Abstracts

In this paper we develop a closed-form expansion of implied volatility and at-the-money variance skew. The expansion is simple, transparent, and easy to implement. Moreover, in contrast to the majority of the previous literature, the purpose of this work is not obtain an asymptotic expansion valid only for extreme maturities or strikes. Here, our aim is to develop a general implied volatility expansion useful in the large region of the volatility surface where options are actually traded. In this respect, the present work is most closely related to, at least in spirit, the most-likely-path approximation of Gatheral (2006), the singular expansion of Jäckel (2009) as well as the recent works of Drimus (2011), Poulsen and Ribeiro (2012), and Nicolato and Sloth (2013).
We illustrate the use of the implied volatility expansion for the popular Heston (1993) stochastic volatility model and the Variance Gamma model of Madan and Seneta (1990). Nevertheless, the expansion can be used for a broad class of option pricing models displaying both jumps and stochastic volatility. The neat simplicity of the expansion and its broad applicability across various models is due to the fact that it is based on 'summary statistics' such as moments and crossmoments of the underlying model components. Moreover, the expansion brings about the key insight that implied volatility can be understood as an expected volatility adjusted for risks of movements in the underlying asset price or its volatility.
The decomposition of the volatility smile clarifies how the analytical features of the underlying model are translated into implied volatilities, highlighting how a model responds to moves of its parameters. Furthermore, the decomposition allows for systematic comparison of complex option pricing models by quantifying how option risks affect the volatility smile across different models. We explore two domains of application of the expansion. First, we propose an expansion-based control variate for option pricing based on Fourier transform methods to improve convergence properties of the embedded numerical integration. This approach results in a significant speed-up which reduces the number of required function evaluations by more than half. Secondly, we propose a fast, first-order model calibration to at-the-money volatilities and skews. Besides a gain in computational speed, this approach requires fewer data points compared to the usual procedure of calibrating to the whole surface of option prices.