Fractional Black-Scholes Option Pricing, Volatility Calibration and Implied Hurst Exponents

Abstract

This paper addresses several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black-Scholes market. In particular, we discuss how the fractional Black-Scholes model admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and also that one-year implied volatility is independent of Hurst exponent and equivalent to fractional volatility. Building on these observations, we introduce a novel 8-parameter fractional Black-Scholes inspired, or FBSI, model. This deterministic volatility surface model is based on the fractional Black-Scholes framework and uses Gatheral’s (2004) SVI pamaterisation for the fractional volatility skew and a quadratic parameterisation for the Hurst exponent skew. The issue of arbitrage-free calibration for the FBSI model is addressed in depth and it is proven in general that any FBSI volatility surface will be free from calendar-spread arbitrage. The FBSI model is empirically tested on implied volatility data on a South African equity index as well as the USDZAR exchange rate. Results show that the FBSI model fits the equity index implied volatility data very well and that a more flexible Hurst exponent parameterisation is needed to accurately fit the USDZAR implied volatility surface data.

Keywords: Fractional brownian motion, Hurst exponent, implied volatility calibration, implied volatility surface

JEL Classification: C59, G12, G13, G17