Robust Calibration of Jump-Type Asset Price Models

Since the development of the Black-Scholes option pricing formula for an asset price process following a geometric Brownian motion, ever sophisticated models have been introduced in order to improve the performance of derivative pricing. The use of models based on jump processes, and Lévy processes in particular, has recently become widespread as a tool for modeling market fluctuations, both in derivatives valuation and risk assessment. Research has mainly been concerned with finding efficient analytical and numerical valuation procedures given a certain stochastic model for the asset price process. However, for practical purposes an indispensable first step is to retrieve the parameters of the unknown underlying process so that it reproduces derivative prices observed in the market. A general statement of the corresponding calibration problem reads: Given a set of European option prices, find model parameters such that the discounted asset price process is a martingale and the observed option prices are given by the discounted risk-neutral expectations of their terminal payoffs. Our calibration problem belongs to the wider class of inverse problems, and as such it might not be properly posed. Numerical difficulties arise especially from the lack of stability, so various regularization methods have been developed to tackle it. The method we are interested in is based on relative entropy. This non-parametric calibration approach has been developed for local volatility models by Avellaneda et al. (1997) and Avellaneda (1998), and applied in the exponential Lévy setting by Cont and Tankov (2002). A first aim of our project is to extend this calibration approach to a setting where the dynamics of the underlying are described by a Lévy-driven stochastic volatility process of the Ornstein-Uhlenbeck type. This class of models was shown to generically reproduce many distinctive features of financial return series by introducing a plausible pattern of time inhomogeneity, and hence represents a considerable improvement with respect to the exponential Lévy framework. A further aim of the project is to develop numerical methods for the pricing of more general path-dependent options in this setting as well. As a natural generalization, the class of models employing affine processes will be treated. The results of the project will give valuable tools for investors and risk managers handling derivative securities by improving the quality of their pricing methods and risk measures.

Since the development of the Black-Scholes option pricing formula for an asset price process following a geometric Brownian motion, ever sophisticated models have been introduced in order to improve the performance of derivative pricing. The use of models based on jump processes, and Lévy processes in particular, has recently become widespread as a tool for modeling market fluctuations, both in derivatives valuation and risk assessment. Research has mainly been concerned with finding efficient analytical and numerical valuation procedures given a certain stochastic model for the asset price process. However, for practical purposes an indispensable first step is to retrieve the parameters of the unknown underlying process so that it reproduces derivative prices observed in the market. A general statement of the corresponding calibration problem reads: Given a set of European option prices, find model parameters such that the discounted asset price process is a martingale and the observed option prices are given by the discounted risk-neutral expectations of their terminal payoffs. Our calibration problem belongs to the wider class of inverse problems, and as such it might not be properly posed. Numerical difficulties arise especially from the lack of stability, so various regularization methods have been developed to tackle it. The method we are interested in is based on relative entropy. This non-parametric calibration approach has been developed for local volatility models by Avellaneda et al. (1997) and Avellaneda (1998), and applied in the exponential Lévy setting by Cont and Tankov (2002). A first aim of our project is to extend this calibration approach to a setting where the dynamics of the underlying are described by a Lévy-driven stochastic volatility process of the Ornstein-Uhlenbeck type. This class of models was shown to generically reproduce many distinctive features of financial return series by introducing a plausible pattern of time inhomogeneity, and hence represents a considerable improvement with respect to the exponential Lévy framework. A further aim of the project is to develop numerical methods for the pricing of more general path-dependent options in this setting as well. As a natural generalization, the class of models employing affine processes will be treated. The results of the project will give valuable tools for investors and risk managers handling derivative securities by improving the quality of their pricing methods and risk measures.