Option Pricing under Fast-varying and Rough Stochastic Volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.Comment: arXiv admin note: text overlap with arXiv:1604.0010

Small-time asymptotics for fast mean-reverting stochastic volatility models

In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment generating function computation in the particular case of the Heston model.Comment: Published in at http://dx.doi.org/10.1214/11-AAP801 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic formulae for implied volatility in the Heston model

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.Comment: Presentation in Section 2 has been improved. Theorem 3.1 has been slightly generalised. Figures 2 and 3 now include the at-the-money point

On spatially irregular ordinary differential equations and a pathwise volatility modelling framework

This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's squared volatility. In the new framework, corresponding processes are strictly increasing, solve random ODEs, and are composed with geometric Brownian motion to obtain price processes. The new framework has no dependence on stochastic calculus, so processes can be studied on a pathwise basis using probability-free ODE techniques and functional analysis. The ODEs considered depend on continuous driving functions which are `spatially irregular', meaning they need not have any spatial regularity properties such as Holder continuity. They are however strictly increasing in time, thus temporally asymmetric. When sensible initial values are chosen, IVP solutions are also strictly increasing, and the IVPs' solution set is shown to contain all differentiable bijections on the non-negative reals. This enables the modelling of any non-negative volatility path which is not zero over intervals, via the time derivative of solutions. Despite this generality, new well-posedness results establish the uniqueness of solutions going forwards in time, and continuity of the IVPs' solution map. Motivation to explore this framework comes from its connection with the Heston volatility model. The framework explains how Heston price processes can converge to an interval-valued generalisation of the NIG Levy process, and reveals a deeper relationship between integrated CIR processes and the IG Levy process. Within this framework, a `Riemann-Liouville-Heston' martingale model is defined which generalises these relationships to fractional counterparts. Implied volatilities from this model are simulated, and exhibit features characteristic of leading `rough' volatility models.Comment: The author's PhD thesis. Major extension of v2. 211 pages, 22 figure

Option pricing under fast-varying long-memory stochastic volatility

Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the option price in the regime where the volatility process is fast mean-reverting. The volatility process is modeled as a smooth and bounded function of a fractional Ornstein-Uhlenbeck process. We give the expression for the implied volatility, which has a fractional term structure

Rough volatility models: small-time asymptotics and calibration

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.Open Acces

Asymptotics of forward implied volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential L\'evy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.Comment: 37 pages, 13 figure

SMALL-TIME ASYMPTOTICS FOR FAST MEAN-REVERTING STOCHASTIC VOLATILITY MODELS

This is the published version, also available here: http://dx.doi.org/10.1214/11-AAP801.In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the “fast variable” lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in Feng, Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126–141] by a moment generating function computation in the particular case of the Heston model