## Stochastic Calculus for Finance II: Continuous-Time Models, Volume 11Stochastic 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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### 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 |

537 | |

545 | |