Efficient Monte Carlo Algorithm Using Antithetic Variate and Brownian Bridge Techniques for Pricing the Barrier Options with Rebate Payments

The down-and-out call barrier option with rebate payment on dividend-paying stock is simulated using a new version of the Monte Carlo algorithm. The standard Monte Carlo method for simulating such an option suffers from two sources of errors: Hitting time error inherent from time stepping and the Monte Carlo statistical error. We present a modified version of Monte Carlo method that can reduce these errors efficiently using the Brownian bridge technique for the hitting time error and the antithetic variate approach for the statistical error. We found that the Brownian bridge technique is responsible for improving the order of convergence in hitting time from one half to one and the antithetic variate technique can speed up the Monte Carlo simulation by reducing the variance of the computed payoff, giving almost twice as much accuracy. The standard error and the coefficient of variation are applied in order to measure the effectiveness of the volatility of the underlying option.