Explaining the Volatility Surface: A Closed-Form Solution to Option Pricing in a Fractional Jump-Diffusion Market

23 Pages Posted: 6 Mar 2012 Last revised: 26 Oct 2012

See all articles by Stefan Rostek

Stefan Rostek

University of Tuebingen - Faculty of Economics and Business Administration

Date Written: June 8, 2012

Abstract

This paper prices European options in a framework that captures both non-normality of returns and serial correlation within financial time series. The underlying security dynamics are driven by a jump-diffusion process where the diffusion part is fractional Brownian motion while jumps exhibit a double-exponential distribution. These model characteristics suffice to overcome most of the evident drawbacks of the classical Black-Scholes setting, while the parsimony of my model still ensures analytical tractability.

Due to market incompleteness, I suggest an equilibrium model à la Brennan (1979). I derive a closed-from solution to the problem, which contains the Black-Scholes pricing formulae and the formulae of Kou (2002) as limit cases.

As an intuitive illustration of the model's power, I choose the phenomenon of volatility surfaces: I show that the derived formulae are able to reflect observable patterns of real market data as the model entails a smile over moneyness as well as a non-flat term structure of implied Black-Scholes volatilities.

Keywords: closed-form solution, equilibrium model, fractional Brownian motion, jump-diffusion

JEL Classification: G12, G13

Suggested Citation

Rostek, Stefan, Explaining the Volatility Surface: A Closed-Form Solution to Option Pricing in a Fractional Jump-Diffusion Market (June 8, 2012). Paris December 2012 Finance Meeting EUROFIDAI-AFFI Paper. Available at SSRN: https://ssrn.com/abstract=2016915 or http://dx.doi.org/10.2139/ssrn.2016915

Stefan Rostek (Contact Author)

University of Tuebingen - Faculty of Economics and Business Administration ( email )

Mohlstrasse 36
D-72074 Tuebingen, 72074
Germany

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