Closed-Form Smile Expansions For The Mixing Setup
Elisa Nicolato (Aarhus University, Denmark)
Joint work with David Sloth Pedersen

Wednesday June 4, 16:30-17:00 | session 6.7 | Stochastic Volatility | room I

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.

A Bayesian analysis of modelling stock returns with time-changed Lévy processes
Kun Zhang (Warwick Business School, UK)
Joint work with Yan Wang

Wednesday June 4, 17:00-17:30 | session 6.7 | Stochastic Volatility | room I

Jump-diffusion asset pricing models are theoretically attractive, in which the diffusion component aims to capture frequent-but-small movements and the jump component models infrequent-but-large movements. However, jump-diffusion models cannot provide satisfied estimation results, and empirical literature have suggested that diffusion can be abandoned if Lévy processes with infinite activities are used. In this paper, we develop a Bayesian inference method to verify this argument. We then extend this method to estimate time-changed Lévy processes, where stochastic volatility is incorporated by stochastic time-change. The accuracy and robustness of this estimation method are assessed based on simulation data. With international indices data, we demonstrate that time-changed Lévy models, e.g. VGSV and CGMYSV, have better goodness of fit than benchmark models including Lévy models and jump-diffusion models. At last, the estimation method is further extended to jointly estimate multivariate time-changed Lévy models. This attempt facilitates the use of time-changed Lévy models in portfolio selection problems.