May 24th, 2016, 1:46 pm
QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: AlanIf the initial condition has compact support, I will guess the solution function tends to zero as you approach infinity in any spatial direction.Some initial brainstorming ..I don't normally do log coordinates but I intend to for a number of reasons. Typically x = log(S1), y = log(S2) so we get a PDE with constant coefficients on in the plane. There are no boundary conditions and we just have a Cauchy problem. I want to flag some assumptions:1. initial condition has compact support is nice, but a call does not have this property in S1 and S2 but maybe things are different in x and y variables?2. Boundary conditions in S1 and S2 have a 'isomorphic' mapping to x and y coordinates?e.g. V(0,S2,t) = IC(S2, t) gets transformed to V(-INF, y, t) = IC(y,t).3. Just ride roughshod over the PDE and put V = 0 everywhere at the faraway infinities?4. What's the magic number for truncation e.g. [-1,1] X [-1,1]5 Domain transformation a feasible option instead of 3.6. I don't need rotation.Fuzzy questions but I got to start somewhere.Re 3.If a put or call, then in the original coordinates, you know some (numerical) boundary conditions that will work,lets say in a bounded computational domain [$](S_{min},S_{max})[$]. So, after the coordinate transformation, I would just apply these same conditions at [$](X_{min},X_{max})[$] where [$]X(S)[$] is the coord. transformand [$]X_{min}= X(S_{min})[$], etc.Re 4.If it is a log transformation, you typically set the cutoffs by the 'sigmas'; for example [$]X_{max} = \log S_0 + 4 \sigma \sqrt{T}[$], etc,where [$]S_0[$] is a hotspot value where you want an answer. This makes the probability that the 'particle', starting at [$]X_0 = \log S_0[$],will reach/cross the numerical boundaries before time [$]T[$] expires, small, and so have little effect on the answer.Re 5.Any other coordinate transformation should work similarly, say [$]X(S) = S/(S+S_0)[$]. There is going to be a numerical bc at [$]X_{max} = 1 - \epsilon[$];I would probably use a derivative condition at that one. For me, I have found that sometimes I can take [$]X_{min} = 0[$] andsometimes I need [$]X_{min} = \epsilon[$], depending on the problem. Here [$]\epsilon[$] is a grid point spacing.
Last edited by
Alan on May 23rd, 2016, 10:00 pm, edited 1 time in total.