Forward Volatility Dynamics in Stochastic Volatility Models Driven by a Gamma Process

Abstract

Pricing models within the Black-Scholes framework assume that the volatility of the underlying security remains constant over the life of the derivative, which cannot explain long-observed characteristics of the implied volatility surface such as volatility smile and skew. With the introduction of Stochastic Volatility (StochVol/SV), the volatility of the underlying state variable is postulated to follow a random process, which is aimed at addressing some of the shortcomings of the Black-Scholes. An insightful guide into the world of volatility modelling is provided. A practical application to the FX volatility modelling is offered by the universal volatility model, a combination of StochVol and LocalVol (SLV) in conjunction with jumps if deemed necessary. As for some of the more complicated exotics, which incorporate a multi-callable feature, the implications for the shape of the forward volatility arising from modelling the volatility based on the choice of a stochastic process need to be considered. Deeply rooted in Mandelbrot’s theory of fractal geometry, the ideas that prices do not vary continuously and that real-time trading unfolds at a variable speed, as discussed, inspired the reflection on the deformation of time in financial markets and finding a way to mathematically formulate it through the Gamma clock. The present document explores the nature of the forward volatility smile in StochVol models driven by Brownian Motions on a Gamma clock. The main exposition begins with the derivation of the functional form of the implied volatility within a particular StochVol model. The second section postulates the Gamma clock StochVol model and derives a series representation of the Gamma probability density functions (PDF) in terms of the Dirac delta function and its derivatives (Gamma expansion) by drawing upon the theory of oscillatory integrals. The third section demonstrates how the proposed model uses the additional degrees of freedom provided by the Gamma process to calibrate to the forward volatility market in addition to fitting the Spot FX Smile and highlights the implications for the shape of the forward volatility smile arising from the usage of different models and/or calibration schemes. The fourth section advocates a natural extension of the base model that entails the introduction of two different correlated Gamma processes to allow for a possible time lag between the discontinuous movements of the Spot FX and the potential subsequent jumps of its volatility. The implied volatility function is derived through a two-dimensional Gamma expansion in the spirit of the base model. The fifth section is concerned with demonstrating that the Bivariate Gamma process offers a unique insight into the phenomenon of time dilation (or contraction) in financial markets and how the model can be used to gain a certain degree of control over the steepness of the forward volatility skew. The main result central to this paper is considered to be the link between the distortions experienced by the passage of financial time and the co-dependence of the random times at which the underlying state variable and its volatility jump, coupled with the realisation that this information could potentially be decoded from glimpses at the implied forward volatility.