Precise asymptotics: robust stochastic volatility models

Abstract

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of [Bayer et al; A regularity structure for rough volatility, 2017]. In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama-Kawabi on rough path space. When applied to rough volatility models, e.g. in the setting of [Forde-Zhang, Asymptotics for rough stochastic volatility models, 2017], one obtains precise asymptotic for European options which refine known large deviation asymptotics. Contents 1. Introduction 2. Notation 3. Preliminaries on Black-Scholes asymptotics 4. Unified large, moderate and rough deviations 4.1. Basic large deviation assumption (A1) 4.2. Moment assumption (A2) and large deviation option pricing 5. Exact call price formula 5.1. The case of SDEs and (classical) StochVol 5.2. The case of rough volatility 5.3. Assumptions (A3-5): robust model specification and control theory 5.4. The call price formula 6. Precise asymptotic pricing 6.1. Precise large deviations 6.2. Precise moderate deviations 7. The case of RoughVol 8.

A tourist's guide to regularity structures and singular stochastic PDEs

“…Inahama and Kawabi extended the approach to a rough paths setting in [42], and Friz, Gassiat and Pigato made a first use of this type of ideas in a regularity structures setting in [25]. The present result holds for all subcritical stochastic singular PDEs, with the above straightforward proof.…”

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].…”

Pathwise moderate deviations for option pricing

“…Essentially, this assumption ensures that some integrability of the stock price e Xt is guaranteed (otherwise Call prices may not be defined), which is not automatically granted by large and moderate deviations results. We refer the reader to [15,Lemma 4.7] for some easy-tocheck condition.…”