Short-time near-the-money skew in rough fractional volatility models

Abstract

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < 1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t 1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

Deep PPDEs for Rough Local Stochastic Volatility

“…so that Θ t is clearly F t -measurable and satisfies (9). We can therefore write the option price as…”

“…We assume that the mean reversion κ, the Hurst parameter H and the initial volatility V 0 are given, while the volatility of volatility ν, the correlation ρ and the long-term variance level V ∞ need to be calibrated. The reason for this choice is practical (the dimension of the optimisation problem is reduced), but can be easily justified: a good proxy for the initial variance V 0 is given by the short-term at-the-money smile, and the Hurst parameter can be calibrated by the maturity-decay of the at-the-money skew of the implied volatility smile; proper and rigorous explanations, via asymptotic limits and expansions, can be found in [3,9,29,30,37,44,52]. We perform a slice by slice calibration, via a minimisation of the difference between market smiles and of model smiles.…”

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

“…On the theoretical side, Jacquier, Pakkanen, and Stone [JPS18] prove a pathwise large deviations principle for a rescaled version of the log stock price process. In this same direction, Bayer, Friz, Gulisashvili, Horvath and Stemper [BFGHS17], Horvath, Jacquier and Lacombe [HJL18] and most recently Friz, Gassiat and Pigato [FGP18] (to name a few) extend the large deviations principle to a wider class of rough volatility models. On the practical side, competitive simulation methods are developed in Bennedsen, Lunde and Pakkanen [BLP15], Horvath, Jacquier and Muguruza [HJM17] and McCrickerd and Pakkanen [MP18].…”

A comparison principle between rough and non-rough Heston models—with applications to the volatility surface

“…While the behavior of implied volatility in rough models has mainly been studied by numerical computation, analytic results have been obtained e.g. in [FZ17, GJRS18, FGS19] (short-and/or long-maturity asymptotics) and [Fuk17,BFG`19] (short-time asymptotics of at-the-money skew).…”