A multifractional option pricing formula

Abstract

Fractional Brownian motion has become a standard tool to address long-range dependence in ¯nancial time series. However, a constant memory parameter is too restrictive to address di®erent market conditions. Here, we model the price °uctuations using a multifractional Brownian motion assuming that the Hurst exponent is a time-deterministic function. Through the multifractional Itô calculus, both the related transition density function and the analytical European Call option pricing formula are obtained. The empirical performance of the multifractional Black–Scholes model is tested by calibration of option market quotes for the SPX index and o®ers best ¯t than its counterparts based on standard and fractional Brownian motions.