Hi there,Let's say I want to calculate the delta. I can price my instrument using Monte Carlo through PV = PV(S), where S is spot of the underlying.So I calculateDelta = [PV(S+epsilon) - PV(S)] / epsilonMy question is, is it standard practice to reuse the same random numbers in both price calculations?I would imagine it is (although this is not what I have seen so far), not only because it would be computationally cheaper, but if my calculations below are correct we would also get a (much) more accurate estimate of the risk.Let's call sigma_P the standard error of PV(S), and sigma_P' the standard error of PV(S+epsilon). Whether or not I re-use the same random numbers, these will be very similar, so let's call both "sigma".Generally speaking,Var[PV(S+epsilon) - PV(S)] = sigma^2 + sigma^2 - 2*rho*sigma^2If I don't reuse my random numbers, the two quantities are practically independent, so rho = 0, which gives me the maximum standard error = sqrt(2)*sigmaHowever, if I do reuse the random numbers, rho = 1, which gives me zero (!) standard error.Have I missed something, or have I just realised something that is well-known? Or maybe nobody cares?Thank you for your answers.

QuoteMy question is, is it standard practice to reuse the same random numbers in both price calculations?use the same numbers again or calculate 3(actually 5if you are using antithetic techniques to reduce variance) paths simultaneously. these areS, (S+dS(+)), (S-dS(+)), (S+dS(-)), (S-dS(-))where S+dS(+) is the path using +epsilon, S+dS(-) is the path from using -epsilon.

Thanks Dave.I think you are confusing two related but different things.My goal is not necessarily to reduce the std error of the price.On the face of it, reusing the same numbers for both sets of paths (or 3 or 4 sets of paths if it's a gamma, etc) _eliminates_ the std error of my risk calculation.So, first you start with S, evolve it to find 10000 paths (potentially using variance reduction techniques, that's irrelevant here) and price your instrument, with a certain standard error.Then, you start with S+eps, evolve it with the same numbers, find 10000 different (but perfectly correlated) paths, and price your instrument again, with the same standard error.As a consequence of this procedure, my estimate of the delta has zero standard error. Is this correct?

sorry my point was you just need to do it on the fly and the only reason i mention variance reduction was because i though you would be doing it anyway.

I see, yes fair point thanks.

Yes, it is essential to use the same random numbers or in most cases the simulation noise will overwhelm the delta.Depending on the complexity of the simulation and instrument, it's better to track the price conditional on the first move rather than computing a price at S+epsilon and S-epsilon. For example, suppose S = $100 and the average derivative price for all paths is $10. You can take the conditional average for all paths in which S hits $100.10 before $99.90, or all the paths where the first move is up or other criterion as appropriate. Since it's no extra work to track conditional means, it's a good idea to use different definitions of move, and to try different intervals.If things are simple and your simulation is good, you'll get very similar delta estimates in all cases. But many problems, either technical with your simulation or financial with your product, will show up as divergent deltas.

QuoteIf things are simple and your simulation is good, you'll get very similar delta estimates in all cases. But many problems, either technical with your simulation or financial with your product, will show up as divergent deltas. The major problems with FD method for MC is that 1) it is ad hoc in my opinion (see remark (*)) 2) results are biased 3) how do we choose epsilon? 4) not for discontinuous payoff functions.If it fails for you, plan B is pathwise method or likelihood ratio method. If you want robust and mathematically justified approximation to delta in small dimensions then PDE/FDM is the way to go. It is very efficient as well. //remark (*)When you think about it, what are we doing? We are taking divided differences in S of paths that are *not* of bounded variation, and hence nowhere differentiable on [0,T]. Instinct alone tells us to expect problems. So when taking limit epsilson --> 0 we approach a derivative which does not exist, yes? Have I left something out?