A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

Abstract

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.

Series representAtion of the Pricing Formula for the EuropeaN Option Driven by Space-Time Fractional Diffusion

“…It features a more complete structure than the simple composition of the time and space fractional models as it exhibits non trivial phenomena including larges jumps and memory effects, which can not be understood as a simple market time re-parametrization of an α-stable process. Let us also mention that it has also found many applications in real systemsfinancial processes representing one of the most promising fields, where the fractional diffusion and generally fractional calculus has been successfully applied [2,12,20,22,21,35,41,42].…”

Applications of Hilfer-Prabhakar Operator to Option Pricing Financial Model

“…The first generalization of Black-Scholes model [5,6] to the model with stable distributions was the Finite moment log-stable option pricing model introduced by Carr and Wu [7]. Later several other models were introduced that included timefractional [13,21,10] derivatives, space-fractional derivatives, [8] space-time fractional derivatives [20,1,2,3,4] derivatives of fractional order [22], etc. It has been shown that the orders of fractional derivatives play the role of risk redistribution parameters in price and time [20].…”

An idea of risk-neutral momentum and market fear

“…Another empirical work concerning option implied measurement of the Hurst exponent was conducted by Flint and Mare (2016). Funahashi and Kijima (2017) further contributes to this fractal literature by modeling the relationships between the short and long ends of the term structure of volatility to estimate the implied Hurst exponent.…”