Easy Gram–Charlier Valuations of Options

Abstract

The Black–Scholes model for pricing a European option written on a lognormal returns process was easy. Unfortunately, real world markets don’t try to make life easy for financial econometricians. Returns processes are more complicated than lognormal diffusions, and an array of derivative products with much more complex payoffs than European calls are now traded. Many of those are path dependent, making Monte Carlo simulation the only viable valuation technique. But, Monte Carlo can require many replications and substantial execution time to achieve reasonable accuracy for a single set of parameters. In this article, Popovic and Goldsman offer a hybrid approach that can achieve accurate pricing very efficiently. A base set of Monte Carlo runs is constructed and then a Gram–Charlier density approximation is fitted to the histogram of simulation results (which may first need to be transformed to be better adapted to the approximation). A series of examples based on pricing Asian options illustrates how the procedure performs and demonstrates its flexibility in handling non-lognormal underlying returns processes.