Multivariate Option Pricing Models: Some Extensions of the αVG Model
Florence M.Y. Guillaume
Mathematics and Computer Science, Eurandom (Eindhoven University of Technology), Eindhoven, Netherlands

Luciano and Semeraro [1] proposed to model multivariate asset log-returns by a time-changed Brownian motion where the time change consists of the weighted sum of a common and an idiosyncratic subordinator. In order to obtain a multivariate subordinator of the same class as its components, Luciano and Semeraro imposed some constraints on the subordinator parameters. Under this restricted setting, the marginal characteristic functions become independent of the common subordinator setting. Hence, the risk-neutral calibration requires liquid multivariate derivative quotes which are often unavailable.

We first propose to extend the original model by relaxing the conditions imposed on the subordinator parameters leading to marginal characteristic functions which become dependent on the whole model parameter set. Hence, the calibration of this generalized model does not require anymore any correlation fit and can be performed either on the volatility surfaces only, or by adding a penalty which measures the goodness of fit of the correlation into the option surface calibration optimizer.

All these multivariate exponential Lévy models lead to asset log-returns correlations which are time independent. Moreover they are usually not able to replicate quoted option prices in both the strike and time to maturity dimensions during investor's fear periods. The former characteristic is in contradiction with the market reality since the implied correlation is a function of the time horizon [2]. Hence, we propose a new class of models, the so-called Sato two factors models obtained by a Sato time changed Brownian motion. Given the non-stationarity of Sato processes through time, the linear dependence structure between the asset log-returns becomes a function of time.

References:

[1] Luciano, E. and Semeraro, P. (2010). Multivariate time changes for Lévy asset models: Characterization and calibration. Journal of Computational and Applied Mathematics, 233, 1937-1953.

[2] Chicago Board Options Exchange (2009). S&P 500 Implied Correlation Index. Working paper, Chicago.

Keywords: Multivariate option pricing models; Lévy processes; Sato processes; Calibration

Biography: Florence Guillaume is a postdoctoral fellow at Eurandom within the Multivariate Risk Modelling group. She holds a joint Ph.D. degree in Mathematics from the Catholic University of Leuven and the Eindhoven University of Technology. Her research interests include exotic credit derivatives pricing and volatility and liquidity modeling.