Momentum and reversion in risk neutral martingale probabilities

Abstract

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.

Keywords:

• Continuous time models
• Derivative pricing models
• Equity options
• Financial modelling
• Levy process
• Markov processes
• Methodology of pricing derivatives
• Non-Gaussian option pricing

JEL Classification:

• G10
• G12
• G13

Acknowledgments

We thank Erhan Çinlar for bringing the work of Hunt to our attention. We also thank Peter Carr, Ernst Eberlein, Martijn Pistorius, Wim Schoutens and Marc Yor for helpful discussions on the subject matter of this paper. Errors remain our responsibility.