Study of European Style Options Under Ito-McKean Brownian Motion with Azzalini Skew-Normal Distribution

Abstract

In this article, we deal European style option, with arbitrary payooff which
includes both put and call options, on an asset whose price evolves as Ito-
McKean skew Brownian motion with Azzalini skew-normal distribution. Ini-
tially, we investigate a condition which leads the Ito-McKean skew Brownian
motion to be a martingale. Next, we price the option and show that if the pay-
off function is convex then so is the price function. After this, we show if the
payoff is fnite then the price function satises a partial differential equation
with respect to time. Further, we provide a necessary and suffcient 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 in-
vestigate the optimal exercise boundary, discrete time hedging strategies etc.
of the option.