Derivative security pricing: techniques, methods and applications
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
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Berlin ; Heidelberg
Springer
[2015]
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| Series: | Dynamic modeling and econometrics in economics and finance
21 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 616 Seiten Diagramme |
| ISBN: | 9783662459058 |
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MARC
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| 245 | 1 | 0 | |a Derivative security pricing |b techniques, methods and applications |c Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos |
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Contents
Part I The Fundamentals of Derivative Security Pricing
1 The Stock Option Problem. 3
1.1 Introduction. 3
1.2 The European Call Option. 3
2 Stochastic Processes for Asset Price Modelling . 7
2.1 Introduction. 7
2.2 Markov Processes. 9
2.3 The Time Evolution of Conditional Probabilities. 13
2.4 Processes with Continuous Sample Paths . 14
2.4.1 Brownian Motion. 15
2.4.2 The Cauchy Process . 16
2.5 The Dirac Delta Function. 17
2.6 The Fokker-Planck and Kolmogorov Equations. 19
2.7 Appendix. 24
2.8 Problems.- —. 34
3 An Initial Attempt at Pricing an Option . 37
3.1 Option Pricing as a Discounted Cash Flow Calculation. 37
3.2 Our First Glimpse of a Martingale . 42
3.3 Our First Glimpse of the Feynman -Kae Formula. 44
3 A Appendix. 47
3.5 Problems.*. 50
4 The Stochastic Differential Equation. 55
4.1 Introduction. 55
4.2 A First Encounter with the Stochastic Differential Equation. 56
4.3 Three Examples of Markov Processes. 59
4.3.1 The Wiener Process. 59
4.3.2 The Omstein-UhlenbeckProcess. 63
4.3.3 The Poisson Process. 65
ix
X
Contents
4.4 Autocovariance Behaviour and White Noise. 66
4.5 Modelling Uncertain Price Dynamics. 70
4.6 Proceeding to the Continuous Time Limit. 72
4.7 The Stochastic Integral. 75
4.8 An Example of Stochastic Integral. 79
4.9 The Proper Definition of the Stochastic Differential Equation . 81
4.10 The Stratonovich Stochastic Integral. 82
4.11 Appendix. 84
4.12 Problems . 89
5 Manipulating Stochastic Differential Equations
and Stochastic Integrals . 93
5.1 The Basic Rules of Stochastic Calculus . 93
5.2 Some Basic Stochastic Integrals . 96
5.3 Higher Dimensional Stochastic Differential Equations. 99
5.3.1 The Two-Noise Case. 101
5.3.2 The Three-Noise Case. 104
5.4 The Kolmogorov Equation for an n-Dimensional
Diffusion System. 105
5.5 The Differential of a Stochastic Integral. 106
5.6 Appendix. 107
5.7 Problems . 108
6 Ito’s Lemma and Its Applications . Ill
6.1 Introduction. Ill
6.2 Ito’s Lemma. 112
6.2.1 Introduction. 112
6.2.2 S tatement and Proof of Ito ’ s Lemma. 113
6.3 Applications of Ito’s Lemma. 114
6.3.1 Function of a Geometric Stock Price Process. 115
6.3.2 The Lognormal Asset Price Process. 115
6.3.3 Exponential Functions. 118
6.3.4 CalculatingE[ex^] . 118
6.3.5 The Omstein—Uhlenbeck Process . 122
6.3.6 Brownian Bridge Processes . 123
6.3.7 White Noise and Colored Noise Processes. 126
6.4 A More Formal Statement of Ito’s Lemma. 129
6.5 Ito’s Lemma in Several Variables. 131
6.5.1 Correlated Wiener Processes. 131
6.5.2 Independent Wiener Processes. 133
6.6 The Stochastic Differential Equation Followed
by the Quotient of Two Diffusions. 135
6.7 Problems . 137
Contents xi
7 The Continuous Hedging Argument 145
7.1 The Continuous Hedging Argument: The
Black-Scholes Approach. 145
7.2 Interpreting the No-Arbitrage Condition . 148
7.3 Alternative Hedgmg Portfolios: The Merton’s Approach. 150
7.4 Self Financing Strategy: The Modem Approach. 151
7.5 Appendix. 155
7.6 Problems . 156
8 The Martingale Approach. 157
8.1 Martingales. 157
8.1.1 Introduction. 157
8.1.2 Examples of Martingales. 158
8.1.3 The Exponential Martingale. 159
8.1.4 Quadratic Variation Processes. 161
8.1.5 Semimartingales. 164
8.2 Changes of Measure and Girsanov’s Theorem. 164
8.3 Girsanov’s Theorem for Vector Processes. 170
8.4 Derivation of Black-Scholes Formula by Girsanov’s
Theorem. 174
8.5 The Pricing Kernel Representation . 178
8.6 The Feynman- Kac Formula. 179
8.7 Appendix. 183
8.8 Problems . 189
9 The Partial Differential Equation Approach Under
Geometric Brownian Motion. 191
9.1 Introduction. 191
9.2 The Transition Density Function for Geometric
Brownian Motion. 192
9.3 The Fourier Transform. 193
9.4 Solutions for Specific Payoff Functions . 198
9.4.1 The Kolmogorov Equation. 198
9.4.2 The European Digital Option. 199
9.4.3 The European All֊or֊Nothing Option . 200
9.4.4 The European Call Option. 200
9.5 Interpreting the General Pricing Relation.* - - -.-. 201
9.6 Appendix. 203
10 Pricing Derivative Securities: A General Approach. 207
10.1 Risk Neutral Valuation. 207
10.2 The Market Price of Risk. 208
10.2.1 Tradable Asset. 209
10.2.2 Non-tradable Asset . 210
211
214
217
219
224
230
234
235
235
235
235
236
237
239
244
249
251
251
252
253
259
265
266
269
270
273
273
274
276
278
282
286
288
295
295
296
304
305
308
310
10.3 Pricing Derivative Securities Dependent on Two Factors .
10.3.1 Two Traded Assets.
10.3.2 Two Traded Assets-Vector Notation.
10.3.3 One Traded Asset and One Non-traded Asset.
10.4 The General Case.
10.5 Appendix.
10.6 Problems .
Applying the General Pricing Framework
11.1 Introduction.
11.2 One-Factor Examples.
11.2.1 Stock Options.
11.2.2 Foreign Currency Options.
11.2.3 Futures Options.
11.3 Options on Two Underlying Factors.
11.4 Appendix.
11.5 Problems .
Jump-DiiTusion Processes.
12.1 Introduction.
12.2 Mathematical Description of the lump Process.
12.2.1 Absolute lumps.
12.2.2 Proportional lumps .
12.2.3 A General Process of Dependent Jump Size .
12.3 Ito’s Lemma for Jump-Diffusion Processes .
12.4 Appendix.
12.5 Problems .
Option Pricing Under Jump-Diffusion Processes.
13.1 Introduction.
13.2 Constructing a Hedging Portfolio.
13.3 Pricing the Option .
13.4 General Form of the Solution.
13.5 Alternative Ways of Completing the Market
13.6 Large Jumps.
13.7 Appendix.
Partial Differential Equation Approach Under Geometric
Jump-Diffusion Process . .
14.1 The Integro-Partial Differential Equation.
14.2 The Fourier Transform.
14.3 Evaluating the Kernel Function Under a Log-Normal
Jump Distribution.
14.4 Option Valuation Under a Log-Normal Jump Distribution
14.5 Using the Expectation Operator to Evaluate
the Option Under Log-Normal Jumps.
14.6 Appendix.
Contents
Xlll
15 Stochastic Volatility . 315
15.1 Introduction. 315
15.2 Modelling Stochastic Volatility . 319
15.3 Option Pricing Under Stochastic Volatility. 322
15.4 The Mean Reverting Volatility Case. 324
15.5 The Heston Model. 326
15.6 Appendix. 330
15.7 Problems . 344
16 Pricing the American Feature. 349
16.1 Introduction. 349
16.2 The Conventional Approach Based on Compound Options. 350
16.3 A General Formulation. 353
16.3.1 The Free Boundary Value Problem. 353
16.3.2 Transforming the Partial Differential Equation. 355
16.3.3 Applying the Fourier Transform. 357
16.3.4 Inverting the Fourier Transform. 361
16.4 An Approximate Solution. 365
16.5 Appendix. 368
16.6 Problems . 369
17 Pricing Options Using Binomial Trees . 371
17.1 Introduction. 371
17.2 The Binomial Model. 372
17.2.1 The Binomial Stock Price Process. 372
17.2.2 Option Pricing in the One-Period Model. 374
17.2.3 Two Period Binomial Option Pricing. 377
17.2.4 n-Period Binomial Option Pricing. 378
17.3 The Continuous Limit. 380
17.3.1 The Limiting Binomial Distribution. 381
17.3.2 The Black-Scholes Partial Differential
Equation as the Limit of the Binomial. 382
17.3.3 The Binomial as a Discretisation
of the Black—Scholes Partial Differential Equation — 383
17.4 Choice of the Parameters utd. 385
18 Volatility Smiles. 389
18.1 Introduction. 389
18.2 Stochastic Volatility as the Origin of the Smile . 391
18.3 Calibrating Deterministic Models to the Smile. 394
18.4 Problems . 401
XIV
Contents
Part II Interest Rate Modelling
19 Allowing for Stochastic Interest Rates
in the Black—Scholes Model . 405
19.1 Introduction. 405
19.2 The Hedging Portfolio. 407
19.3 Solving for the Option Price. 410
19.4 Appendix. 414
19.5 Problems . 415
20 Change of Numeraire. 419
20.1 A Change of Numeraire Theorem. 419
20.2 The Radon—Nikodym Derivative. 423
20.3 Option Pricing Under Stochastic Interest Rates. 424
20.4 Change of Numeraire with Multiple Sources of Risk . 427
20.5 Problems . 430
21 The Paradigm Interest Rate Option Problem. 431
21.1 Interest Rate Caps, Floors and Collars. 431
21.1.1 Interest Rate Caps. 431
21.1.2 Interest Rate Floors. 432
21.1.3 Interest Rate Collars. 433
21.2 Payoff Structure of Interest. Rate Caps and Floors . 434
21.3 Relationship to Bond Options. 435
21.4 The Inherent Difficulty of the Interest Rate Option Problem. 435
22 Modelling Interest Rate Dynamics . 439
22.1 The Relationship Between Interest Rates, Bond
Prices and Forward Rates. 439
22.2 Modelling the Spot Interest Rate. 443
22.3 Motivating the Feller (or Square Root) Process. 447
22.4 Fubini’s Theorem . 452
22.5 Modelling Forward Rates. 454
22.5.1 From Forward Rate to Bond Price Dynamics. 457
22.5.2 A Specific Example. . 459
22.6 Appendix . 463
22.7 Problems . 464
23 Interest Rate Derivatives: One Factor Spot Rate Models. 469
23.1 Introduction . 469
23.2 Arbitrage Models of the Term Structure. 470
23.3 The Martingale Representation. 473
23.4 Some Specific Term Structure Models. 475
23.4.1 The Vasicek Model . 475
23.4.2 The Hull-White Model. 479
23.4.3 The Cox-Ingersoll—Ross (CIR) Model. 481
Calculation of the Bond Price from the Expectation Operator . 484
23.5
Contents
XV
23.6 Pricing Bond Options. 486
23.7 Solving the Option Pricing Equation . 492
23.7.1 The Hull White Model. 492
23.7.2 The CIR Model. 494
23.8 Rendering Spot Rate Models Preference
Free-Calibration to the Currently Observed Yield Curve. 496
23.9 Appendix. 499
23.10 Problems . 503
24 Interest Rate Derivatives: Multi-Factor Models . 505
24.1 Hull-White Two-Factor Model . 505
24.1.1 Bond Price. 506
24.1.2 Option Prices. 510
24.2 The General Framework. 513
24.2.1 Bond Pricing. 513
24.2.2 Bond Option Pricing. 517
24.3 The Affine Class of Models. 522
24.3.1 The Two-Factor Case. 522
24.4 Problems . 526
25 The Heath-Jarrow-Morton Framework . 529
25.1 Introduction. 529
25.2 The Basic Structure. 531
25.3 The Arbitrage Pricing of Bonds. 533
25.4 Arbitrage Pricing of Bond Options. 536
25.5 Forward-Risk-Adjusted Measure . 538
25.6 Reduction to Markovian Form. 539
25.7 Some Special Models. 542
25.7.1 The Hull-White Extended Vasicek Model . 542
25.7.2 The General Spot Rate Model. 543
25.7.3 The Forward Rate Dependent Volatility Model. 544
25.8 Heath-Jarrow-Morton Multi-Factor Models. 549
25.9 Relating Heath-Jarrow-Morton to Hull-White
Two֊Factor Models. 552
25.10 The Covariance Structure Implied
by the Heath-Jarrow-Morton Model. 555
25.10.1 The Covariance Structure of the Forward
Rate Changes. . 556
25.10.2 The Covariance Structure of the Forward Rate. 557
25.11 Appendix. 559
25.12 Problems . 565
XVI
Contents
26 The LIBOR Market Model 569
26.1 Introduction. 569
26.2 The Brace-Musiela Paraméterisation
of the Heath—Jarrow—Morton Model. 570
26.3 The LIBOR Process. 573
26.4 Lognormal LIBOR Rates. 577
26.5 Pricing Caps. 581
26.6 Pricing Swaptions. 583
26.6.1 Swaps. 583
26.6.2 Swaptions. 586
26.7 Pricing a Caplet Under Gaussian Forward Rate Dynamics. 589
26.8 Appendix. 594
26.9 Problems . 603
References. 605
Index. 611 |
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| id | DE-604.BV042532792 |
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| indexdate | 2024-11-06T09:03:31Z |
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| isbn | 9783662459058 |
| language | English |
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| physical | xvi, 616 Seiten Diagramme |
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| publisher | Springer |
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| series | Dynamic modeling and econometrics in economics and finance |
| series2 | Dynamic modeling and econometrics in economics and finance 21 |
| spelling | Chiarella, Carl 1944-2016 Verfasser (DE-588)121195724 aut Derivative security pricing techniques, methods and applications Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos Berlin ; Heidelberg Springer [2015] © 2015 xvi, 616 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Dynamic modeling and econometrics in economics and finance 21 Volume 21 Finanzinnovation (DE-588)4124975-6 gnd rswk-swf Finanzinnovation (DE-588)4124975-6 s DE-604 He, Xue-zhong Verfasser (DE-588)17182380X aut Nikitopoulos, Christina Sklibosios Verfasser (DE-588)129986372 aut Erscheint auch als Online-Ausgabe 978-3-662-45906-5 Dynamic modeling and econometrics in economics and finance 21 (DE-604)BV012605915 21 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027966978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chiarella, Carl 1944-2016 He, Xue-zhong Nikitopoulos, Christina Sklibosios Derivative security pricing techniques, methods and applications Dynamic modeling and econometrics in economics and finance Finanzinnovation (DE-588)4124975-6 gnd |
| subject_GND | (DE-588)4124975-6 |
| title | Derivative security pricing techniques, methods and applications |
| title_auth | Derivative security pricing techniques, methods and applications |
| title_exact_search | Derivative security pricing techniques, methods and applications |
| title_full | Derivative security pricing techniques, methods and applications Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos |
| title_fullStr | Derivative security pricing techniques, methods and applications Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos |
| title_full_unstemmed | Derivative security pricing techniques, methods and applications Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos |
| title_short | Derivative security pricing |
| title_sort | derivative security pricing techniques methods and applications |
| title_sub | techniques, methods and applications |
| topic | Finanzinnovation (DE-588)4124975-6 gnd |
| topic_facet | Finanzinnovation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027966978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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