On Pricing Exponential Square Root Barrier Knockout European Options

A barrier option is one of the most popular exotic options which 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 an exponential double bariier knockout option. In terms of the underlying Brownian motion W(t), the logarithm of their barriers for a stock price process S(t) assumed to be geometric Brownian motion are straight line boundaries, and so their protection is not uniform over time. To remedy this problem, we propose square root curved boundatires *** for the underlying process W(t). Since the standard deviation of Brownian motion is proportional to **, 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 bariier knockout European call options. With these formulae, it takes very short time (t < 10** sec) to compute numerical values, whereas it takes much longer times to perform Monte Carlo simulations to determine option premiums.