Stochastic Calculus for Finance II,9780387401010

Stochastic Calculus for Finance II

by
ISBN13: 9780387401010
ISBN10: 0387401016
Format: Hardcover
Pub. Date: 2004-06-30
Publisher(s): Springer Verlag
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Summary

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.

Author Biography

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.

Table of Contents

1 General Probability Theory
1(48)
1.1 Infinite Probability Spaces
1(6)
1.2 Random Variables and Distributions
7(6)
1.3 Expectations
13(10)
1.4 Convergence of Integrals
23(4)
1.5 Computation of Expectations
27(5)
1.6 Change of Measure
32(7)
1.7 Summary
39(2)
1.8 Notes
41(1)
1.9 Exercises
41(8)
2 Information and Conditioning
49(34)
2.1 Information and o-algebras
49(4)
2.2 Independence
53(13)
2.3 General Conditional Expectations
66(9)
2.4 Summary
75(2)
2.5 Notes
77(1)
2.6 Exercises
77(6)
3 Brownian Motion
83(42)
3.1 Introduction
83(1)
3.2 Scaled Random Walks
83(10)
3.2.1 Symmetric Random Walk
83(1)
3.2.2 Increments of the Symmetric Random Walk
84(1)
3.2.3 Martingale Property for the Symmetric Random Walk
85(1)
3.2.4 Quadratic Variation of the Symmetric Random Walk
85(1)
3.2.5 Scaled Symmetric Random Walk
86(2)
3.2.6 Limiting Distribution of the Scaled Random Walk
88(3)
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
91(2)
3.3 Brownian Motion
93(5)
3.3.1 Definition of Brownian Motion
93(2)
3.3.2 Distribution of Brownian Motion
95(2)
3.3.3 Filtration for Brownian Motion
97(1)
3.3.4 Martingale Property for Brownian Motion
98(1)
3.4 Quadratic Variation
98(9)
3.4.1 First-Order Variation
99(2)
3.4.2 Quadratic Variation
101(5)
3.4.3 Volatility of Geometric Brownian Motion
106(1)
3.5 Markov Property
107(1)
3.6 First Passage Time Distribution
108(3)
3.7 Reflection Principle
111(4)
3.7.1 Reflection Equality
111(1)
3.7.2 First Passage Time Distribution
112(1)
3.7.3 Distribution of Brownian Motion and Its Maximum
113(2)
3.8 Summary
115(1)
3.9 Notes
116(1)
3.10 Exercises
117(8)
4 Stochastic Calculus
125(84)
4.1 Introduction
125(1)
4.2 Itô's Integral for Simple Integrands
125(7)
4.2.1 Construction of the Integral
126(2)
4.2.2 Properties of the Integral
128(4)
4.3 Ito's Integral for General Integrands
132(5)
4.4 Itô-Doeblin Formula
137(16)
4.4.1 Formula for Brownian Motion
137(6)
4.4.2 Formula for Ito Processes
143(4)
4.4.3 Examples
147(6)
4.5 Black-Scholes-Merton Equation
153(11)
4.5.1 Evolution of Portfolio Value
154(1)
4.5.2 Evolution of Option Value
155(1)
4.5.3 Equating the Evolutions
156(2)
4.5.4 Solution to the Black-Scholes-Merton Equation
158(1)
4.5.5 The Greeks
159(3)
4.5.6 Put-Call Parity
162(2)
4.6 Multivariable Stochastic Calculus
164(8)
4.6.1 Multiple Brownian Motions
164(1)
4.6.2 Itô-Doeblin Formula for Multiple Processes
165(3)
4.6.3 Recognizing a Brownian Motion
168(4)
4.7 Brownian Bridge
172(11)
4.7.1 Gaussian Processes
172(3)
4.7.2 Brownian Bridge as a Gaussian Process
175(1)
4.7.3 Brownian Bridge as a Scaled Stochastic Integral
176(2)
4.7.4 Multidimensional Distribution of the Brownian Bridge
178(4)
4.7.5 Brownian Bridge as a Conditioned Brownian Motion
182(1)
4.8 Summary
183(4)
4.9 Notes
187(2)
4.10 Exercises
189(20)
5 Risk-Neutral Pricing
209(54)
5.1 Introduction
209(1)
5.2 Risk-Neutral Measure
210(11)
5.2.1 Girsanov's Theorem for a Single Brownian Motion
210(4)
5.2.2 Stock Under the Risk-Neutral Measure
214(3)
5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure
217(1)
5.2.4 Pricing Under the Risk-Neutral Measure
218(1)
5.2.5 Deriving the Black-Scholes-Merton Formula
218(3)
5.3 Martingale Representation Theorem
221(3)
5.3.1 Martingale Representation with One Brownian Motion
221(1)
5.3.2 Hedging with One Stock
222(2)
5.4 Fundamental Theorems of Asset Pricing
224(10)
5.4.1 Girsanov and Martingale Representation Theorems
224(2)
5.4.2 Multidimensional Market Model
226(2)
5.4.3 Existence of the Risk-Neutral Measure
228(3)
5.4.4 Uniqueness of the Risk-Neutral Measure
231(3)
5.5 Dividend-Paying Stocks
234(6)
5.5.1 Continuously Paying Dividend
235(2)
5.5.2 Continuously Paying Dividend with Constant Coefficients
237(1)
5.5.3 Lump Payments of Dividends
238(1)
5.5.4 Lump Payments of Dividends with Constant Coefficients
239(1)
5.6 Forwards and Futures
240(8)
5.6.1 Forward Contracts
240(1)
5.6.2 Futures Contracts
241(6)
5.6.3 Forward-Futures Spread
247(1)
5.7 Summary
248(2)
5.8 Notes
250(1)
5.9 Exercises
251(12)
6 Connections with Partial Differential Equations
263(32)
6.1 Introduction
263(1)
6.2 Stochastic Differential Equations
263(3)
6.3 The Markov Property
266(2)
6.4 Partial Differential Equations
268(4)
6.5 Interest Rate Models
272(5)
6.6 Multidimensional Feynman-Kac Theorems
277(3)
6.7 Summary
280(1)
6.8 Notes
281(1)
6.9 Exercises
282(13)
7 Exotic Options
295(44)
7.1 Introduction
295(1)
7.2 Maximum of Brownian Motion with Drift
295(4)
7.3 Knock-out Barrier Options
299(9)
7.3.1 Up-and-Out Call
300(1)
7.3.2 Black-Scholes-Merton Equation
300(4)
7.3.3 Computation of the Price of the Up-and-Out Call
304(4)
7.4 Lookback Options
308(12)
7.4.1 Floating Strike Lookback Option
308(1)
7.4.2 Black-Scholes-Merton Equation
309(3)
7.4.3 Reduction of Dimension
312(2)
7.4.4 Computation of the Price of the Lookback Option
314(6)
7.5 Asian Options
320(11)
7.5.1 Fixed-Strike Asian Call
320(1)
7.5.2 Augmentation of the State
321(2)
7.5.3 Change of Numeraire
323(8)
7.6 Summary
331(1)
7.7 Notes
331(1)
7.8 Exercises
332(7)
8 American Derivative Securities
339(36)
8.1 Introduction
339(1)
8.2 Stopping Times
340(5)
8.3 Perpetual American Put
345(11)
8.3.1 Price Under Arbitrary Exercise
346(3)
8.3.2 Price Under Optimal Exercise
349(2)
8.3.3 Analytical Characterization of the Put Price
351(2)
8.3.4 Probabilistic Characterization of the Put Price
353(3)
8.4 Finite-Expiration American Put
356(5)
8.4.1 Analytical Characterization of the Put Price
357(2)
8.4.2 Probabilistic Characterization of the Put Price
359(2)
8.5 American Call
361(7)
8.5.1 Underlying Asset Pays No Dividends
361(2)
8.5.2 Underlying Asset Pays Dividends
363(5)
8.6 Summary
368(1)
8.7 Notes
369(1)
8.8 Exercises
370(5)
9 Change of Numeraire
375(28)
9.1 Introduction
375(1)
9.2 Numeraire
376(5)
9.3 Foreign and Domestic Risk-Neutral Measures
381(11)
9.3.1 The Basic Processes
381(2)
9.3.2 Domestic Risk-Neutral Measure
383(2)
9.3.3 Foreign Risk-Neutral Measure
385(2)
9.3.4 Siegel's Exchange Rate Paradox
387(1)
9.3.5 Forward Exchange Rates
388(2)
9.3.6 Garman-Kohlhagen Formula
390(1)
9.3.7 Exchange Rate Put-Call Duality
390(2)
9.4 Forward Measures
392(5)
9.4.1 Forward Price
392(1)
9.4.2 Zero-Coupon Bond as Numeraire
392(2)
9.4.3 Option Pricing with a Random Interest Rate
394(3)
9.5 Summary
397(1)
9.6 Notes
398(1)
9.7 Exercises
398(5)
10 Term-Structure Models 403(58)
10.1 Introduction
403(2)
10.2 Affine-Yield Models
405(1)
10.2.1 Two-Factor Vasicek Model
406(1)
10.2.2 Two-Factor CIR Model
420(1)
10.2.3 Mixed Model
422(1)
10.3 Heath-Jarrow-Morton Model
423(1)
10.3.1 Forward Rates
423(1)
10.3.2 Dynamics of Forward Rates and Bond Prices
425(1)
10.3.3 No-Arbitrage Condition
426(1)
10.3.4 HJM Under Risk-Neutral Measure
429(1)
10.3.5 Relation to Affine-Yield Models
430(1)
10.3.6 Implementation of HJM
432(3)
10.4 Forward LIBOR Model
435(1)
10.4.1 The Problem with Forward Rates
435(1)
10.4.2 LIBOR and Forward LIBOR
436(1)
10.4.3 Pricing a Backset LIBOR Contract
437(1)
10.4.4 Black Caplet Formula
438(1)
10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities
440(1)
10.4.6 A Forward LIBOR Term-Structure Model
442(5)
10.5 Summary
447(3)
10.6 Notes
450(1)
10.7 Exercises
451(10)
11 Introduction to Jump Processes 461(66)
11.1 Introduction
461(1)
11.2 Poisson Process
462(1)
11.2.1 Exponential Random Variables
462(1)
11.2.2 Construction of a Poisson Process
463(1)
11.2.3 Distribution of Poisson Process Increments
463(1)
11.2.4 Mean and Variance of Poisson Increments
466(1)
11.2.5 Martingale Property
467(1)
11.3 Compound Poisson Process
468(1)
11.3.1 Construction of a Compound Poisson Process
468(1)
11.3.2 Moment-Generating Function
470(3)
11.4 Jump Processes and Their Integrals
473(1)
11.4.1 Jump Processes
474(1)
11.4.2 Quadratic Variation
479(4)
11.5 Stochastic Calculus for Jump Processes
483(1)
11.5.1 Itô-Doeblin Formula for One Jump Process
483(1)
11.5.2 Itô-Doeblin Formula for Multiple Jump Processes
489(3)
11.6 Change of Measure
492(1)
11.6.1 Change of Measure for a Poisson Process
493(1)
11.6.2 Change of Measure for a Compound Poisson Process
495(1)
11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion
502(3)
11.7 Pricing a European Call in a Jump Model
505(1)
11.7.1 Asset Driven by a Poisson Process
505(1)
11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process
512(11)
11.8 Summary
523(2)
11.9 Notes
525(1)
11.10 Exercises
525(2)
A Advanced Topics in Probability Theory 527(1)
A.1 Countable Additivity
527(3)
A.2 Generating σ-algebras
530(1)
A.3 Random Variable with Neither Density nor Probability Mass Function
531(2)
B Existence of Conditional Expectations 533(2)
C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing 535(2)
References 537(8)
Index 545

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