Research On A New Method Of Control Variates In Option Pricing

We assume that prices of stocks follow the geometric Brownian motionin this paper. According to Risk-Neutral Pricing Theorem, the price of European option can be transformed to the expectation of the payoff function. Monte Carlo methods play an important role in these problems because we could not find the analytical expectation of these payoff functions in most situations. So it becomes more and more important to improve the efficiency of Monte Carlo methods. Variance reduction and generating more effective paths through quasi-Monte Carlo methods are two main ways to improve the efficiency. It has been proved that the method of control variates is a good choice to reduce variance. The main idea of the method of control variates is to replace payoff function by a new function with smaller variance, for which there exists an analytical expectation. A typical control variate is applied in the pricing of Asian option, in which the geometric mean price is used as the classical control variate. Researchers introduced a new control variate, combined with conditional Monte Carlo, in pricing Asian and basket options in recent years. And they’ve got a great decrease in variance.In this paper we study this new control variate, and combine it with randomized quasi-Monte Carlo methods. A key point is to generate paths that leads geometric mean price less than strike price in pricing Asian and basket options. Comparing with Monte Carlo methods, quasi-Monte Carlo methods, classical control variate with quasi-Monte Carlo methods, our aim is to analyze the properties of this new control variate. The results of simulating showed that the new control variate works much better than the other three methods. And the variance reduction factor(VRF) of the new control variate increases when the strike price K decreases in both situations. Increase of T could lead to the increase of VRF when pricing basket options. We also try to find the relationship between classical and new control variates.Similar conclusions are got when we use the new methods combined with pathwise methods in estimating the delta, an important Greeks of Asian option.