Numerical Methods in Finance - Bordeaux, June 2010

von: René Carmona, Pierre Del Moral, Peng Hu, Nadia Oudjane

Springer-Verlag, 2012

ISBN: 9783642257469

Sprache: Englisch

474 Seiten, Download: 8233 KB

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Numerical Methods in Finance - Bordeaux, June 2010

	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

Numerical Methods in Finance - Bordeaux, June 2010

von: René Carmona, Pierre Del Moral, Peng Hu, Nadia Oudjane

Numerical Methods in Finance - Bordeaux, June 2010

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