Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model

TL;DR

This paper develops an approximate pricing model for European call options where volatility follows an exponential Ornstein-Uhlenbeck process, capturing stochastic volatility effects in financial markets.

Contribution

It introduces a novel two-dimensional market model combining log-Brownian motion and Ornstein-Uhlenbeck processes, with an approximate pricing formula under specific volatility conditions.

Findings

The model accurately captures implied volatility patterns observed in Dow Jones data.

Approximate prices are valid for large volatility fluctuations and slow mean reversion.

The approach provides practical insights into option pricing with stochastic volatility.

Abstract

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility. We study the resulting European call price and its implied volatility for a range of parameters consistent with daily Dow Jones Index data.