My idea is that we can convert the local vol surface to implied vol surface at any time and any stock price level.For instance, say we're simulating the process at T1, T2, T3, ... , Tn, given S0 and vanilla prices at all strikes and maturities (basically, the local vol function). In order to simulate S1, we can use Alan's method below:From the local vol surface from t=0..T1, we get the call prices of all strikes starting at T0, maturing at T1. Take the negative of 1st deri of C(T1, S0) w.r.t. K, divided by discount factor. This is the cummulative function of S1 so we can simulate S1.Now given S1, and using the same local vol surface from t = T1 ... T2, we get call prices of all strikes starting at T1, maturing at T2. Take the negative of 1st deri of C(T2, S1) w.r.t. K, divided by discount factor. This is the cummulative function of S2 given S1 so we can simulate S2.And so on.I think this is essentially equivalent with simulating S in very small time steps, using the local vol surface. Would you agree here?

QuoteOriginally posted by: BonDoes all these discussion imply that although we can find the law of S from T_0 to T_1, we still can't find the law of S from T_1 to T_2, given S_1? My information, of course, is all vanilla prices over all strikes and maturities.That's correct -- you can't because the "true law" may contain ofther factors F besides S.The law of S from T_0 to T_1 you are observing (to the extent that such a law exists at all, which is a separate issue) is really p(S | S0, F0). This is just a marginal with limited information.A simple example is when the true process has stochastic volatility, so that F0 = V0.You said "Now given S1, and using the same local vol surface from t = T1 ... T2, we get call prices of all strikes starting at T1, maturing at T2".In general, when T1 actually rolls around and the stock price is indeed S1, this procedure (carried out at T0) won't price the calls to the (new) market at T1 because other factors, like V0, will have changed to V1. This is the fundamental problem with the local vol. idea. If unrecognized, it leads to mispriced exotics and incorrect hedging of vanillas (although the vanillas are priced correctly).See my answer here I would agree that the simulation procedure with very small time steps is an equivalent way to extract the (limited) informationthat is present in the local vol. surface. But this information is still limited to what you can deduce from p(St | S0, F0), for 0 < t < Tmax,where Tmax is the longest expiration observed. As discussed, that information is limited to(1) correct vanilla prices observed at T0, and the(2) correct answer to "vanilla-type" questions, like what is the risk-adjusted probability that St > K. For questions beyond those, the answers are likely to be incorrect and perhaps grossly misleading.

Hi Alan,Actually, I am assuming the local vol process holds until the maturity of my option.dS / S = (r-q)dt * sigma(S, t) dWUnder this assumption, I think we can indeed back out the forward starting option from the generated local vol surface, in a sort of "Sticky Surface" way.If we simulate the above process under very small dt interval from time 0 till T1, we should get the same result as the method you describe. Now using this surface, we COULD simulate using small dt from T1 to T2. In another words, this surface can give an implied vol surface for T2, viewed at T1 and S1. Now with this imp vol surface, we can back out the CDF correct? Therefore we can repeat the process in step 1 and move on.Am I missing something here? I'm fairly convinced this will work. Not saying the LV process is a realistic one, but I need some fast way of simulating the process other than small time steps. Of course, the conversion from local vol surface to "forward starting" implied vol surface doesn't seem trival. Do you have any sources for that?

No sources, but if a Monte Carlo is too slow for your forward starting option, a PDE approach should work very quickly.As far as mssing something, you may be missing how unrealistic the results will be -- perhaps worse than just estimating the value with a simple Black-Scholes model.

QuoteOriginally posted by: BonUnder this assumption, I think we can indeed back out the forward starting option from the generated local vol surface, in a sort of "Sticky Surface" way.Alan is quite right: just because you "can" doesn't mean you "should". Your forward vol surface evolution will be disastrously bad and, because you are using a fancy model, you will have a false sense of confidence about it all. That is a recipe for losing a whole lot of money.The upside is that your counterparties will love you...

But won't it be the same as simulating a forward start option using small dt steps? My goal is not to get a forward starting price. It is to find an algorithm that can avoid simulating a long dated option without resorting to tiny time steps, yet the distribution remains the same. I don't see how my method below yields a different result. May I be enlightened on this regard?

QuoteOriginally posted by: BonI don't see how my method below yields a different result. May I be enlightened on this regard?All mathematically correct methods will give you the same results given your local vol model - MC, PDE, Tree, or anything else you can cook up. As Alan suggests, MC is probably the slowest of these - I imagine PDE or tree is going to be orders of magnitude faster. Your method will certainly give consistent results; the issue here is that local vol is known to be a particularly bad model for forward volatility and I (and, probably, Alan, though it's not my place to speak for him) wanted to make sure you knew the risks you are running. Forward-starting option prices and hedges are acutely model-dependent.