Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application

TLDR

The paper investigates pricing European call options on stocks whose variance rates change randomly. The authors model stock returns and stochastic volatility as continuous‑time diffusion processes, show that a risk‑free hedge requires the stock and two options, simplify by assuming volatility risk can be diversified away and is uncorrelated with returns, and compute option prices via Monte Carlo simulations applied to real market data. Because random volatility is not tradable, a unique option price cannot be obtained from a risk‑free hedge alone; instead, an equilibrium asset‑pricing model is needed, yielding prices that depend on the risk premium of the random volatility and are expressed as an integral of the Black‑Scholes formula weighted by the variance distribution.

Abstract

In this paper, we examine the pricing of European call options on stocks that have vari? ance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral ofthe Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.