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Steven Shreve: Stochastic Calculus and Finance - 1997

The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory and probability theory. In this course, we shall use it for both these purposes. In the binomial asset pricing model, we model stock prices in discrete time, assuming that at each step, the stock price will change to one of two possible values. Let us begin with an initial positive stock price S0. There are two positive numbers, d and u, with 0 | Steven Shreve Stochastic Calculus and Finance Prasad Chalasani Carnegie Mellon University chal@cs.cmu.edu Somesh Jha Carnegie Mellon University sjha@cs.cmu.edu THIS IS A DRAFT please do not distribute Copyright Steven E. Shreve 1996 July 25 1997 Contents 1 Introduction to Probability Theory 11 1.1 The Binomial Asset Pricing Model. 11 1.2 Finite Probability Spaces. 16 1.3 Lebesgue Measure and the Lebesgue Integral. 22 1.4 General Probability Spaces. 30 1.5 Independence. 40 1.5.1 Independence of sets. 40 1.5.2 Independence of CT -algebras. 41 1.5.3 Independence of random variables. 42 1.5.4 Correlation and independence. 44 1.5.5 Independence and conditional expectation. 45 1.5.6 Law of Large Numbers. 46 1.5.7 Central Limit Theorem. 47 2 Conditional Expectation 49 2.1 A Binomial Model for Stock Price Dynamics. 49 2.2 Information. 50 2.3 Conditional Expectation . 52 2.3.1 An example. 52 2.3.2 Definition of Conditional Expectation. 53 2.3.3 Further discussion of Partial Averaging. 54 2.3.4 Properties of Conditional Expectation. 55 2.3.5 Examples from the Binomial Model. 57 2.4 Martingales. 58 1 2 3 Arbitrage Pricing 59 3.1 Binomial Pricing. 59 3.2 General one-step APT. 60 3.3 Risk-Neutral Probability Measure . 61 3.3.1 Portfolio Process. 62 3.3.2 Self-financing Value of a Portfolio Process A. 62 3.4 Simple European Derivative Securities. 63 3.5 The Binomial Model is Complete. 64 4 The Markov Property 67 4.1 Binomial Model Pricing and Hedging. 67 4.2 Computational Issues. 69 4.3 Markov Processes. 70 4.3.1 Different ways to write the Markov property . 70 4.4 Showing that a process is Markov . 73 4.5 Application to Exotic Options . 74 5 Stopping Times and American Options 77 5.1 American Pricing. 77 5.2 Value of Portfolio Hedging an American Option. 79 5.3 Information up to a Stopping Time. 81 6 Properties of American Derivative Securities 85 6.1 The properties. 85 6.2 Proofs of the Properties. 86 6.3 Compound European Derivative Securities. 88 6.4 Optimal Exercise of .