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

22 Pages Posted: 13 Jun 2016

See all articles by Emlyn James Flint

Emlyn James Flint

Legae Peresec; Department of Actuarial Science, University of Cape Town

Eben Mare

Independent

Date Written: June 10, 2016

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

Suggested Citation

Flint, Emlyn James and Mare, Eben, Fractional Black-Scholes Option Pricing, Volatility Calibration and Implied Hurst Exponents (June 10, 2016). Available at SSRN: https://ssrn.com/abstract=2793927 or http://dx.doi.org/10.2139/ssrn.2793927

Emlyn James Flint (Contact Author)

Legae Peresec ( email )

15 Cavendish Street
Claremont
Cape Town, Western Cape 7700
South Africa
27117227556 (Phone)

HOME PAGE: http://www.legaeperesec.co.za

Department of Actuarial Science, University of Cape Town ( email )

Actuarial Science Section, University of Cape Town
Private Bag X3, Rondebosch
Cape Town, Western Cape 7701
South Africa
+27 21 650 2475 (Phone)

Eben Mare

Independent ( email )

No Address Available
United States

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