I have problem understanding estimating delta using pathwise method and likelihood Ratio method.Suppose option value is C=E[f(x)] where f(x) is payoff function and expectation is taking over density g(x). then delta is dE[f(x)]/ds(0)Delta in pathwise method is E[df(x)/ds(0)]Delta in LR method is E(f(x)*score function] where score = dg(x)/ds(0)/g(x).So basically, pathwise method differentiate payoff function first and then estimate the expected value of the differentiated payoff. LR differentiate density function first and estimate expected value of the product of differentiated log density and original payoff.But I thought both f(x) and g(x) is function of s(0). So when you differentiate E[f(x)] wrt s(0), you have to apply chain rule to f(x)*g(x). You can't differentiate just f(x) or just g(x).

Hmm . . . I only understand about half of your post, but if you want to estimate delta using MC-sim, it is just:[V(S+h)-V(S-h)]/2hIt is best if you use the same random draws as when you calculate V(S) when calculating V(S+h) and V(S-h).

if you write out the expectation as integral and differentiate under the integral sign you get an expression. if you a change of variables you get another one. that's the difference between pathwise andd likelihood ratio