I am not sure which forum this belongs to, but since it seems to be a fairly common interview question, and since I couldn't find a previous reference to this in any of the fora, I am asking it here:What are the types of problems which lend themselves and do not lend themselves to Monte Carlo/Finite Difference/Trees. The only example I could find is that American Options are better modeled with trees, because early exercise is easier to incorporate in that case. I was wondering what other examples there are, and if there was a general intuition behind it.I also had more specific questions about finite differences:- Is anything used in practice besides a Crank-Nicholson scheme?- Are there many examples of problems that lend themselves to finite differences but do not have analytical/closed form solutions?

QuoteOriginally posted by: hazeriderI am not sure which forum this belongs to, but since it seems to be a fairly common interview question, and since I couldn't find a previous reference to this in any of the fora, I am asking it here:What are the types of problems which lend themselves and do not lend themselves to Monte Carlo/Finite Difference/Trees. The only example I could find is that American Options are better modeled with trees, because early exercise is easier to incorporate in that case. I was wondering what other examples there are, and if there was a general intuition behind it.I'd say American Options are best handled with finite differences in general, and then put Trees and Monte Carlo (least squares) as equal seconds depending on your trade-off between complexity and repeatability. Finite differences on the LCP has ways of handling truly American exercise, while the others provide the Bermudan approximation.Quote- Is anything used in practice besides a Crank-Nicholson scheme?Yes, I routinely use the Implicit Euler as well as CN, and my toolkit will even handle explicit. I actually prefer IE because I can use more steps to get accuracy, and I don't want to worry about the oscillations.Quote- Are there many examples of problems that lend themselves to finite differences but do not have analytical/closed form solutions?I'd say yes, but I rarely look for closed-form solutions. Some examples that come to mind:1.) Arithmetically averaged Asians2.) Harmonically averaged Asians3.) Discrete and unevenly stepped-up barriers4.) Volatility swaps

QuoteOriginally posted by: exotiqI'd say yes, but I rarely look for closed-form solutions. Some examples that come to mind:1.) Arithmetically averaged Asians2.) Harmonically averaged Asians3.) Discrete and unevenly stepped-up barriers4.) Volatility swapsNot sure what harmonically averaged Asians are. But as far as I know, there is no closed-form expression for Arithmetically averaged Asians, and geometrically averaged Asians are used as control variates in Monte Carlo for that case.

Sure, Control Variates are great when you use Monte Carlo. I generally prefer FD to MC (even with CV) because of the smooth values for greeks, and reserve MC for high-dimension problems.The harmonic average c of a and be is defined as: 2/c = 1/a + 1/b (I think).In theory, an asian could be averaged this way, and I could see this as useful in a dollar-cost averaging kind of way, but I haven't seen these trade.

I've seen harmonic asian averages come up for fx options, where you're dealing with X^{F->D} == 1/X^{D->F}.

Exotiq indicated that he reserved FD methods for "higher dimension problems". I be interested to hear you opinion on what constitutes "higher dimension".In my esperience, it seems that FD is only efficient for 3 or fewer dimensions. Above this, MC tends to be faster. With exotics on baskets, where you might be modeling a running maximum, the current index level, stochastic volatility, interest rates, etc., you can easily blow through 3 or 4 dimensions!Comments?

Actually, I said I reserved Monte Carlo for higher dimension problems, where Finite Differences is no longer as feasible.I'd agree that three is a good maximum dimension for FD, and most of the problems I deal with either have three or fewer dimensions, while others have 10 or more and MC is clearly the better mthod.Most FD problems I do are two dimensional (+time), say underlying and stochastic vol, an asian averaging dimension, a pair, etc. I have also been looking at a three dimensional stochastic covariance problem, and am curious about FD in 4/5 dimensions, as well as hybrid methods. I have also seen other tricks like sparse grids and finite elements which may be able to handle an extra dimension or two.For just about any basket problem or string model, I do not even consider methods other than MC, but I'd love to learn new ideas...

If one can avoid MC methods, one should. For some path dependent problems, one cannot. For some high dimensional problems, one cannot. High dimensions: 2d + time is solvable routinely by finite differences. 3d + time is marginal for finite differences unless one has some "cheapo" available. Maybe or maybe not"Cheapo" is a technical term: It may be some symmetery or invariance which allows one to use, e.g, FFT's or to reduce the dimension by one. More commonly, it may be relative unimportance of one of the dimensions which allows one to use, very few grid points, as in 3 (high, low and middle) for the dimension.At first glance, trees are simply one of the many explicit finite difference schemes. At second glance (if done correctly) they are an exactly arbitrage free model which is an approximation of the arbitrage free model one wishes to implement. Because they are arbitrage free, if they are calibrated exactly to vanillas and the discount curve, they can be surprisingly accuarate and robust.(At third glance, one should ask why not modify the common finite difference schemes to be arbitrage free ... this would be analogous to the finite difference schemes that were developed to exactly preserve conservation laws and higher invariants).

Pat,Thanks for your insights. So if I am asked at an interview, the answer as to when to use each is:- MC: some path dependent problems, high dimensionality problems- Trees: one-dimensional, arbitrage-free required problems, easy to calibrate- Finite differences: for 2d+time problemsI do have a couple more questions though:- Isn't the arbitrage-freeness embedded in the PDE used for the FD methods?- Is there any reason to use something other than a Crank-Nicholson scheme in FD methods?

Exotiq,I have a question on the definition of "high dimensional". I thought that in an asian option the time dimension counted as more than one dimension. Ie if there were 100 averaging points then the time dimension would count as 100 dimensions. This is because the joint distribution (I don't like the word correlation becuase it doesn't capture the full co-dependence) between every averaging point must be taken into account. This that makes asian options (say arithmetic) very high dimensional and thus more suited to monte carlo with (as previuosly mentioned) say a geometric asian option used as a control variate.What are other peoples thoughts on this? And if it can be done with FD, how is the joint distribution between all the averaging points( in the above case a 100 dimensional problem) accounted for.Cheers,Yurtle.

QuoteOriginally posted by: YurtleExotiq,I have a question on the definition of "high dimensional". I thought that in an asian option the time dimension counted as more than one dimension. Ie if there were 100 averaging points then the time dimension would count as 100 dimensions. This is because the joint distribution (I don't like the word correlation becuase it doesn't capture the full co-dependence) between every averaging point must be taken into account. This that makes asian options (say arithmetic) very high dimensional and thus more suited to monte carlo with (as previuosly mentioned) say a geometric asian option used as a control variate.What are other peoples thoughts on this? And if it can be done with FD, how is the joint distribution between all the averaging points( in the above case a 100 dimensional problem) accounted for.Cheers,Yurtle.not exactly, a new variable is created from the average of those 100 points, so the average creates one extra variable.

QuoteOriginally posted by: exotiqActually, I said I reserved Monte Carlo for higher dimension problems, where Finite Differences is no longer as feasible.I'd agree that three is a good maximum dimension for FD, and most of the problems I deal with either have three or fewer dimensions, while others have 10 or more and MC is clearly the better mthod.Most FD problems I do are two dimensional (+time), say underlying and stochastic vol, an asian averaging dimension, a pair, etc. I have also been looking at a three dimensional stochastic covariance problem, and am curious about FD in 4/5 dimensions, as well as hybrid methods. I have also seen other tricks like sparse grids and finite elements which may be able to handle an extra dimension or two.For just about any basket problem or string model, I do not even consider methods other than MC, but I'd love to learn new ideas...out of curiousity, how are you modeling stochastic covariance?

In the limit that the discretization goes to zero, the finite difference method exactly solves the PDE and is arb free. But in real life, the discretization error is non zero. In a fairly crude tree, or in an exactly-arbitrage free finite difference scheme, our answer may be surprisingly good compared to our discretization error .... this because if our method is arb free, then our method is an EXACT implementation of an arb free model. Who's to say that the continuous model is better than our discrete one?

QuoteOriginally posted by: PatIn the limit that the discretization goes to zero, the finite difference method exactly solves the PDE and is arb free. But in real life, the discretization error is non zero. In a fairly crude tree, or in an exactly-arbitrage free finite difference scheme, our answer may be surprisingly good compared to our discretization error .... this because if our method is arb free, then our method is an EXACT implementation of an arb free model. Who's to say that the continuous model is better than our discrete one?not I said the fly...I am sold on it for many problems.

Actually, I was trying yesterday to construct a MC in the forward bond measure that could also be used to sample some path dependencies...it is looking more complicated than I envisioned.