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Monte Carlo Method
Monte Carlo Monte Carlo Monte Carlo H2 H2

Monte Carlo  Monte Carlo

dS = &mu;S dt + &sigma;S dWt dS = &mu;S dt + &sigma;S dWt



Antithetic paths
A simple technique is, for every sample path obtained, to take its antithetic path &mdash; that is given a path $$\{\varepsilon_1,\dots,\varepsilon_M\}$$ to also take $$\{-\varepsilon_1,\dots,-\varepsilon_M\}$$. Since the variables $$\varepsilon_i$$ and $$-\varepsilon_i$$ form an antithetic pair, a large value of one is accompanied by a small value of the other. This suggests that an unusually large or small output computed from the first path may be balanced by the value computed from the antithetic path, resulting in a reduction in variance. Not only does this reduce the number of normal samples to be taken to generate N paths, but also, under same conditions, such as negative correlation between two estimates, reduces the variance of the sample paths, improving the accuracy.