Study of European style options under Itô-McKean Brownian motion with Azzalini skew-normal distribution

Abstract

In this article, we deal European style option, with arbitrary payoff which includes both put and call options, on an asset whose price evolves as It?-McKean skew Brownian motion with Azzalini skew-normal distribution. Initially, we investigate a condition which leads the It?-McKean skew Brownian motion to be a martingale. Next, we price the option and show that if the payoff function is convex then so is the price function. After this, we show if the payoff is finite then the price function satisfies a partial differential equation with respect to time. Further, we provide a necessary and sufficient condition for the price function to satisfy Feymann-Kac type equation. Next, we study Black-Scholes type equation and give expressions for the delta hedge. Finally, we study the particular case of an European call option in order to compare some of our results with the existing literature. Our results can be used to investigate the optimal exercise boundary, discrete time hedging strategies etc. of the option.