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22-12-2016 | Research Article

Does the Hurst index matter for option prices under fractional volatility?

Authors: Hideharu Funahashi, Masaaki Kijima

Published in: Annals of Finance | Issue 1/2017

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Abstract

This study examines the effect of fractional volatility on option prices. To this end, we develop an approximation method for the pricing of European-style contingent claims when volatility follows a fractional Brownian motion. Through extensive numerical experiments, we confirm that the decrease in the smile amplitude under fractional volatility is much slower than that under the standard stochastic volatility model. We also show that the Hurst index under fractional volatility has a crucial impact on option prices when the maturity is short and speed of mean reversion is slow. On the contrary, the impact of the Hurst index on option prices reduces for long-dated options.

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See Mandelbrot (1997) for the introduction of fBMs in finance.

See, e.g., Sottinen (2001) and Cheridito (2003) for related problems. Hu and Öksendal (2003) develop a no-arbitrage model by introducing Wick integrals, but this model cannot add a natural economic interpretation.

Alos and Yang (2014) derive an approximation formula of European option prices by using a different methodology when volatility follows a fractional Heston model.

To consider the long-memory feature of volatility, we restrict ourselves to the case that \(0.5 \le H <1\) under the physical measure \(\mathbb {P}\) until Sect. 3.2.

It seems reasonable to assume a mean-reverting process for the evolution of volatility over a long period of time under the physical measure.

To be precise, this formulation is a truncated version of the Mandelbrot–Van Ness representation of fBMs. In the next section, we consider its full version. See Comte and Renault (1998) for details.

Alternatively, as a market practice, assuming that \(\bar{\eta }_t\) is a deterministic function (e.g., piecewise constant) of time t, \(\bar{\eta }_t\) can be used to fit the option prices observed in the market.

They consider the mean-reverting volatility process as \(\mathrm{d}\sigma _t = \kappa (\widetilde{\theta }- \sigma _t ) \mathrm{d}t + \gamma \sigma _t \mathrm{d}w_t\). Hence, the parameter \(\theta \) in our model corresponds to \(\kappa \widetilde{\theta }\) in their model.

However, because the convergence speed is very slow in the fractional Monte Carlo simulation, we stop our simulations with 1,000,000 trials. The Monte Carlo simulation for fBMs is difficult to perform because of the non-Markovian nature (see, e.g., Kijima and Tam 2013).

This observation suggests that the fractional volatility model may have a strong impact on the prices of path-dependent options such as Asian and barrier options.

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Title: Does the Hurst index matter for option prices under fractional volatility?
Authors: Hideharu Funahashi
Masaaki Kijima
Publication date: 22-12-2016
Publisher: Springer Berlin Heidelberg
Published in: Annals of Finance / Issue 1/2017
Print ISSN: 1614-2446
Electronic ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-016-0289-1

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