# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

5
The Black-Scholes Model
This chapter presents the basic Black-Scholes model of arbitrage pricing in continuous time, as well as extensions to a nonparametric multivariate Markov setting. We first introduce the Brownian model of uncertainty and continuous security trading, and then derive partial differential equations for the arbitrage-free prices of derivative securities. The classic example is the Black-Scholes option-pricing formula. Chapter 6 extends to a non-Markovian setting using more general techniques.
A. Trading Gains for Brownian Prices
We fix a probability space (Ω, F, P). A process is a measurable function on Ω x [0, ∞) into R. (For a definition of measurability with respect to a product space of this variety, see Appendix C.) The value of a process X at time t is the random variable variously written as Xt, X(t), or X, t): Ω → R. A standard Brownian motion is a process B defined by the properties:
 a. B0 = 0 almost surely; b. for any times t and s > t, Bs - Bt is normally distributed with mean zero and variance s - t; c. for any times t0, …, tn such that 0 ≤ t0 < t1 < … < tn < ∞, the random variables B(t0), B(t1) - B(t0), …, B(tn) - B(tn-1) are independently distributed; and d. for each ω in Ω, the sample path t ↦ B(ω, t) is continuous.

It is a nontrivial fact, whose proof has a colorful history, that the probability space (Ω, F, P) can be constructed so that there exist standard

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Dynamic Asset Pricing Theory

• Title Page iii
• Contents vii
• Preface xiii
• I - Discrete-Time Models 1
• 1 - Introduction to State Pricing 3
• 2 - The Basic Multiperiod Model 21
• 3 - The Dynamic Programming Approach 49
• 4 - The Infinite-Horizon Setting 65
• II - Continuous-Time Models 81
• 5 - The Black-Scholes Model 83
• 6 - State Prices and Equivalent Martingale Measures 101
• 7 - Term-Structure Models 135
• 8 - Derivative Pricing 167
• 9 - Portfolio and Consumption Choice 203
• 10 - Equilibrium 235
• 11 - Corporate Securities 259
• 12 - Numerical Methods 293
• Appendixes 321
• Bibliography 373
• Symbol Glossary 445
• Author Index 447
• Subject Index 457
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