Option pricing under fast-varying long-memory stochastic volatility

Josselin Garnier and Knut Solna

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Abstract: Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean-reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein-Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure.

Date: 2016-03, Revised 2018-04
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