Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer
Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic Processes 3 1.1 Discrete-Time Stochastic Processes 3 1.1.1 Conditional Expectations and Filtrations 3 1.2 Discrete-Time Markov Chains 6 1.2.1 An Introductory Example 6 1.2.2 Definitions and Examples 7 1.2.3 Chapman-Kolmogorov Equations 10 1.2.4 Long-Range Behavior 12 1.3 Discrete-Time Martingales 12 1.3.1 Definitions and Examples 12 1.3.2 Stopping Times and Optional Stopping Theorem 17 1.3.3 Doob's Decomposition 21 2 Some Classes of Continuous-Time Stochastic Processes 23 2.1 Continuous-Time Stochastic Processes 23 2.1.1 Generalities 23 2.1.2 Continuous-Time Martingales 24 2.2 The Poisson Process and Continuous-Time Markov Chains... 24 2.2.1 The Poisson Process 27 2.2.2 Two-State Continuous Time Markov Chains 31 2.2.3 Birth-and-Death Processes '... 33 2.3 Brownian Motion 33 2.3.1 Definition and Basic Properties 34 2.3.2 Random Walk Approximation. 35' 2.3.3 Second Order Properties 36 2.3.4 Markov Properties 36 2.3.5 First Passage Times of a Standard Brownian Motion... 38 2.3.6 Martingales Associated with Brownian Motion 39 2.3.7 First Passage Times of a Drifted Brownian Motion 42 2.3.8 Geometric Brownian Motion 43
xiv Contents 3 Elements of Stochastic Analysis 45 3.1 Stochastic Integration 45 3.1.1 Integration with Respect to a Symmetric Random Walk.. 45 3.1.2 The Ito Stochastic Integral for Simple Processes 46 3.1.3 The General Ito Stochastic Integral 49 3.1.4 Stochastic Integral with Respect to a Poisson Process... 51 3.1.5 Semimartingale Integration Theory (*) 51 3.2 Ito Formula 53 3.2.1 Introduction 53 3.2.2 Ito Formulas for Continuous Processes 54 3.2.3 Ito Formulas for Processes with Jumps (*) 57 3.2.4 Brackets (*) 60 3.3 Stochastic Differential Equations (SDEs) 62 3.3.1 Introduction 62 3.3.2 Diffusions 63 3.3.3 Jump-Diffusions (*) 69 3.4 Girsanov Transformations 71 3.4.1 Girsanov Transformation for Gaussian Distributions... 71 3.4.2 Girsanov Transformation for Poisson Distributions 73 3.4.3 Abstract Bayes Formula 75 3.5 Feynman-Kac Formulas (*) 75 3.5.1 Linear Case 75 3.5.2 Backward Stochastic Differential Equations (BSDEs)... 76 3.5.3 Nonlinear Feynman-Kac Formula 77 3.5.4 Optimal Stopping 78 Part II Pricing Equations 4 Martingale Modeling 83 4.1 General Setup 85 4.1.1 Pricing by Arbitrage 86 4.1.2 Hedging 95 4.2 Markovian Setup 102 4.2.1 Factor Processes 103 4.2.2 Markovian Reflected BSDEs and Obstacles PIDE Problems 104 4.2.3 Hedging Schemes 106 4.3 Extensions 108 4.3.1 More General Numeraires 108 4.3.2 Defaultable Derivatives Ill 4.3.3 Intermittent Call Protection...' 119 4.4 From Theory to Practice 121 4.4.1 Model Calibration 121 4.4.2 Hedging 121 5 Benchmark Models 123 5.1 Black-Scholes and Beyond 123
xv 5.1.1 Black-Scholes Basics 123 5.1.2 Heston Model 126 5.1.3 Merton Model 127 5.1.4 Bates Model 127 5.1.5 Log-Spot Characteristic Functions in Affine Models... 127 5.2 Libor Market Model of Interest-Rate Derivatives 130 5.2.1 Black Formula 130 5.2.2 Libor Market Model 132 5.2.3 Caps and Floors 133 5.2.4 Adding Correlation 134 5.2.5 Swaptions -. - 136 5.2.6 Model Simulation 137 5.3 One-Factor Gaussian Copula Model of Portfolio Credit Risk.... 138 5.3.1 Credit Derivatives 139 5.3.2 Gaussian Copula Model 140 5.4 Benchmark Models in Practice 144 5.4.1 Implied Parameters 144 5.4.2 Implied Delta-Hedging 146 5.5 Vanilla Options Fourier Transform Pricing Formulas 150 5.5.1 Fourier Calculus 150 5.5.2 Black-Scholes Type Pricing Formula 151 5.5.3 Carr-Madan Formula 153 Part III Numerical Solutions 6 Monte Carlo Methods 161 6.1 Uniform Numbers 161 6.1.1 Pseudo-Random Generators 162 6.1.2 Low-Discrepancy Sequences 164 6.2 Non-uniform Numbers 166 6.2.1 Inverse Method 166 6.2.2 Gaussian Pairs 167 6.2.3 Gaussian Vectors 169 6.3 Principles of Monte Carlo Simulation 170 6.3.1 Law of Large Numbers and Central Limit Theorem... 170 6.3.2 Standard Monte Carlo Estimator and Confidence Interval. 170 6.4 Variance Reduction 171 6.4.1 Antithetic Variables 171 6.4.2 Control Variates 172 6.4.3 Importance Sampling 173 6.4.4 Efficiency Criterion 174 6.5 Quasi Monte Carlo 175 6.6 Greeking by Monte Carlo 176 6.6.1 Finite Differences 176 6.6.2 Differentiation of the Payoff 177 6.6.3 Differentiation of the Density 177
6.7 Monte Carlo Algorithms for Vanilla Options 178 6.7.1 European Call, Put or Digital Option 178 6.7.2 Call on Maximum, Put on Minimum, Exchange or Best of Options 179 6.8 Simulation of Processes 182 6.8.1 Brownian Motion 182 6.8.2 Diffusions 184 6.8.3 Adding Jumps 186 6.8.4 Monte Carlo Simulation for Processes 188 6.9 Monte Carlo Methods for Exotic Options 188 6.9.1 Lookback Options S 190 6.9.2 Barrier Options 192 6.9.3 Asian Options 193 6.10 American Monte Carlo Pricing Schemes 194 6.10.1 Time-0 Price 195 6.10.2 Computing Conditional Expectations by Simulation... 196 Tree Methods 199 7.1 Markov Chain Approximation of Jump-Diffusions 199 7.1.1 Kushner's Theorem 199 7.2 Trees for Vanilla Options 201 7.2.1 Cox-Ross-Rubinstein Binomial Tree 201 7.2.2 Other Binomial Trees 206 7.2.3 Kamrad-Ritchken Trinomial Tree 206' 7.2.4 Multinomial Trees 207 7.3 Trees for Exotic Options 208 7.3.1 Barrier Options 208 7.3.2 Bermudan Options 209 7.4 Bidimensional Trees 210 7.4.1 Cox-Ross-Rubinstein Tree for Lookback Options 210 7.4.2 Kamrad-Ritchken Tree for Options on Two Assets 210 Finite Differences 213 8.1 Generic Pricing PIDE 213 8.1.1 Maximum Principle 214 8.1.2 Weak Solutions 215 8.2 Numerical Approximation 216 8.2.1 Finite Difference Methods 216 8.2.2 Finite Elements and Beyond 218 8.3 Finite Differences for European Vanilla Options 220 8.3.1 Localization and Discretization in Space 220 8.3.2 Theta-Schemes in Time..." 222 8.3.3 Adding Jumps 226 8.4 Finite Differences for American Vanilla Options 229 8.4.1 Splitting Scheme 229 8.5 Finite Differences for Bidimensional Vanilla Options 230
xvii 8.5.1 ADI Scheme 231 8.6 Finite Differences for Exotic Options 233 8.6.1 Lookback Options 233 8.6.2 Barrier Options 234 8.6.3 Asian Options 235 8.6.4 Discretely Path Dependent Options 237 9 Calibration Methods 243 9.1 The Ill-Posed Inverse Calibration Problem 243 9.1.1 Tikhonov Regularization of Nonlinear Inverse Problems.. 244 9.1.2 Calibration by Nonlinear Optimization 247 9.2 Extracting the Effective Volatility.\ : r 247 9.2.1 Dupire Formula 248 9.2.2 The Local Volatility Calibration Problem 250 9.3 Weighted Monte Carlo 254 9.3.1 Approach by Duality 256 9.3.2 Relaxed Least Squares Approach 257 Part IV Applications 10 Simulation/Regression Pricing Schemes in Diffusive Setups 261 10.1 MarketModel 262 10.1.1 Underlying Stock 262 10.1.2 Convertible Bond 264 10.2 Pricing Equations and Their Approximation 265 10.2.1 Stochastic Pricing Equation 266 10.2.2 Markovian Case 267 10.2.3 Generic Simulation Pricing Schemes 268 10.2.4 Convergence Results 270 10.3 American and Game Options 272 10.3.1 No Call 272 10.3.2 No Protection 274 10.3.3 Numerical Experiments 275 10.4 Continuously Monitored Call Protection 277 10.4.1 Vanilla Protection 278 10.4.2 Intermittent Vanilla Protection 280 10.4.3 Numerical Experiments 282 10.5 Discretely Monitored Call Protection. 283 10.5.1 "/ Last" Protection 284 10.5.2 "/ Out of the Last d" Protection 285 10.5.3 Numerical Experiments.... : 287 10.5.4 Conclusions 291 11 Simulation/Regression Pricing Schemes in Pure Jump Setups... 293 11.1 Generic Markovian Setup 294 11.1.1 Generic Simulation Pricing Scheme 295 11.2 Homogeneous Groups Model of Portfolio Credit Risk 296
xviii Contents 11.2.1 Hedging in the Homogeneous Groups Model 297 11.2.2 Simulation Scheme 299 11.3 Pricing and Greeking Results in the Homogeneous Groups Model. 299 11.3.1 Fully Homogeneous Case 300 11.3.2 Semi-Homogeneous Case 302 11.4 Common Shocks Model of Portfolio Credit Risk 305 11.4.1 Example 308 11.4.2 Marshall-Olkin Representation 309 11.5 CVA Computations in the Common Shocks Model 310 11.5.1 Numerical Results 312 11.5.2 Conclusions >.. 319 Part V Jump-Diffusion Setup with Regime Switching (**) 12 Backward Stochastic Differential Equations 323 12.1 General Setup 323 12.1.1 Semimartingale Forward SDE 326 12.1.2 Semimartingale Reflected and Doubly Reflected BSDEs.. 328 12.2 Markovian Setup 334 12.2.1 Dynamics 336 12.2.2 Mapping with the General Set-Up 338 12.2.3 Cost Functionals. 339 12.2.4 Markovian Decoupled Forward Backward SDE 340 12.2.5 Financial Interpretation 342 12.3 Study of the Markovian Forward SDE 343 12.3.1 Homogeneous Case 344 12.3.2 Inhomogeneous Case 348 12.4 Study of the Markovian BSDEs 351 12.4.1 Semigroup Properties 354 12.4.2 Stopped Problem 355 12.5 Markov Properties 358 13 Analytic Approach 359 13.1 Viscosity Solutions of Systems of PIDEs with Obstacles 359 13.2 Study of the PIDEs 362 13.2.1 Existence 362 13.2.2 Uniqueness..363 13.2.3 Approximation 365 14 Extensions 369 14.1 Discrete Dividends 369 14.1.1 Discrete Dividends on a Derivative 369 14.1.2 Discrete Dividends on Underlying Assets 371 14.2 Intermittent Call Protection 373 14.2.1 General Setup 374 14.2.2 Marked Jump-Diffusion Setup 377 14.2.3 Well-Posedness of the Markovian RIBSDE 379
xix 14.2.4 Semigroup and Markov Properties 382 14.2.5 Viscosity Solutions Approach 384 14.2.6 Protection Before a Stopping Time Again 385 Part VI Appendix 15 Technical Proofs (**) 391 15.1 Proofs of BSDE Results 391 15.1.1 Proof of Lemma 12.3.6 391 15.1.2 Proof of Proposition 12.4.2 392 15.1.3 Proof of Proposition 12.4.3 396 15.1.4 Proof of Proposition 12.4.7 -.-." 397 15.1.5 Proof of Proposition 12.4.10 399 15.1.6 Proof of Theorem 12.5.1 400 15.1.7 Proof of Theorem 14.2.18 403 15.2 Proofs of PDE Results 405 15.2.1 Proof of Lemma 13.1.2 405 15.2.2 Proof of Theorem 13.2.1 405 15.2.3 Proof of Lemma 13.2.4 410 15.2.4 Proof of Lemma 13.2.8 416 16 Exercises 421 16.1 Discrete-Time Markov Chains 421 16.2 Discrete-Time Martingales 421 16.3 The Poisson Process and Continuous-Time Markov Chains... 423 16.4 Brownian Motion 423 16.5 Stochastic Integration 424 16.6 Ito Formula 424 16.7 Stochastic Differential Equations 425 17 Corrected Problem Sets 427 17.1 Exit of a Brownian Motion from a Corridor 427 17.2 Pricing with a Regime-Switching Volatility 428 17.3 Hedging with a Regime-Switching Volatility 431 17.4 Jump-to-Ruin 434 References 441 Index... 453