Hi all,Can someone explain me what are the benefits of MC resolution methods compared to PDE for equity options?Thank you very much....

ThanksIn fact I was speaking about multi asset option such basket or everest.... There are no closed formulas, so we have to choose a financial model...Would you use MC or PDE?

Poor Lapin, these are his first messages on the forum and he finds himself in a one-to-one conversation with MP! For 1 space dimension PDE is best.For 2 space dimensions PDE is still best. For 3 space dimensions some people still use PDE but most people switch to MC.

Lapin,I would say, if your payoff function is somewhat like dirac (digital), I would suggest monte carlo. If your payoff can be thought smooth enough compared to dirac, then both finite difference and monte carlo are ok.One more, if it's path dependent option, mc is much easier.

I think MobPsycho made a good point. Many people have more faith in Monte Carlo, because it's often easier to understand. And I can cite many examples of quants who worked for weeks on analytic models, only to find that the actual computation was simulated anyway.It's unfortunately easy to make big dollar errors with either approach. With an analytic model you can make a mistake in your derivation or make an innocent-looking assumption that kills you. With Monte Carlo you can either (a) use too low a dimension and miss a crucial pricing parameter or (b) use too high a dimension and get lost.My general answer would be if you understand your underlying better than your derivative, Monte Carlo is good. If you understand your derivative better than your underlying, analytic is good. In most cases, where you understand neither very well, use both or be sorry.Here are some answers I think are bad:(1) If I can solve it, use analytic, otherwise simulate. Even if you can't write down a closed form solution, you should be able to come up with a good approximation with calculable error (these are often very complicated functions). So you should always be able to choose either method. Too many people are so proud of an analytic solution that they use them when they shouldn't.(2) Go with whichever one you're better at. Quants have to be proficient at both.(3) Go with the one that the (non-quant) customer wants. They haven't a clue. You can always dress one up to look like the other ("oh, I'm not simulating there, I'm adding terms in a power series"). You have to put out a model you believe in because you're going to make a lot of mistakes and they might as well be your own. You can learn from your own mistakes.

There are a couple of things in favour of Monte Carlo:1. It is a very general method, works for all asset price dynamics.Differential equations can only be used under restricted conditions on the asset price dynamics:dS(t)=mu(t,S(t))dt + sigma(t,S(t))dW(t)where the drift mu=mu(t,s) and volatility sigma=sigma(t,s) are deterministic functions of time t and "space" s\in R_+^d. .In fact you can go a little further and add in more "factors" in addition to the asset price vector (interest rate r, volatility sigma,...) at the expense of higher dimension which leads to2. The curse of dimensionality. If the dimension d is large (d>=4) you'll have problems with PDEs. You are discretizing a d-dimensional rectangle and the grid points grow like n^d, where n is the number of grid points on each edge of the rectangle. This problem is devastating. Monte Carlo on the other hand scales better to higher dimension. Clearly the simulation of higher dimensional asset vectors is the slower the higher the dimension. However I have simulated 40 dimensional Libor paths and generated 100000 paths in minutes (a true full factor 40 dimensional model). In dimension d=40 no PDE can be attempted.The main reason why you would want a PDE is that American options cannot easily be priced with Monte Carlo.

as far as I've seen PDE/Trees are better adapted to American contracts, the American MC's exist but are so-so ...on the other hand, MC's are better adapted to path-dependent contracts, again path-dependent trees/ PDEs exist but are harder to implement ...

However I simulated 40 dimensional Libor paths and generated 100000 paths in minutes (a true full factor 40 dimensional model). TRC, Out of curiosity how did you confirm that the final answer was the converged one? I would say, if your payoff function is somewhat like dirac (digital), I would suggest monte carlo. If your payoff can be thought smooth enough compared to dirac, then both finite difference and monte carlo are ok.In most of the discussions I read, there's little mention of the impact of B.C.'s on the efficacy of the method. May be the ramp function is a given and implicit in all discussions? I've asked similar questions before....and am still trying to understand it....Monk, with reference to the delta case, am I right in saying that your observations/suggestions are based on practicality as opposed to something inherent in the MC methodology? My intuitive guess is that for spiked payoff profiles using MC helps 'cause your essentially avoiding the singularity? The same cannot be said of FD since the BC couples the interior, ie. you can't escape pathological payoffs influencing your sys of eqns....right??

TRC, I wasn't inquiring the specifics of the problem, but rather how one gathers confidence in one's method for such difficult problems. Thanks.WB.

I always test the model on known analytic formulas (caplets for example).This will show you if your code is set up correctly.From theory you know that Monte Carlo will converge to the proper expectation and your question is now how close your particular result is.For this you study the standard deviation of you sample of simulated values which gives you a probabilistic estimate (confidence intervals).However there is no absolute certainty.With Quasi Monte Carlo there is in principle the possibility of a deterministic error estimate. But I doubt one can pull it off inpractice since some very unpleasant quantities have to be computedand then the estimate is probably not very precise.