Hi i been looking around the net for a Finite difference script/routine (For Matlab, c++, python) which specifically can depict the early exercise boundary for an American put with the following inputs:

S = 100
K = 100
r = 0.08
volatility = 0.2
dividend = 0.04
T = 3.0
dt = 0.03

I want the routine to yield a Boundary value for every dt from 0 up to T. ? Can someone suggest a working finite difference routine/script (For Matlab,c++,python) which is able to do that ?

Is the put price around 8.94397963647 ?


If so then I have these values below..,.. the grid is however *not* uniform with dt=0.03

My value is very close "  8.94366" 
Did you attain those values using FDM ? 
Also, I maybe asking for a lot here but,  your plot is exactly what I want but I would like the dots to be more connected (a line between them maybe? ) so the boundary the depicted would be almost like a continuous function , is that possible?

I think I figured it out in mathematica!!

Most FDM don't model the boundary B(t) directly as such (at least, the common-or-garden ones). You can use the Landau y = S/B(t) and then compute both V and B(t) as a nonlinear FDM.

In the easiest FDM you check the early exercise at each time level up to T. May something can be gotten out of it. Haven't tried it myself. But FDM is not handy IMO. It's derivative and your really need integrals

B(t) satisfies an integral equation directly It's mentioned in section 9.7 of Alan's book and Andersen et al has done it. 
AFAIR outrun posted but I think they gone to the big postmortem in the sky


https://papers.ssrn.com/sol3/papers.cfm ... id=2547027

My value is very close "  8.94366" 
Did you attain those values using FDM ? 

Great, then it looks we have used the same parameter definitions!
No, I didn't use FDM, it's code I had lying round.
As you notice the dots get spaced closer together near T=0 which is the region that needs more attention. Cuch has had some good results with transforming the T axis in the context of FDM as well, (right cuch?)

would it look different if You had used FDM to create it?

would it look different if You had used FDM to create it?

I think it will look the same although I have no ideas what the precision would be with dt=0.03. I would expect anything between a 1E-5 and 2 deviation at some point on the curve.

is your method extremely accurate?. Just want to get a feel for your boundary compared to FDM or any other numerical method.

It's highly accurate for most cases (I expect 6 digits here), but there are exstreme-ish parameter configurations where it blows up. I would still need to quantify that region before I would use it in production.

It's the method described in this paper "High Performance American Option Pricing, Andersen et al, 2015". It references a lot of papers on various boundary estimate done in the past.

The main novelties of this paper seem to result in a numerical algorithm that converged exponential-ish in time instead of e.g. square root (tree) or linear.

It's highly accurate for most cases (I expect 6 digits here), but there are exstreme-ish parameter configurations where it blows up. I would still need to quantify that region before I would use it in production.

It's the method described in this paper "High Performance American Option Pricing, Andersen et al, 2015". It references a lot of papers on various boundary estimate done in the past.

The main novelties of this paper seem to result in a numerical algorithm that converged exponential-ish in time instead of e.g. square root (tree) or linear.

Ah I see! So would you then argue that it is more accurate than the standard FDM?

It's highly accurate for most cases (I expect 6 digits here), but there are exstreme-ish parameter configurations where it blows up. I would still need to quantify that region before I would use it in production.

It's the method described in this paper "High Performance American Option Pricing, Andersen et al, 2015". It references a lot of papers on various boundary estimate done in the past.

The main novelties of this paper seem to result in a numerical algorithm that converged exponential-ish in time instead of e.g. square root (tree) or linear.

Ah I see! So would you then argue that it is more accurate than the standard FDM?

I think so -for this case- but all methods have strengths and weaknesses. E.g. FDM is more flexible and easier. 
I tried to study and plot error vs computation time against some *tree* methods and convergence was very fast for parameters similar to yours. In this plot the methods of the paper is "And" (the blue dots), and I only have two blue dots before it hit machine precision. It's still on my (low priority) todo list to run it with high precision math engine and see how  error vs computation time relate if we have more digits.

It's highly accurate for most cases (I expect 6 digits here), but there are exstreme-ish parameter configurations where it blows up. I would still need to quantify that region before I would use it in production.

It's the method described in this paper "High Performance American Option Pricing, Andersen et al, 2015". It references a lot of papers on various boundary estimate done in the past.

The main novelties of this paper seem to result in a numerical algorithm that converged exponential-ish in time instead of e.g. square root (tree) or linear.

Ah I see! So would you then argue that it is more accurate than the standard FDM?

I think so -for this case- but all methods have strengths and weaknesses. E.g. FDM is more flexible and easier. 
I tried to study and plot error vs computation time against some *tree* methods and convergence was very fast for parameters similar to yours. In this plot the methods of the paper is "And" (the blue dots), and I only have two blue dots before it hit machine precision. It's still on my (low priority) todo list to run it with high precision math engine and see how  error vs computation time relate if we have more digits.

I agree all have strengths and weaknesses , but just on the info you gave me it looks extremely accurate! I JUST WISH you could have produced the same values with a simple FDM  so could compare both  

Ah I see! So would you then argue that it is more accurate than the standard FDM?

I think so -for this case- but all methods have strengths and weaknesses. E.g. FDM is more flexible and easier. 
I tried to study and plot error vs computation time against some *tree* methods and convergence was very fast for parameters similar to yours. In this plot the methods of the paper is "And" (the blue dots), and I only have two blue dots before it hit machine precision. It's still on my (low priority) todo list to run it with high precision math engine and see how  error vs computation time relate if we have more digits.

I agree all have strengths and weaknesses , but just on the info you gave me it looks extremely accurate! I JUST WISH you could have produced the same values with a simple FDM  so could compare both  

It'll be a good thing to try yourself?

As mentioned, there is no 'simple' FDM for this IMO. And since this is a nonlinear problem then accuracy is first-order, all things being equal. Or use front-tracking. LIke this

http://docs.lib.purdue.edu/cgi/viewcont ... ext=cstech

It's trickier than usual FDM but is doable.  Especially with MOL and ODE solvers.

As mentioned, there is no 'simple' FDM for this IMO. And since this is a nonlinear problem then accuracy is first-order, all things being equal. Or use front-tracking. LIke this

http://docs.lib.purdue.edu/cgi/viewcont ... ext=cstech

It's trickier than usual FDM but is doable.  Especially with MOL and ODE solvers.

I wish I had their code so I could try myself, ..