Numerical Methods in Finance (eBook)

Preface 6
References 12
Contents 14
Contributors 16
Part I Particle Methods in Finance 19
An Introduction to Particle Methods with Financial Applications 20
1 Introduction 21
2 Option Prices and Feynman-Kac Formula 22
2.1 Discrete Time Models 22
2.1.1 European Barrier Option 23
2.1.2 Asian Option 23
2.2 Continuous Time Models 24
3 Interacting Particle Approximations 25
3.1 Feynman-Kac Semigroups 25
3.2 Interacting Particle Methodologies 28
3.3 Path Space Models 30
3.3.1 Genealogical Tree Based Algorithms 30
3.3.2 Backward Markov Chain Model 32
3.4 Parallel Island Particle Models 35
4 Application in Credit Risk Analysis 36
4.1 Change of Measure for Rare Events and Feynman-Kac Formula 37
4.2 On the Choice of the Potential Functions 38
5 Sensitivity Computation 40
5.1 Likelihood Ratio: Application to Dynamic Parameter Derivatives 40
5.2 Tangent Process: Application to Initial State Derivatives 43
6 American-Style Option Pricing 49
6.1 Description of the Model 49
6.2 A Perturbation Analysis 51
6.3 Particle Approximations 52
7 Pricing Models with Partial Observation Models 54
7.1 Abstract Formulation and Particle Approximation 54
7.2 Optimal Stopping with Partial Observation 56
7.3 Parameter Estimation in Hidden Markov Chain Models 60
References 64
American Option Valuation with Particle Filters 67
1 Introduction 68
2 Valuation Framework 70
2.1 A Risk–Neutral Stochastic Volatility Model 70
2.2 Simulation Methodology 72
2.3 Latent Volatility 74
2.4 Risk Quantification 75
3 American Options and Particle Filters 78
3.1 Filter Statistics 79
3.2 Pricing Algorithm 79
4 Benchmark Analysis 82
5 Application to Index Options 85
5.1 Data Description 85
5.2 Parameter Estimation 89
5.3 Volatility Risk Premium 91
6 Concluding Remarks 94
References 95
Monte Carlo Methods for Adaptive Disorder Problems 99
1 Introduction 99
2 Problem Formulation 102
2.1 Canonical Setup 102
2.2 Physical Probability P 103
2.3 Bayes Risk 105
3 Filtering 107
3.1 Conditional Moments 108
3.2 Particle Filters 109
3.3 Particle Degeneracy 113
4 Solving the Optimal Stopping Problem 115
4.1 Integrated Algorithm 116
4.2 Error Analysis 118
5 Numerical Examples 119
5.1 Analysis of Particle Filter 119
5.2 Example 1 120
5.3 Example 2 122
6 Extensions 124
6.1 Compound Poisson Process Observations 124
6.2 Jump Markov Signal 125
References 126
Part II Numerical Methods for Backward Conditional Expectations 129
Monte Carlo Approximations of American Options that Preserve Monotonicity and Convexity 130
1 Introduction 131
2 A Brief Review of Algorithms for Valuation of American Options 131
2.1 Snell Envelope 132
2.2 Classes of Algorithms for Valuation of American Options 133
2.3 Carriere Algorithm 134
2.4 Longstaff-Schwartz Algorithm 135
2.5 Primal-Dual Approach 135
2.5.1 Algorithm for the Lower Bound 136
3 Approximation of the Snell Envelope 138
3.1 Properties of U and V 138
3.2 Description of the Algorithm and Justification 139
3.2.1 Algorithm 140
4 Implementation Issues 142
4.1 Geometric Brownian Motion 142
4.1.1 Numerical Illustration for the American Call Option 142
4.1.2 Numerical Illustration for the American Call-on-max Option on Two Assets 143
4.2 N-GARCH Models 145
5 Conclusion 148
6 Auxiliary Results 149
7 Proofs of the Main Results 151
7.1 Proof of Proposition 3.2 151
7.2 Proof of Theorem 3.1 151
7.3 Proof of Corollary 3.1 153
8 Linear Interpolations 153
8.1 Quick Linear Interpolation on Rectangles 156
References 157
Optimal Hedging of American Options in Discrete Time 159
1 Introduction 160
2 Optimal Hedging of American Options 162
3 Choosing a Stopping Time Strategy 163
3.1 Implementation of the Stopping Time Strategy 164
4 Examples of Application 166
4.1 Lévy Models 166
4.1.1 Binomial Tree Model 168
4.1.2 Implementation 168
5 Conclusion 174
6 Proofs of the Main Results 176
6.1 Proof of Lemma 2.1 176
6.2 Proof of Proposition 2.1 177
6.3 Proof of Proposition 2.2 178
6.4 Proof of Proposition 3.1 179
7 Proof of the Perfect Hedging in the Binomial Tree Model 180
References 182
Optimal Delaunay and Voronoi Quantization Schemes for Pricing American Style Options 185
1 Introduction 185
2 Quantized Backward Dynamic Programming Principle 187
3 Optimal Voronoi and Delaunay Quantizations 194
3.1 Optimal Voronoi Quantization 194
3.2 Optimal Delaunay Quantization 196
3.2.1 Brief Comparison of Delaunay and Voronoi Quantization 198
3.3 Quantization Rates 199
4 How to Get Optimal Voronoi and Delaunay Quantizations 200
4.1 Optimal Quadratic Voronoi Quantization 200
4.1.1 Original and Randomized Lloyd's I Algorithm 200
4.1.2 The Competitive Learning Vector Quantization Algorithm 202
4.1.3 Companion Parameters 203
4.1.4 More on Practical Aspects 204
4.2 Dual Quantization 205
4.2.1 Lloyd-Type Algorithm for Dual Quantization 207
4.2.2 CLVQ Like Procedure for Dual Quantization 208
4.2.3 Search for the Matching Delaunay Hyper-triangle 208
5 Application to Cubature Formula for Numerical Integration 209
6 Quantization Tree 210
6.1 Error Bounds 211
6.2 Design of an Optimized Quantization Tree by Simulation 212
6.2.1 Grid Sizes 212
6.2.2 Transition Weight Estimation 212
6.3 Martingale Correction: An Efficient Heuristics 215
7 Numerical Experiments 215
7.1 Swing Options 215
7.2 Bermuda Options 218
7.2.1 Geometric Exchange Option 218
7.2.2 Put-on-the-Min Option 222
References 225
Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods 228
1 Introduction 228
2 Fundamental Results for the Construction of Numerical Algorithms 230
2.1 Definitions and Facts 231
2.2 From Bermudan to American Options 232
2.3 Delta Representations 233
2.3.1 Finite Difference Approach 233
2.3.2 Tangent Process Approach 234
2.3.3 Malliavin Calculus Approach 235
3 Abstract Algorithms 236
3.1 Backward Induction for the Pricing of Bermudan Options 236
3.2 Hedging Strategy Approximation 238
4 Improved Algorithms for the Estimation of Conditional Expectations 239
4.1 The Regression Based Approach 239
4.1.1 Generalities 239
4.1.2 General Comments on the Regression Procedure 241
4.1.3 Drawbacks of Polynomial Regressions 241
4.1.4 The Adaptive Local Basis Approach 242
4.2 The Malliavin Based Approach 245
4.2.1 The Alternative Representation for Conditional Expectations 245
4.2.2 General Comments 246
4.2.3 Simplifications in the Gaussian Case 247
4.2.4 Improved Numerical Methods 248
5 Numerical Experiments 254
5.1 Model and Payoffs 254
5.2 Numerical Results on Prices 255
5.3 Numerical Results on Hedging Policies 262
References 266
Least-Squares Monte Carlo for Backward SDEs 269
1 Introduction 269
2 Least-Squares Monte Carlo for BSDEs 271
2.1 Time Discretization 272
2.2 Approximation of Conditional Expectations 275
3 Martingale Basis Functions 278
4 Numerical Experiments 285
4.1 The Test Example 285
4.2 Numerical Results 287
5 Proof of Theorem 3.1 297
References 299
Pricing American Options in an Infinite Activity Lévy Market: Monte Carlo and Deterministic Approaches Using a Diffusion Approximation 302
1 Introduction 303
1.1 The Approximation of Infinite Activity Lévy Processes 303
1.2 American Options 304
2 Lévy Process Models for Price Processes 304
2.1 Lévy Processes 304
2.2 Lévy Models for Pricing 306
2.2.1 The CGMY Process 307
2.2.2 The Variance Gamma Process 307
2.3 Infinite Activity Processes 308
2.4 The Diffusion Approximation 308
2.4.1 Truncation of Small Jumps 309
3 Numerical Methods 311
3.1 Stochastic Numerical Methods 312
3.1.1 Monte Carlo Methods for Infinite Activity Lévy Processes 312
3.1.2 Monte Carlo Pricing Using Least-Squares 312
3.1.3 Simulation of the Underlying Process Under the Martingale Measure 314
3.2 Deterministic Numerical Methods 315
3.2.1 European Options 315
3.2.2 American Options 318
3.2.3 Variational Formulation 319
3.2.4 Localization 319
3.2.5 Discretization in Space 320
3.2.6 Discretization in Time 321
4 Numerical Results 322
4.1 Monte Carlo Results 322
4.1.1 Summary 324
4.2 Deterministic Numerical Results 325
4.2.1 Computation Time 327
5 Discussion 328
References 331
Fourier Cosine Expansions and Put–Call Relations for Bermudan Options 333
1 Introduction 333
2 Preliminaries 334
2.1 Exponential Lévy Asset Dynamics 335
2.2 The Fourier Cosine Method (COS) for European Options 336
2.3 Truncation Range and Put–Call Relations 337
2.3.1 European Option Results 338
3 Pricing Early-Exercise Options 340
3.1 Pricing Bermudan Options by the COS Method 340
3.1.1 American Options 343
3.2 Error Analysis 344
3.2.1 Local Error 344
4 Pricing Bermudan Call Options Using the Put–Call Relations 346
4.1 The Put–Call Parity 346
4.2 The Put–Call Duality 350
4.3 Error Analysis with the Put–Call Relations 354
5 Numerical Examples 355
5.1 American Options 358
6 Conclusions and Discussion 359
References 359
Part III Numerical Methods for Energy Derivatives 361
A Practical View on Valuation of Multi-Exercise American Style Options in Gas and Electricity Markets 362
1 Introduction 362
2 Option Types with Multiple Exercise Rights 363
2.1 Swing Options 364
2.2 Gas Storage 367
2.3 Hydro Power 369
3 Valuation Methods 372
3.1 Forward Market Optimization 373
3.1.1 Intrinsic Value 373
3.1.2 Rolling Intrinsic Value 374
3.2 Spot Market Optimization 375
3.2.1 Deterministic Value 375
3.2.2 Fair Value 375
3.3 Hedging and Optimal Exercise Strategies 377
3.3.1 Hedging 378
3.3.2 Optimal Exercise Strategies 378
4 Spot Price Models 378
4.1 Electricity Spot Price Model 379
4.2 Natural Gas Spot Price Model 379
5 Some Examples 381
6 Conclusion 386
References 386
Swing Options Valuation: A BSDE with ConstrainedJumps Approach 388
1 Introduction 389
2 BSDE Representation for Impulse Control Problems 390
2.1 A Class of Impulse Control Problems 391
2.2 Link to BSDEs with Constrained Jumps 392
3 Convergence of the Numerical Approximation by Penalization 393
3.1 Approximation by Penalization 394
3.2 Convergence Rate of the Numerical Scheme 397
4 Application to Swing Options Valuation 399
4.1 Swing Options Valuation as an Impulse Control Problem 399
4.2 Numerical Valuation Algorithm 401
4.3 Pricing Results 405
4.3.1 Special Case of American Options: nmax = 1 405
4.3.2 Swing Options with nmax = 2 405
References 408
Swing Option Pricing by Optimal Exercise Boundary Estimation 410
1 Introduction 411
2 Algorithm Presentation 411
2.1 Notations and Hypothesis 412
2.2 Bellman Equation 413
2.3 First Intuition 413
2.4 Algorithm Description 414
3 Implementation 420
4 Numerical Results 424
4.1 Quality Criteria 424
4.1.1 Computation Time 424
4.1.2 Accuracy 424
4.1.3 Precision 424
4.1.4 Strategies Stability 425
4.2 Numerical Results 426
5 Conclusion 428
References 428
Gas Storage Hedging 429
1 Introduction 429
2 Recall on American and Bermudan Options and Delta Hedging 431
2.1 Formulas 431
2.2 Classical Longstaff-Schwartz and Conditional Delta 433
3 Gas Storage Valuation and Hedging Methodology 435
3.1 Price Model 435
3.1.1 Future Price Model 435
3.1.2 Tangent Process 436
3.2 Gas Storage Modelization 436
3.2.1 Dynamic Programming and Daily Hedging for Gas Storage 437
3.2.2 Cash Flow Simulation and Delta Hedging 439
4 Numerical Results 443
4.1 Market Representation 443
4.2 Gas Storage Description 444
4.3 Comparison between Finite Difference and Tangent Process 445
4.3.1 Fast Storage Results 446
4.3.2 Seasonal Storage Results 448
References 452
Sensitivity Analysis of Energy Contracts by Stochastic Programming Techniques 454
1 Motivation 455
2 A Review of Quantization Discretization and Stochastic Dual Dynamic Programming Approach 456
2.1 Discretization 456
2.2 Stochastic Dual Dynamic Programming Algorithm 457
3 Price Model 459
4 Sensitivity Analysis 461
4.1 Danskin's Theorem and Its Applications 461
4.2 Convergence of Sensitivity Estimate 464
5 Algorithm and Numerical Tests 469
5.1 Algorithm 469
5.2 Comparison of Methods 470
5.2.1 Danskin+Quantization Tree+State Space Discretization 470
5.2.2 Danskin+PDE+State Space Discretization 471
5.2.3 Finite Difference (fd.)+PDE+State Space Discretizaton 472
5.3 Swing Option 472
5.4 Small Commodity Portfolio Case Study 474
6 Appendix: Implicit Scheme of Finite Difference for One Dimension PDE 476
References 477