On Pricing Exponential Square Root Barrier Knockout European Options

Abstract

A barrier option is one of the most popular exotic options which is designed to give a protection against unexpected wild fluctuation of stock prices. Protection is given to both the writer and holder of such an option. Kunitomo and Ikeda [1992] analytically obtained a pricing formula for exponential double barrier knockout options. Since the logarithm of their proposed barriers for the stock price process S( t), which is assumed to be geometric Brownian motion, are nothing but straight line boundaries, the protection provided by them is not uniform over time. To remedy this problem, we propose square root curved boundaries plus minus b times root t for the underlying Brownian motion process W(t). Since the standard deviation of Brownian motion is proportional to root of t, these boundaries (after transformation) can provide uniform protection throughout the life time of the option. We will apply asymptotic expansions of certain conditional probabilities obtained by Morimoto [1999] to approximate pricing formulae for exponential square root double barrier knockout European call options. These formulae allow us to compute numerical values in a very short time (t to the minus 6 seconds), whereas it takes much longer to perform Monte Carlo simulations to determine option premiums.