Option pricing and hedging in incomplete market driven by Normal Tempered Stable process with stochastic volatility

Abstract

This paper develops a distribution class, termed Normal Tempered Stable, by subordinating a drifted Brownian motion through a strictly increasing Tempered Stable process that generalizes the Variance Gamma and the Normal Inverse Gaussian and is used to model the logarithm asset returns. The newly added parameter is to create subclasses for all the distributions discovered in financial market. The empirical test suggests that time series of Technology stock returns in US market reject both the Variance Gamma distribution and the Normal Inverse Gaussian distribution and admit instead another subclass of the Normal Tempered Stable distribution. Furthermore, we introduce stochastic volatilities into the Normal Tempered Stable process and derive explicit formulae for option pricing and hedging by means of the characteristic function based methods. To answer the question of how well different models work in practice, we investigate four models adopting data on daily equity option prices and obtain several findings from the numerical results. To sum up, the Normal Tempered Stable process with stochastic volatility is able to adequately capture implied volatility dynamics and seen as a superior model relative to the jump-diffusion stochastic volatility model, based on the construction methodology that incorporates more sophisticated and flexible jump structure and the systematic and realistic treatment of volatility dynamics. The Normal Tempered Stable model turns out to have the competitive performance in an efficient manner given that it only requires three parameters.