Peer-Reviewed Research

BACKGROUND : Contingent claims on underlying assets are typically priced under a framework
that assumes, inter alia, that the log returns of the underlying asset are normally distributed.
However, many researchers have shown that this assumption is violated in practice. Such
violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis
and long memory. This paper considers the pricing of contingent claims when the underlying
is assumed to display long memory, an issue that has heretofore not received much attention.
AIM : We address several theoretical and practical issues in option pricing and implied volatility
calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter
fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider
in depth the issue of calibration. One of the main benefits of such a model is that it allows one
to decompose implied volatility into an independent long-memory component – captured by
an implied Hurst exponent – and a conditional implied volatility component. Such a
decomposition has useful applications in the areas of derivatives trading, risk management,
delta hedging and dynamic asset allocation.
SETTING : The proposed FBSI volatility model is calibrated to South African equity index options
data as well as South African Rand/American Dollar currency options data. However, given
the focus on the theoretical development of the model, the results in this paper are applicable
across all financial markets.
METHODS : The FBSI model essentially combines a deterministic function form of the 1-year
implied volatility skew with a separate deterministic function for the implied Hurst exponent,
thus allowing one to model both observed implied volatility surfaces as well as decompose
them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.
RESULTS : It is shown that a fractional Black–Scholes model always admits a non-constant
implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied
volatility is independent of the Hurst exponent and equivalent to fractional volatility.
Furthermore, we show that the FBSI model fits the equity index implied volatility data very
well but that a more flexible Hurst exponent parameterisation is required to fit accurately the
currency implied volatility data.
CONCLUSIONS : The FBSI model is an arbitrage-free deterministic volatility model that can
accurately model equity index implied volatility. It also provides one with an estimate of the
implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.