Rough volatility models: small-time asymptotics and calibration

Abstract

Inspired by the work of Al`os, Le ́on and Vives [ALV07] and Fukasawa [Fuk17], who showed that a volatility process driven by a fractional Brownian motion generates the power law at-the-money volatility skew observed in financial market data, Gatheral, Jaisson and Rosenbaum [GJR18a] spawned a class of models now known as rough volatility models. We study the asymptotic behaviour of such models, and investigate how convolutional neural networks can be used for their calibration. Chapter 1 serves as an introduction. We begin with implied volatility, and then intro- duce a number of model classes, starting with local volatility models and ending with rough volatility models, and discuss their associated asymptotic behaviour. We also introduce the theoretical tools used to prove the main results. In Chapter 2 we study the small-time behaviour of the rough Bergomi model, introduced by Bayer, Friz, and Gatheral [BFG16]. We prove a pathwise large deviations principle for a small-noise version of the model, and use this result to establish the small-time behaviour of the rescaled log stock price process. This, in turn, allows us to characterise the small-time implied volatility behaviour of the model. Using the same theoretical framework, we are also able to establish the small-time implied volatility behaviour of the lognormal fSABR model of Akahori, Song, and Wang [ASW17]. In Chapter 3 we present small-time implied volatility asymptotics for realised variance (RV) options for a number of (rough) stochastic volatility models via a large deviations principle. We interestingly discover that these (rough) volatility models, together with others proposed in the literature, generate linear smiles around the money. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we develop an approximation scheme for the density of the realised variance, which in turn allows the volatility swap density to be expressed in closed form. Lastly, we investigate different constructions of multi-factor models and how their construction affects the convexity of 4 the implied volatility smile. Remarkably, we identify a class of models that can generate non-linear smiles around-the-money. Additionally, we establish small-noise asymptotic behaviour of a general class of VIX options in the large strike regime. In Chapter 4, which is self-contained, we give an introduction to machine learning and neural networks. We investigate the use of convolutional neural networks to find the H ̈older exponent of simulated sample paths of the rough Bergomi model, a method which performs extremely well and is found to be robust when applied to trajectories of a fractional Brownian motion and an Ornstein-Uhlenbeck process. We then propose a novel calibration scheme for the rough Bergomi model based on our results.