Linear stochastic volatility models

TL;DR

This paper studies linear stochastic volatility models, deriving formulas for asset price densities and option prices, including new results for Heston models, unifying several well-known models under a common framework.

Contribution

It introduces a unified approach to linear stochastic volatility models, providing explicit formulas for densities and option prices, with novel results for Heston models.

Findings

Closed-form density functions for certain models

Explicit option pricing formulas

New insights into Heston model properties

Abstract

In this paper we investigate general linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process. This class contains among others Black-Scholes model, a log-normal stochastic volatility model and Heston stochastic volatility model. For a linear stochastic volatility model we derive representations for the probability density function of the arbitrage price of a financial asset and the prices of European call and put options. A closed-form formulae for the density function and the prices of European call and put options are given for log-normal stochastic volatility model. We also obtain present some new results for Heston and extended Heston stochastic volatility models.