Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model

TL;DR

This paper introduces a linear approximation approach for option pricing under Ornstein-Uhlenbeck stochastic volatility models, deriving explicit formulas and validating them with market data for improved calibration and fit.

Contribution

It presents a novel linear approximation method for exponential Ornstein-Uhlenbeck and Stein-Stein models, enabling explicit characteristic functions and efficient calibration.

Findings

Good fit with market implied volatility surfaces

Accurate calibration using first four cumulants

Linear approximation simplifies complex stochastic models

Abstract

We consider the problem of option pricing under stochastic volatility models, focusing on the linear approximation of the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. Indeed, we show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the exact expression of the characteristic function associated to the risk neutral probability density. This expression allows us to compute option prices exploiting a formula derived by Lewis and Lipton. We analyze in detail the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We also compute the analytical expressions of the first four cumulants, that are crucial to implement a simple two steps calibration procedure. It has been tested against a data set of options traded on the Milan Stock Exchange. The data analysis that we present reveals a good fit with the market implied surfaces and corroborates the accuracy of the linear approximation.