Implied Volatility Structure in Turbulent and Long-Memory Markets

Abstract

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We consider fractional stochastic volatility models that extend the classic Black–Scholes model for asset prices. The models are general and motivated by recent empirical results regarding the behavior of realized volatility. While such models retain the semimartingale property for the asset price the associated European option pricing problem becomes complex, with no explicit solution. In a number of canonical scaling regimes it is possible, however, to derive asymptotic and sparse representations for the option price and the associated implied volatility, that are parameterized by a few effective parameters and that involve power law dependencies on time to maturity. These effective parameters may depend in a complicated way on the volatility model, but they can be easily estimated from the observation of a few option prices. The effective parameters associated with a particular underlying asset can be calibrated with respect to liquid contracts written on this asset and then used for pricing less liquid contracts written on the same underlying asset. Therefore, the effective parameters provide a robust link between financial products written on a particular underlying asset.

Keywords

• stochastic volatility
• long-memory process
• fractional process
• volterra process
• asymptotics
• time scales