A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility
Mizuho Securities Co. Ltd., Tokyo 100-0004, Japan
Master of Finance Program, Tokyo Metropolitan University, Tokyo 100-0005, Japan
Author to whom correspondence should be addressed.
Received: 28 October 2017 / Revised: 22 November 2017 / Accepted: 22 November 2017 / Published: 25 November 2017
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
, (ii) is proven to be satisfied by a rough volatility model with
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.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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MDPI and ACS Style
Funahashi, H.; Kijima, M. A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility. Fractal Fract 2017, 1, 14.
Funahashi H, Kijima M. A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility. Fractal and Fractional. 2017; 1(1):14.
Funahashi, Hideharu; Kijima, Masaaki. 2017. "A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility." Fractal Fract 1, no. 1: 14.
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