Martingale measures for NIG Lévy processes with applications to mathematical finance

Abstract

In this study we investigate equivalent martingale measures for exponential NIG- Lévy processes. Lévy processes in particular exponential NIG Lévy models give rise to an incomplete market, thus leading to a continuum of equivalent martingale measures that can be used for risk-neutral pricing. We need to find a suitable Q-equivalent martingale measure. Therefore we consider Esscher transform method to drive a Q-equivalent martingale measure. This approach preserves structure of probability distribution. We see that we chose one specific Q-equivalent martingale measure as a representative of all the martingale measures. The density of martingale measure includes also jumps in process. In this manner, we get the chance to model jump behaviours which one seen in stock market prices. Traditional diffusion type models are insufficient to model for price jump behaviour. We prefer NIG-Lévy models instead of diffusion type models. After defining the process that captures the asset return behaviour, under the absence of arbitrage assumption, the set of equivalent martingale measures can be constructed, so, we obtain the a set of arbitrage free prices for derivatives. We use national and international index and stock returns data to calibration NIG-Lévy models. Using with Esscher transform method, we compute equivalent martingale densities. Finally, we also consider pricing of derivatives on securities that their returns based on exponential NIG-Lévy process and simulation of NIG distribution. © EuroJournals Publishing, Inc. 2009.