Financial modeling: a backward stochastic differential equations perspective
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2013
|
| Schriftenreihe: | Springer Finance : Textbooks
|
| Schlagwörter: | |
| Links: | http://deposit.dnb.de/cgi-bin/dokserv?id=4255340&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890423&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIX, 459 S. graph. Darst. |
| ISBN: | 9783642371127 |
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Datensatz im Suchindex
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|---|---|
| adam_text | IMAGE 1
CONTENTS
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
XIII
HTTP://D-NB.INFO/1031344721
IMAGE 2
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
IMAGE 3
CONTENTS 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
IMAGE 4
XVI CONTENTS
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 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
7 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
8 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
IMAGE 5
CONTENTS 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 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
MARKET MODEL 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 Z 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 . . . . 2 9
3
11.1 GENERIC MARKOVIAN SETUP 294
11.1.1 GENERIC SIMULATION PRICING SCHEME 295
11.2 HOMOGENEOUS GROUPS MODEL OF PORTFOLIO CREDIT RISK 296
IMAGE 6
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
IMAGE 7
CONTENTS 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 PROOFOFLEMMA 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
|
| any_adam_object | 1 |
| author | Crépey, Stéphane 1971- |
| author_GND | (DE-588)1037488725 |
| author_facet | Crépey, Stéphane 1971- |
| author_role | aut |
| author_sort | Crépey, Stéphane 1971- |
| author_variant | s c sc |
| building | Verbundindex |
| bvnumber | BV040911097 |
| classification_rvk | QP 700 SK 820 SK 980 |
| ctrlnum | (OCoLC)856794375 (DE-599)DNB1031344721 |
| dewey-full | 332.6450151922 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.6450151922 |
| dewey-search | 332.6450151922 |
| dewey-sort | 3332.6450151922 |
| dewey-tens | 330 - Economics |
| discipline | Informatik Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV040911097 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:26:59Z |
| institution | BVB |
| isbn | 9783642371127 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025890423 |
| oclc_num | 856794375 |
| open_access_boolean | |
| owner | DE-11 DE-521 DE-945 DE-384 DE-83 DE-19 DE-BY-UBM |
| owner_facet | DE-11 DE-521 DE-945 DE-384 DE-83 DE-19 DE-BY-UBM |
| physical | XIX, 459 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer Finance : Textbooks |
| spellingShingle | Crépey, Stéphane 1971- Financial modeling a backward stochastic differential equations perspective Derivat Wertpapier (DE-588)4381572-8 gnd Hedging (DE-588)4123357-8 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastisches Differentialgleichungssystem (DE-588)4225826-1 gnd Simulation (DE-588)4055072-2 gnd Preisbildung (DE-588)4047103-2 gnd |
| subject_GND | (DE-588)4381572-8 (DE-588)4123357-8 (DE-588)4017195-4 (DE-588)4225826-1 (DE-588)4055072-2 (DE-588)4047103-2 |
| title | Financial modeling a backward stochastic differential equations perspective |
| title_auth | Financial modeling a backward stochastic differential equations perspective |
| title_exact_search | Financial modeling a backward stochastic differential equations perspective |
| title_full | Financial modeling a backward stochastic differential equations perspective Stéphane Crépey |
| title_fullStr | Financial modeling a backward stochastic differential equations perspective Stéphane Crépey |
| title_full_unstemmed | Financial modeling a backward stochastic differential equations perspective Stéphane Crépey |
| title_short | Financial modeling |
| title_sort | financial modeling a backward stochastic differential equations perspective |
| title_sub | a backward stochastic differential equations perspective |
| topic | Derivat Wertpapier (DE-588)4381572-8 gnd Hedging (DE-588)4123357-8 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastisches Differentialgleichungssystem (DE-588)4225826-1 gnd Simulation (DE-588)4055072-2 gnd Preisbildung (DE-588)4047103-2 gnd |
| topic_facet | Derivat Wertpapier Hedging Finanzmathematik Stochastisches Differentialgleichungssystem Simulation Preisbildung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4255340&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025890423&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT crepeystephane financialmodelingabackwardstochasticdifferentialequationsperspective |