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 payoffs (based on results of Breeden and Litzenberger and Carr and Madan). For the numerical study, we develop different popular exponential Lévy 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 that computes directly the optimal value of the model parameters in terms of the second to the th market standardized moments under the different Lévy models under investigation. Hence, the proposed calibration can be performed almost instantaneously. Furthermore, for the models considered in this paper, the method does not require any search algorithm and hence any starting value for the model parameters and avoids the problem of becoming stuck in local minima.

Keywords:

• Model calibration
• Financial engineering; Lévy process
• Options applications

Classification:

• C6
• C60
• C63

Acknowledgments

The authors thank Peter Carr, Santiago Garcia and Dilip Madan for their helpful comments.

Notes

Florence Guillaume is a postdoctoral fellow of the Fund for Scientifc Research, Flanders (Belgium) (F.W.O.).

Note that it is possible that for other models (with a more complicated characteristic function), this system transformation becomes quite challenging, or even impossible, analytically. In that case, we might have to resort to some search algorithm to solve the nonlinear moment system.

An out-of-the-money call option is characterized by and an out-of-the-money put option by .

For the standard calibration problem (1.2), we consider the same set of options as that used to determine the market implied moments.