# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

12
Numerical Methods

THIS CHAPTER REVIEWS three numerical approaches to pricing securities in a continuous-time setting: “binomial” approximation, Monte Carlo simulation, and finite-difference solution of the associated partial differential equation.

A. Central Limit Theorems

It is well known that a normal random variable can be represented as the limit of normalized sums of Bernoulli trials, that is, i.i.d. binomial random variables. This idea, a version of the Central Limit Theorem, leads to the characterization given in this section of the Black-Scholes option-pricing formula (equation [5.11]) as the limit of the binomial option-pricing formula (equation [2.16]), letting the number of trading periods per unit of time go to infinity. Aside from making an interesting connection between the discrete- and continuous-time settings, this also suggests a numerical recipe for calculating continuous-time arbitrage-free derivative security prices.

A sequence {Xn} of random variables converges in distribution to a random variable X, denoted XnX, if, for any bounded continuous function f : R → R, we have E[f(Xn)] → E[f(X)]. We could allow X and each of X1 ,X2,… to be defined on different probability spaces. A standard version of the Central Limit Theorem reads along the following lines. A random variable is standard normal if it has the standard normal cumulative distribution function.

Central Limit Theorem. Suppose Y1Y2, … is a sequence of independent and identically distributed random variables on a probability space, each with expected

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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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