Abstract
In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H < 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.
Highlights
In the finance literature, there has been a general consensus that volatility is highly persistent
There are numerous pieces of evidence that the price dynamics of financial products are consistent with fractional Brownian motion volatility models with Hurst index H > 1/2, which implies that the volatility has a long memory
An important finding in this paper is that, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist
Summary
There has been a general consensus that volatility is highly persistent. E.g., [1] for the existence of the long memory features in stock market volatilities Inconsistent with this stylized fact, [2] recently find that the log-volatility behaves essentially as an fBM with H close to zero at any reasonable time scale, by estimating the volatility from high frequency data. This puzzle (the word “puzzle” is used in the context of using a fractional volatility model in option pricing, but not used in the context of finance in general) has not been resolved, one possible explanation may be the smoothing effect by sampling intervals of data. The definitions of the functions contained in our approximation formula are given in Appendix A
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