Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview
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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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