# ShreveStochastic Calculus for Finance II

Continuous-Time Models

1. Auflage 2004. Corr. 2. printing 2010, 550 Seiten, Gebunden, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1016 g Reihe: Springer Finance
ISBN: 978-0-387-40101-0
Verlag: Springer

## ShreveStochastic Calculus for Finance II

"A wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions. In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach.It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." --SIAM

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## Weitere Infos & Material

1 General Probability Theory1.1 In.nite Probability Spaces1.2 Random Variables and Distributions1.3 Expectations1.4 Convergence of Integrals1.5 Computation of Expectations1.6 Change of Measure1.7 Summary1.8 Notes1.9 Exercises 2 Information and Conditioning2.1 Information and s-algebras2.2 Independence2.3 General Conditional Expectations2.4 Summary2.5 Notes2.6 Exercises 3 Brownian Motion3.1 Introduction3.2 Scaled Random Walks3.2.1 Symmetric Random Walk3.2.2 Increments of Symmetric Random Walk3.2.3 Martingale Property for Symmetric Random Walk3.2.4 Quadratic Variation of Symmetric Random Walk3.2.5 Scaled Symmetric Random Walk3.2.6 Limiting Distribution of Scaled Random Walk3.2.7 Log-Normal Distribution as Limit of Binomial Model3.3 Brownian Motion3.3.1 Definition of Brownian Motion3.3.2 Distribution of Brownian Motion3.3.3 Filtration for Brownian Motion3.3.4 Martingale Property for Brownian Motion3.4 Quadratic Variation3.4.1 First-Order Variation3.4.2 Quadratic Variation3.4.3 Volatility of Geometric Brownian Motion3.5 Markov Property3.6 First Passage Time Distribution3.7 Re.ection Principle3.7.1 Reflection Equality3.7.2 First Passage Time Distribution3.7.3 Distribution of Brownian Motion and Its Maximum3.8 Summary3.9 Notes3.10 Exercises 4 Stochastic Calculus4.1 Introduction4.2 It o's Integral for Simple Integrands4.2.1 Construction of the Integral4.2.2 Properties of the Integral4.3 It o's Integral for General Integrands4.4 It o-Doeblin Formula4.4.1 Formula for Brownian Motion4.4.2 Formula for It o Processes4.4.3 Examples4.5 Black-Scholes-Merton Equation4.5.1 Evolution of Portfolio Value4.5.2 Evolution of Option Value4.5.3 Equating the Evolutions4.5.4 Solution to the Black-Scholes-Merton Equation4.5.5 TheGreeks4.5.6 Put-Call Parity4.6 Multivariable Stochastic Calculus4.6.1 Multiple Brownian Motions4.6.2 It o-Doeblin Formula for Multiple Processes4.6.3 Recognizing a Brownian Motion4.7 Brownian Bridge4.7.1 Gaussian Processes4.7.2 Brownian Bridge as a Gaussian Process4.7.3 Brownian Bridge as a Scaled Stochastic Integral4.7.4 Multidimensional Distribution of Brownian Bridge4.7.5 Brownian Bridge as Conditioned Brownian Motion4.8 Summary4.9 Notes4.10 Exercises 5 Risk-Neutral Pricing5.1 Introduction5.2 Risk-Neutral Measure5.2.1 Girsanov's Theorem for a Single Brownian Motion5.2.2 Stock Under the Risk-Neutral Measure5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure5.2.4 Pricing Under the Risk-Neutral Measure5.2.5 Deriving the Black-Scholes-Merton Formula5.3 Martingale Representation Theorem5.3.1 Martingale Representation with One Brownian Motion5.3.2 Hedging with One Stock5.4 Fundamental Theorems of Asset Pricing5.4.1 Girsanov and Martingale Representation Theorems5.4.2 Multidimensional Market Model5.4.3 Existence of Risk-Neutral Measure5.4.4 Uniqueness of the Risk-Neutral Measure5.5 Dividend-Paying Stocks5.5.1 Continuously Paying Dividend5.5.2 Continuously Paying Dividend with Constant Coeffcients5.5.3 Lump Payments of Dividends5.5.4 Lump Payments of Dividends with Constant Coeffcients5.6 Forwards and Futures5.6.1 Forward Contracts5.6.2 Futures Contracts5.6.3 Forward-Futures Spread5.7 Summary5.8 Notes5.9 Exercises 6 Connections with Partial Differential Equations6.1 Introduction6.2 Stochastic Differential Equations6.3 The Markov Property6.4 Partial Differential Equations6.5 Interest Rate Models6.6 Multidimensional Feynman-Kac Theorems6.7 Summary6.8 Notes6.9 Exercises 7 Exotic Options7.1 Introduction

Shreve, Steven

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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