Rough volatility models: small-time asymptotics and calibration

Henry Stone, Antoine Jacquier, Mikko Pakkanen, Engineering And Physical Sciences Research Council
2020
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
more » ... s 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 i [...]
doi:10.25560/81768 fatcat:mke4gdxdxvgufod3iq5kg5a5xa