Implied Volatility Structure in Turbulent and Long-Memory Markets

We consider fractional stochastic volatility models that extend the classic Black–Scholesmodel for asset prices. The models are general and motivatedby recent empirical resultsregarding the behavior of realized volatility. While such models retain the semimartingaleproperty for the asset price the associated European optionpricing problem becomescomplex, with no explicit solution. In a number of canonicalscaling regimes it is possible,however, to derive asymptotic and sparse representations for the option price and theassociated implied volatility, that are parameterized by afew effective parameters andthat involve power law dependencies on time to maturity. These effective parameters maydepend in a complicated way on the volatility model, but theycan be easily estimatedfrom the observation of a few option prices. The effective parameters associated witha particular underlying asset can be calibrated with respect to liquid contracts writtenon this asset and then used for pricing less liquid contractswritten on the sameunderlying asset. Therefore, the effective parameters provide a robust link betweenfinancial products written on a particular underlying asset.