A Moment Matching Market Implied Calibration

Abstract

This paper provides a new market implied calibration based on a moment matching methodology where the moments of the risk-neutral density function are inferred from at-the-money and out-the-money European vanilla option quotes. In particular, we derive a model independent risk-neutral formula for the moments of the asset log-return distribution function by expanding power returns as a weighted sum of vanilla option payos (based on results of Breeden, D. and Litzenberger, R. (1978) and Carr, P. and Madan, D.B. (2002)). For the numerical study, we work out dierent popular exponential Levy models, namely the VG, NIG and Meixner models. The new calibration methodology rests on closed-form formulae only: it is shown that the moment matching system can be transformed into a system of algebraic equations which computes directly the optimal value of the N model parameters in terms of the second to the (N 1)th market standardized moments under the dierent Levy models under investigation. Hence, the proposed calibration can be performed almost instantaneously. Furthermore, the method is not requiring a starting value for the model parameters and avoids the problem of getting stuck in local minima.

Keywords: calibration, moment matching, exponential Levy models