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

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.

Final version 17 pages, 5 figures, 3 tables. Accepted for publication on Int. J. Theoretical Appl. Finance

References in corpus (4)

• Asymptotic formulae for implied volatility in the Heston model
• A Generalized Fourier Transform Approach to Risk Measures
• A Review of Volatility and Option Pricing
• Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

Cited by in corpus (3)

• A Generalized Fourier Transform Approach to Risk Measures
• Graphical modeling of stochastic processes driven by correlated errors
• Maximum likelihood approach for several stochastic volatility models