The Impact of Deviations from Random Walk in Security Prices on Option Prices

Abstract

This paper examines the effects of deviations from random walk in asset prices on option prices. Several approaches can be taken to model asset price processes as non-random walk processes. We choose to model the equity prices as fractional Brownian motions (FBM). Though FMB is not the most ideal model for the behavior of security prices, it offers many advantages namely it allows for short-term and long-term memory in asset prices. This paper uses Monte Carlo simulations to determine option prices under the assumption that equity prices follow FBM. The simulated prices and their implied volatilities are then compared to prices that one would obtain using the Black-Scholes-Merton option price model. The results show that even a small deviation from random walk can have a significant impact on option prices. The volatilities implied by the simulated prices generally increase as the call options go deeper into the money when security prices have short-term memory. In other words, short-term memory in security prices contributes to the presence of the so-called smile effect in implied volatilities.