Sampling Efficiency Criterion Quasi Monte Carlo Greeking by Monte Carlo Finite Differences Differentiation of the Payoff Differentiation of the Density 177

5 6.7 Monte Carlo Algorithms for Vanilla Options European Call, Put or Digital Option Call on Maximum, Put on Minimum, Exchange or Best of Options Simulation of Processes Brownian Motion Diffusions Adding Jumps Monte Carlo Simulation for Processes Monte Carlo Methods for Exotic Options Lookback Options S Barrier Options Asian Options American Monte Carlo Pricing Schemes Time-0 Price Computing Conditional Expectations by Simulation Tree Methods Markov Chain Approximation of Jump-Diffusions Kushner's Theorem Trees for Vanilla Options Cox-Ross-Rubinstein Binomial Tree Other Binomial Trees Kamrad-Ritchken Trinomial Tree 206' Multinomial Trees Trees for Exotic Options Barrier Options Bermudan Options Bidimensional Trees Cox-Ross-Rubinstein Tree for Lookback Options Kamrad-Ritchken Tree for Options on Two Assets 210 Finite Differences Generic Pricing PIDE Maximum Principle Weak Solutions Numerical Approximation Finite Difference Methods Finite Elements and Beyond Finite Differences for European Vanilla Options Localization and Discretization in Space Theta-Schemes in Time..." Adding Jumps Finite Differences for American Vanilla Options Splitting Scheme Finite Differences for Bidimensional Vanilla Options 230

6 xvii ADI Scheme Finite Differences for Exotic Options Lookback Options Barrier Options Asian Options Discretely Path Dependent Options Calibration Methods The Ill-Posed Inverse Calibration Problem Tikhonov Regularization of Nonlinear Inverse Problems Calibration by Nonlinear Optimization Extracting the Effective Volatility.\ : r Dupire Formula The Local Volatility Calibration Problem Weighted Monte Carlo Approach by Duality Relaxed Least Squares Approach 257 Part IV Applications 10 Simulation/Regression Pricing Schemes in Diffusive Setups MarketModel Underlying Stock Convertible Bond Pricing Equations and Their Approximation Stochastic Pricing Equation Markovian Case Generic Simulation Pricing Schemes Convergence Results American and Game Options No Call No Protection Numerical Experiments Continuously Monitored Call Protection Vanilla Protection Intermittent Vanilla Protection Numerical Experiments Discretely Monitored Call Protection "/ Last" Protection "/ Out of the Last d" Protection Numerical Experiments.... : Conclusions Simulation/Regression Pricing Schemes in Pure Jump Setups Generic Markovian Setup Generic Simulation Pricing Scheme Homogeneous Groups Model of Portfolio Credit Risk 296

7 xviii Contents Hedging in the Homogeneous Groups Model Simulation Scheme Pricing and Greeking Results in the Homogeneous Groups Model Fully Homogeneous Case Semi-Homogeneous Case Common Shocks Model of Portfolio Credit Risk Example Marshall-Olkin Representation CVA Computations in the Common Shocks Model Numerical Results Conclusions > Part V Jump-Diffusion Setup with Regime Switching (**) 12 Backward Stochastic Differential Equations General Setup Semimartingale Forward SDE Semimartingale Reflected and Doubly Reflected BSDEs Markovian Setup Dynamics Mapping with the General Set-Up Cost Functionals Markovian Decoupled Forward Backward SDE Financial Interpretation Study of the Markovian Forward SDE Homogeneous Case Inhomogeneous Case Study of the Markovian BSDEs Semigroup Properties Stopped Problem Markov Properties Analytic Approach Viscosity Solutions of Systems of PIDEs with Obstacles Study of the PIDEs Existence Uniqueness Approximation Extensions Discrete Dividends Discrete Dividends on a Derivative Discrete Dividends on Underlying Assets Intermittent Call Protection General Setup Marked Jump-Diffusion Setup Well-Posedness of the Markovian RIBSDE 379

8 xix Semigroup and Markov Properties Viscosity Solutions Approach Protection Before a Stopping Time Again 385 Part VI Appendix 15 Technical Proofs (**) Proofs of BSDE Results Proof of Lemma Proof of Proposition Proof of Proposition Proof of Proposition " Proof of Proposition Proof of Theorem Proof of Theorem Proofs of PDE Results Proof of Lemma Proof of Theorem Proof of Lemma Proof of Lemma Exercises Discrete-Time Markov Chains Discrete-Time Martingales The Poisson Process and Continuous-Time Markov Chains Brownian Motion Stochastic Integration Ito Formula Stochastic Differential Equations Corrected Problem Sets Exit of a Brownian Motion from a Corridor Pricing with a Regime-Switching Volatility Hedging with a Regime-Switching Volatility Jump-to-Ruin 434 References 441 Index