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

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