Deterministic and Stochastic Topics in Computational Finance:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2017]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xix, 461 Seiten Diagramme |
| ISBN: | 9789813203082 9789813203075 |
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| 100 | 1 | |a Calin, Ovidiu L. |d 1971- |e Verfasser |0 (DE-588)128990554 |4 aut | |
| 245 | 1 | 0 | |a Deterministic and Stochastic Topics in Computational Finance |c Ovidiu Calin, Princeton University |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2017] | |
| 264 | 4 | |c © 2017 | |
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| 500 | |a Includes bibliographical references and index | ||
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Record in the Search Index
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| adam_text | Titel: Deterministic and stochastic topics in computational finance
Autor: Calin, Ovidiu
Jahr: 2017
Contents
Preface v
Chapters Diagram ix
List of Notations and Symbols xi
I Introduction 1
1 Determinism or Stochasticity? 3
1.1 Determinism and Semi-determinism ..............................3
1.2 How to Measure ....................................................5
1.3 An Uncertainty Principle ..........................................6
1.4 Market Completeness ..............................................7
1.5 Market Efficiency....................................................7
1.6 Stopping Times......................................................8
1.7 Martingales and Submartingales ..................................9
1.8 Optional Stopping Theorem........................................9
1.9 Can Random be Deterministic?....................................10
1.10 Change of Time Scale..............................................13
2 Calibration to the Market 19
2.1 Deterministic Regression............................................19
2.2 Stochastic Regression ..............................................23
2.3 Calibration..........................................................27
2.4 Alternatives to the Method of Least Squares......................31
2.4.1 The Maximum Likelihood Method........................32
2.4.2 The Maximum Entropy Method ..........................34
2.4.3 The Kullback-Leibler Relative Entropy....................35
2.4.4 The Cross Entropy..........................................35
II Interest Rates and Bonds
3 Modeling Stochastic Rates
3.1 Deterministic versus Stochastic Calculus .
3.2 Langevin s Equation...........
3.3 Equilibrium Models ...........
3.3.1 Rendleman and Bartter Model .
3.3.2 Vasicek Model ..........
3.3.3 Calibration of Vasicek s Model .
3.3.4 Cox-Ingersoll-Ross Model ....
3.4 No-arbitrage Models...........
3.4.1 Ho and Lee Model........
3.4.2 Hull and White Model......
3.5 Nonstationary Models..........
3.5.1 Black, Herman and Toy Model .
4 Bonds, Forward Rates and Yield Curves
4.1 Bonds...................
4.2 Yield....................
4.3 Bootstrap Method............
4.4 Forward Rates ..............
4.5 Single-Factor HJM Models........
4.5.1 Ho-Lee Model...........
4.5.2 Hull and White Model......
4.5.3 Vasicek Model ..........
4.6 Relation Formulas............
4.7 A Simple Spot Rate Model.......
4.8 Bond Price for Ho-Lee Model......
4.9 Bond Price for Vasicek s Model.....
4.10 Bond Price for CIR s Model.......
4.11 Mean Reverting Model with Jumps . .
4.12 A Model with Pure Jumps........
Ill Risk-Neutral Valuation Pricing
5 Modeling Stock Prices
5.1 Constant Drift and Volatility Model
5.2 Correlation of Two Stocks
5.3 When Does a Stock Hit a Given Barrier?
5.4 Probability to Hit a Barrier Before T .
5.5 Multiple Barriers.....
37
39
39
41
42
43
44
46
49
51
51
52
53
53
55
55
57
59
60
62
63
64
67
68
69
71
73
76
77
81
85
87
87
89
92
97
102
5.6 Estimation of Parameters.....................105
5.7 Time-dependent Drift and Volatility...............106
5.8 Models for Stock Price Averages .................108
5.9 Stock Prices with Rare Events ..................115
5.10 Dividend Paying Stocks.......................121
5.11 Currencies..............................121
6 Risk-Neutral Valuation 125
6.1 The Method of Risk-Neutral Valuation..............125
6.2 The Superposition Principle and Applications..........132
6.3 Packages...............................137
6.4 Strike Sensitivity..........................140
6.5 Volatility Sensitivity........................143
6.6 Implied Volatility..........................144
6.7 Asian Forward Contracts .....................146
6.8 Asian Options............................151
6.9 The dk Notations..........................157
6.10 All-or-nothing Look-back Options.................159
6.11 Asset-or-nothing Look-back Options...............160
6.12 Look-back Call Options......................164
6.13 Forward Look-back Contracts...................164
6.14 Immediate Rebate Options ....................166
6.15 Perpetual Look-back Options...................169
6.16 Double Barrier Immediate Option ................169
6.17 Pricing in a Rare Events Environment..............170
6.17.1 Pricing a Call........................171
6.17.2 Pricing a Forward Contract................172
6.18 Pricing with Pareto Distribution.................174
7 Martingale Measures 179
7.1 Martingale Measures........................179
7.1.1 Is the stock price St a martingale?............179
7.1.2 Risk-neutral World and Martingale Measure.......181
7.1.3 Finding the Risk-Neutral Measure............182
7.2 Risk-neutral World Density Functions.....•.........184
7.3 Self-financing Portfolios......................186
7.4 The Sharpe Ratio..........................187
7.5 Risk-neutral Valuation for Derivatives..............190
IV PDE Approach 193
8 Black-Scholes Analysis 195
8.1 Heat Equation ...........................195
8.2 What is a Portfolio?........................198
8.3 Risk-less Portfolios.........................199
8.4 Black-Scholes Equation ......................201
8.5 Delta Hedging............................203
8.6 Tradable Securities.........................204
8.7 Risk-less Investment Revised ...................206
8.8 Solving the Black-Scholes.....................209
8.9 Black-Scholes and Risk-neutral Valuation ............212
8.10 Boundary Conditions........................213
8.11 The Black-Scholes Operator....................214
8.12 Hedging and Black-Scholes ....................216
8.12.1 Hedging Stocks.......................217
8.12.2 Hedging Derivatives....................218
8.12.3 Hedging Bonds.......................219
8.12.4 Particular Cases......................223
8.12.5 The Bond Formula.....................226
8.13 Interest Rate Swaps........................227
8.13.1 Case of Deterministic Rates................227
8.13.2 A Black-Scholes Type Equation..............229
8.13.3 Solving the Equation....................232
8.13.4 Some Particular Cases...................234
8.13.5 Hedging with Swaps....................236
9 Black-Scholes for Asian Derivatives 237
9.1 Weighted Averages..............................237
9.2 Setting up the Black-Scholes Equation..............239
9.3 Weighted Average Strike Call Option...............240
9.4 Boundary Conditions..........................242
9.5 Asian Forward Contracts ......................246
9.6 Put-Call Parity......................249
9.7 Valuation of Arithmetic Asian Price Options 250
10 American Options 253
10.1 Perpetual American Options ...................253
10.1.1 Present Value of Barriers.................253
10.1.2 Perpetual American Calls.................257
10.1.3 Perpetual American Puts.................259
10.2 Perpetual American Log Contract ................261
10.3 Perpetual American Power Contract...............263
10.4 Finitely Lived American Options.................264
10.4.1 American Call .......................264
10.5 One-phase Stefan Problem.....................268
10.6 Free-Boundary of a Call......................275
10.7 The Call as a Free Boundary Problem..............279
10.7.1 Dynamics of the Free-Boundary..............283
10.7.2 Local Analysis near Maturity...............285
10.7.3 The Infinite Horizon Case.................289
10.8 American Put................................292
10.8.1 Properties of the Free-Boundary of a Put........292
10.8.2 The Put as a Free-Boundary Problem..........294
10.8.3 The Simplified Free-Boundary Problem .........295
10.8.4 An Integral Equation for the Free-Boundary.......297
10.8.5 Approximation of the Boundary near Maturity.....301
10.8.6 The Infinite Horizon Case.................303
10.9 MacMillan-Barone-Adesi-Whaley Formula............305
10.9.1 Analytic Approximation to a Call ............305
10.9.2 Analytic Approximation to a Put.............309
V Stochastic Volatility and Return Models 311
11 Heston Model 313
11.1 Heston Equation..........................313
11.1.1 Dynamics of the Model ..................313
11.1.2 Setting up the Heston PDE................316
11.1.3 Simplifying the Heston PDE ...............318
11.2 Boundary Conditions for European Call.............319
11.2.1 PDEs for 7Ti and 7r2 ....................322
11.2.2 Finding 7Ti and tt2.....................323
11.2.3 Solving the Riccati Equation...............326
11.2.4 Finding the Function C(t,w)...............327
11.2.5 The Final Formula.....................328
11.2.6 Concluding Formula....................332
11.3 Delta of a Call...........................333
11.3.1 Interpretation of f2...................334
11.4 American Options .........................335
11.4.1 Calls.............................336
11.4.2 Call Perpetuity.......................338
11.4.3 Analytic Approximation to a Call ............341
12 GARCH Model 347
12.1 GARCH (1,1) Differential Model..................347
12.2 Dynamics of the GARCH Model .................351
12.3 The GARCH PDE.........................352
12.4 Simplifying the PDE........................353
12.5 Solving the Equation........................354
13 AR(1) Model 359
13.1 AR(1)-Differential Model .....................359
13.2 The PDE..............................301
14 Stochastic Return Models 365
14.1 Mean-reverting Ornstein-Uhlenbeck Process
.........365
14.2 A Continuous VAR Process....................300
14.3 Probability Density of St......................370
14.4 The PDE..............................370
14.5 Simplifying the PDE........................378
14.6 Solving the PDE..........................380
14.7 All-or-Nothing Option.......................384
15 Hints and Solutions 391
Appendix 419
A Useful Transforms 419
A.l The Fourier Transform.......................419
A.2 The Laplace Transform ......................421
B Probability Concepts 423
B.l Events and Probability.......................423
B.2 Stochastic Processes................. .... 424
B.3 Expectation................................424
B.4 Conditional Expectations.....................425
B.5 Martingales..........................426
B-6 Submartingales and their Properties .... 427
B.7 Jensen s Inequality........................428
C Elements of Stochastic Calculus 431
C.l The Poisson Process......................431
C.2 The Brownian Motion.........................434
C.3 Exponential Process............436
0.4 Ito s Lemma.............................436
0-5 Girsanov Theorem...........438
C.6 Ito Integral.............................439
C.7 Brownian Motion with Drift....................441
C.8 The Generator of an Ito Diffusion.................442
D Series and Equations 447
D.l Confluent Hypergeometric Equation...............447
D.2 Duhamel s Principle........................448
Bibliography 451
Index 457
|
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| physical | xix, 461 Seiten Diagramme |
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| publisher | World Scientific |
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| spelling | Calin, Ovidiu L. 1971- Verfasser (DE-588)128990554 aut Deterministic and Stochastic Topics in Computational Finance Ovidiu Calin, Princeton University New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2017] © 2017 xix, 461 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index FinancexMathematical models Stochastic analysis Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s Finanzmathematik (DE-588)4017195-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029615900&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Calin, Ovidiu L. 1971- Deterministic and Stochastic Topics in Computational Finance FinancexMathematical models Stochastic analysis Finanzmathematik (DE-588)4017195-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4017195-4 (DE-588)4132272-1 |
| title | Deterministic and Stochastic Topics in Computational Finance |
| title_auth | Deterministic and Stochastic Topics in Computational Finance |
| title_exact_search | Deterministic and Stochastic Topics in Computational Finance |
| title_full | Deterministic and Stochastic Topics in Computational Finance Ovidiu Calin, Princeton University |
| title_fullStr | Deterministic and Stochastic Topics in Computational Finance Ovidiu Calin, Princeton University |
| title_full_unstemmed | Deterministic and Stochastic Topics in Computational Finance Ovidiu Calin, Princeton University |
| title_short | Deterministic and Stochastic Topics in Computational Finance |
| title_sort | deterministic and stochastic topics in computational finance |
| topic | FinancexMathematical models Stochastic analysis Finanzmathematik (DE-588)4017195-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | FinancexMathematical models Stochastic analysis Finanzmathematik Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029615900&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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