I presume all these methods are for 1 underlying and flat vol?

QuoteOriginally posted by: outruna semigroup approach to american optionsclaims a reduction to a 1D integral (3.5) that can be solved numerically. Has anyone thried that?Don't suppose you can show us the integral?I have dond the PDE formulation which is OK but an integral formulation would be interesting because you can integrate by part to get the FB out into the open. Then it starting to look like FEM?

You can get to an integral equation. Is that what it is? That can be done by transforms quite straightforwardly.P

Thanks, got the paper. This reminds me of joint work I did a decade and a half ago. We didn't bother publishing the results because we thought they were a bit too obvious. (Fourier transform a pde to get an integral equation is undergrad stuff!) That was way back when I was a proper applied mathematician, and before I realised that really trivial results would get published in this new field of quant finance! (I expect Cuch feels similar!)P

QuoteOriginally posted by: PaulThanks, got the paper. This reminds me of joint work I did a decade and a half ago. We didn't bother publishing the results because we thought they were a bit too obvious. (Fourier transform a pde to get an integral equation is undergrad stuff!) That was way back when I was a proper applied mathematician, and before I realised that really trivial results would get published in this new field of quant finance! (I expect Cuch feels similar!)PI feel the same as well. When I wrote the C++ book in 2004 I made it advanced because I thought that everyone in finance know about C++ and they would have found my telling them what pointers, memory management to be trivial.And Fourier transform was one of many methods one should learn asap. It's 1st year maths.

QuoteOriginally posted by: outrunit looks like thisbut I'm a bit clueless about all the variables (I haven't coded the method), and it looks like I'm going to have to retype the whole paper... Shall I send it to you?Look like a Volterra integral equation (btw I think the upper limit in the integral should be tau and the integrand should be f(y, t,mu)??0Of course the curse lies in exp(At) and A is the semigroup operator (see "Nineteen dubious ways to compute exponential of a matrix..")Is it fast? //Edit: there is no dependence on 't' on LHS of eq. 3.5 and there is a dependence on t on RHS??

My definition of fast also includes an element of "How fast is it to generalize?" I.e. we spend weeks formulating a fast numerical solution and then the contract changes a teeny bit and we have to start all over again! I think that rules out a lot of schemes and approximations. But if all you want is to price the put then make some look-up tables! How many would you need?! How many dimensionless quantities are there? S/E, (T-t)*sig^2, r/sig^2, D/sig^2.P

For the kind of error levels mentioned, the Ju and Zhong analytic approximation for regular American puts is almost as quick as Black-Scholes - a little bit slower is Ju's exponential approximation - quicker than trees, FDs, MCsI wrote about the lookup-table idea from a paper by Peter Duck et al in one of my Wilmott columns

QuoteOriginally posted by: PaulMy definition of fast also includes an element of "How fast is it to generalize?" I.e. we spend weeks formulating a fast numerical solution and then the contract changes a teeny bit and we have to start all over again! I think that rules out a lot of schemes and approximations. PThis is Suitability, how applicable is a solution to a wide range of parameters. Indeed, you don't want a scheme that breaks down when a constant parameter changes to one that is a function of t or S. This is why I tend to avoid searching for exact solutions (apart from the fact that I have not practiced the knack for a while..)Someone said: begin with the end in mind!

QuoteOriginally posted by: spursfanFor the kind of error levels mentioned, the Ju and Zhong analytic approximation for regular American puts is almost as quick as Black-Scholes - a little bit slower is Ju's exponential approximation - quicker than trees, FDs, MCsOK, but what is the scope of the JU article. Is is constant parameters, 1 factor etc. etc. We should compare apples and apples! Quoteuitloop..but regarding the speed, I thought it was strange for the paper to not mention any numerical examples, comparison of results. etc.Well, what the authors have done is a transformation PDE -> simpler PDE -> IE, the last being quite complicated to solve computationally.The method will probably not work in the general case precisley because of the change of variables.This is also relevant.Matrix Expo

I have a question on front fixing for early exercise. Instead of Landauy = S/B(t)where B is moving boundary I want to usey = S / (S + B(t))but this also leads to a NL PDE in convection term.What would be better is just nonlinearity in the new boundaries but not in the PDE.Any ideas?