Stochastic Calculus for Finance II: Continuous-Time Models, Volume 11 (Google eBook)

Front Cover
Springer, Jun 3, 2004 - Business & Economics - 550 pages
5 Reviews

Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes.

This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time.

Masters level students and researchers in mathematical finance and financial engineering will find this book useful.

Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.

  

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Review: Stochastic Calculus Models for Finance II: Continuous Time Models (Springer Finance)

User Review  - Austin - Goodreads

This is the best, most readable book on this topic (though make no mistake, it is still a graduate level mathematics text). The .pdf of Shreve's lecture notes that eventually became this book have had a loyal following on the net for years. This should be on every quant's shelf. Read full review

Review: Stochastic Calculus Models for Finance II: Continuous Time Models (Springer Finance)

User Review  - Joe Malicki - Goodreads

If I'm going to learn stochastic calculus this rigorously, I want more in-depth treatment of Ito's lemma and the like to know how much to believe it and what the proofs really depend on. If you're not ... Read full review

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Contents

General Probability Theory
1
12 Random Variables and Distributions
7
13 Expectations
13
14 Convergence of Integrals
23
15 Computation of Expectations
27
16 Change of Measure
32
17 Summary
39
18 Notes
41
73 Knockout Barrier Options
299
731 UpandOut Call
300
733 Computation of the Price of the UpandOut Call
304
74 Lookback Options
308
742 BlackScholesMerton Equation
309
743 Reduction of Dimension
312
744 Computation of the Price of the Lookback Option
314
75 Asian Options
320

Information and Conditioning
49
22 Independence
53
23 General Conditional Expectations
66
24 Summary
75
25 Notes
77
Brownian Motion
83
322 Increments of the Symmetric Random Walk
84
323 Martingale Property for the Symmetric Random Walk
85
325 Scaled Symmetric Random Walk
86
326 Limiting Distribution of the Scaled Random Walk
88
327 LogNormal Distribution as the Limit of the Binomial Model
91
332 Distribution of Brownian Motion
95
333 Filtration for Brownian Motion
97
334 Martingale Property for Brownian Motion
98
341 FirstOrder Variation
99
342 Quadratic Variation
101
343 Volatility of Geometric Brownian Motion
106
35 Markov Property
107
36 First Passage Time Distribution
108
37 Reflection Principle
111
372 First Passage Time Distribution
112
373 Distribution of Brownian Motion and Its Maximum
113
38 Summary
115
39 Notes
116
310 Exercises
117
Stochastic Calculus
125
421 Construction of the Integral
126
422 Properties of the Integral
128
43 Itos Integral for General Integrands
132
44 ItoDoeblin Formula
137
442 Formula for Ito Processes
143
443 Examples
147
45 BlackScholesMerton Equation
153
451 Evolution of Portfolio Value
154
452 Evolution of Option Value
155
453 Equating the Evolutions
156
454 Solution to the BlackScholesMerton Equation
158
455 The Greeks
159
456 PutCall Parity
162
462 ItoDoeblin Formula for Multiple Processes
165
463 Recognizing a Brownian Motion
168
47 Brownian Bridge
172
472 Brownian Bridge as a Gaussian Process
175
473 Brownian Bridge as a Scaled Stochastic Integral
176
474 Multidimensional Distribution of the Brownian Bridge
178
475 Brownian Bridge as a Conditioned Brownian Motion
182
48 Summary
183
49 Notes
187
410 Exercises
189
RiskNeutral Pricing
209
522 Stock Under the RiskNeutral Measure
214
523 Value of Portfolio Process Under the RiskNeutral Measure
217
524 Pricing Under the RiskNeutral Measure
218
53 Martingale Representation Theorem
221
532 Hedging with One Stock
222
54 Fundamental Theorems of Asset Pricing
224
542 Multidimensional Market Model
226
543 Existence of the RiskNeutral Measure
228
544 Uniqueness of the RiskNeutral Measure
231
55 DividendPay ing Stocks
234
551 Continuously Paying Dividend
235
552 Continuously Paying Dividend with Constant Coefficients
237
553 Lump Payments of Dividends
238
554 Lump Payments of Dividends with Constant Coefficients
239
56 Forwards and Futures
240
562 Futures Contracts
241
563 ForwardFutures Spread
247
57 Summary
248
58 Notes
250
59 Exercises
251
Connections with Partial Differential Equations
263
63 The Markov Property
266
64 Partial Differential Equations
268
65 Interest Rate Models
272
66 Multidimensional FeynmanKac Theorems
277
67 Summary
280
68 Notes
281
69 Exercises
282
Exotic Options
295
752 Augmentation of the State
321
753 Change of Numeraire
323
76 Summary
331
78 Exercises
332
American Derivative Securities
339
82 Stopping Times
340
83 Perpetual American Put
345
831 Price Under Arbitrary Exercise
346
832 Price Under Optimal Exercise
349
833 Analytical Characterization of the Put Price
351
834 Probabilistic Characterization of the Put Price
353
84 FiniteExpiration American Put
356
841 Analytical Characterization of the Put Price
357
842 Probabilistic Characterization of the Put Price
359
85 American Call
361
852 Underlying Asset Pays Dividends
363
86 Summary
368
87 Notes
369
88 Exercises
370
Change of Numeraire
375
92 Numeraire
376
93 Foreign and Domestic RiskNeutral Measures
381
932 Domestic RiskNeutral Measure
383
933 Foreign RiskNeutral Measure
385
934 Siegels Exchange Rate Paradox
387
935 Forward Exchange Rates
388
936 GarmanKohlhagen Formula
390
94 Forward Measures
392
943 Option Pricing with a Random Interest Rate
394
95 Summary
397
96 Notes
398
10 TermStructure Models
403
102 AffineYield Models
405
1021 TwoFactor Vasicek Model
406
1022 TwoFactor CIR Model
420
1023 Mixed Model
422
103 HeathJarrowMorton Model
423
1032 Dynamics of Forward Rates and Bond Prices
425
1033 NoArbitrage Condition
426
1034 HJM Under RiskNeutral Measure
429
1035 Relation to AffineYield Models
430
1036 Implementation of HJM
432
104 Forward LIBOR Model
435
1042 LIBOR and Forward LIBOR
436
1043 Pricing a Backset LIBOR Contract
437
1044 Black Caplet Formula
438
1045 Forward LIBOR and ZeroCoupon Bond Volatilities
440
1046 A Forward LIBOR TermStructure Model
442
105 Summary
447
106 Notes
450
107 Exercises
451
Introduction to Jump Processes
461
112 Poisson Process
462
1122 Construction of a Poisson Process
463
1124 Mean and Variance of Poisson Increments
466
1125 Martingale Property
467
113 Compound Poisson Process
468
1132 MomentGenerating Function
470
114 Jump Processes and Their Integrals
473
1141 Jump Processes
474
1142 Quadratic Variation
479
115 Stochastic Calculus for Jump Processes
483
1152 ItoDoeblin Formula for Multiple Jump Processes
489
116 Change of Measure
492
1161 Change of Measure for a Poisson Process
493
1162 Change of Measure for a Compound Poisson Process
495
1163 Change of Measure for a Compound Poisson Process and a Brownian Motion
502
117 Pricing a European Call in a Jump Model
505
1172 Asset Driven by a Brownian Motion and a Compound Poisson Process
512
118 Summary
523
119 Notes
525
Advanced Topics in Probability Theory
527
A2 Generating cralgebras
530
A3 Random Variable with Neither Density nor Probability Mass Function
531
B Existence of Conditional Expectations
533
C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing
535
References
537
Index
545
Copyright

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Page 542 - Carr, P., Jarrow, R. and Myneni, R. [1992] Alternative Characterizations of American Put Options Mathematical Finance 2, 87-106.

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About the author (2004)

Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.