Pricing formula for a Barrier call option based on stochastic delay differential equation

Abstract

We derive new explicit pricing formulae for a type of Barrier call option, down and in call option when underlying asset price processes are represented by a stochastic delay differential equation (hereafter “SDDE”). We note the conditional normality of a stochastic integral with respect to a Wiener process to find the joint distribution of the stochastic integral and their minimum. On the basis of this result, we obtain pricing formulae for the Barrier call option which extends ones in the classical Black-Scholes models without delay. Finally, through Monte-Carlo simulations, we demonstrate that our theoretical prices for a Barrier option are correct.