American Options

Cuchulainn · Posted: Mon Mar 23, 2009 5:53 am Post subject:

The results are not bad comparing with the classical parameters used in various papers, e.g. Zvan. But the speed is a problem as with epsilon becoming smaller and smaller the time steps required for the semi-implicit scheme grows very fast. Also, I found that with epsilon small enough (e.g. 0.0001) the results becomes worse, not better as the theory predicts, which shouldn't be the case.

This is my experience as well. The speed is a bit slow and smaller epsilon is not necessarily more accurate.

My idea is to discretise in S and then get a system of ODEs, then use Bulirsch-Stoer. I have not done it myself but I reckon we need a good solver in tine t.

kubila · Joined: 06 Mar 2009 Posts: 3

waiting for your advice...

By the way, I implemented the semi-implicit by the following:
1) split the PDE into two legs as in the european case, i.e. leg one PDE consists only the underlying stock variable; leg two PDE consists only of the volatility variable;
2) set C = r*K;
3) put the penalty term f(P) together with the zero term in the leg one PDE;

The results are not bad comparing with the classical parameters used in various papers, e.g. Zvan. But the speed is a problem as with epsilon becoming smaller and smaller the time steps required for the semi-implicit scheme grows very fast. Also, I found that with epsilon small enough (e.g. 0.0001) the results becomes worse, not better as the theory predicts, which shouldn't be the case.

Cuchulainn · Posted: Thu Mar 19, 2009 7:50 am Post subject:

Good questions. I will get back soon.

MC for early exericise is not fast. Longstaff-Schwarz is also used it seems.

kubila · Joined: 06 Mar 2009 Posts: 3

Hi Daniel,

Thanks a lot for your quick and helpful response. I am sorry to forget to mention that I am looking at pricing American put under Heston SVM. I've checked a couple of papers related to this topic, e.g. Ikonen & Toivanen's paper on operator splitting method; Zvan's penalty method and of course your book of FDM in financial engineering. I plan on a combination of penalty method and the splitting method, as you pointed out in your book chapter 28.

Can you please advise on the following:

0) Seems that the operator splitting method mentioned in Ikonen & Toivanen is different from yours? They introduce an auxilary term lamda and split the PDE into two steps. In the first step a system of linear equations is solved; in the second step, the prices and the lamda are updated to satisfy the constraint. In comparison, you split the PDE into two steps by separating the S variable and Volatility variable. In EACH leg a system of linear equations is solved -- basically chapter 22.

1) How do I split the PDE in the American put case with penalty method? Especially which leg should I put the f(P) term in?

2) How to determine the constant C in the penalty function?

3) What do you mean by semi-discretization in S?

4) What's the most popular method the finance industry uses in pricing American options? FDM or MC or something else?

Again thanks a lot for your time and reply. You are my hero. BTW, when will your Monte Carlo book come out? I've been waiting for it for long.

Best,

Kubila

Cuchulainn · Posted: Sun Mar 15, 2009 11:45 am Post subject:

Kubila,
As far as I know there is no real exact solution to this problem. Many are quasi-exact I think..

The -rS term could be included in the implaicit (linear) part of the equation and then use explicit for f(P). You will need to experiment with epsilon and deltaT will depend on it.

The resulting scheme is not fastest.

A good nonlinear solver (e.g. Bulirsch-Stoer) might be useful. Do a semi-discretisation in S ==> nonlinear ODE system.

There is s/w for BS on www.alglib.net (or org??)

hope this helps

Daniel

kubila · Joined: 06 Mar 2009 Posts: 3

Hi Daniel,

How about the exact solution to American put in the paper written by Geske and Johnson: "The American Put Option Valued Analytically" published in The Journal Of Finance, Dec. 1984?

I read all your books and is now working on the FDM for American Put. I am thinking of using semi-implicit scheme. But how should I split the PDE in this case? Do I group the zero term -r * P with the penalty term f(P) then use the same method as in the European case?

Any comments or response will be greatly appreciated.

Kubila

Cuchulainn · Posted: Fri Nov 21, 2008 9:28 am Post subject:

Nic,
How much time do you still have in order to finish the thesis?

By the looks of it, Zhu's results are not public domain (for whatever reason!) and I personally do not believe in closed solutions that are computable in general.

So, I would use lattices or FDM. The latter will demand a learning period.

Numerical methods are the only solution in this case, in my opinion.

There is also a recent discussion on Wilmott.

hope this helps

Daniel

nix · Joined: 10 Nov 2008 Posts: 2 Location: Durban, South Africa

Hi Daniel

Thanks for responding.

I am totally with you in regards to the practical downfalls of Zhu's exact solution. Its slow and there are too many easier approximations and numerical techniques applied to the original problem to merit this method. Its not really even closed form or analytical - in the sense that you cannot analytically solve the integrals in the solution.

However, for my purposes (i.e. to graduate!) I need to get some results that verify this method works. As I had said before, I tried to use numerical integration (comp simpsons and trap) to the actual solution. I could not seem to get a good enough resolution in terms of the accuracy. It also took hours to run.

I was thinking that I should try to use FDM on the D.E. before it is solved? Its just a bit tricky as one of the boundary conditions is an ode.

I'm not sure what to do! There is also a little bit of an issue with the f function in the paper - I have written to Zhu but with no real response.

What do you suggest?

Kind Regards
Nic

Cuchulainn · Posted: Thu Nov 20, 2008 3:32 pm Post subject:

Hi nix,
Looking for the exact soution is not a good idea in my opinion (others have mentioned this as well). So,it's down to numerical methods:

fdm, as discussed in my book (penalty, front fixing)
trinomial/lattice methods

I reckon these are the most promising.

hope this helps

D

nix · Joined: 10 Nov 2008 Posts: 2 Location: Durban, South Africa

Hi Daniel

I am referring to a post on the Wilmott forum (Numerical Mathods Forum) on the topic: "American Options, Zhu's paper".

I need to reproduce the results in the paper (Zhu - An exact and explicit solution for the valuation of American put options) and have had quite a hard time at doing that. I was using integration methods (comp trap and comp simpsons) to solve for the "U" function (eq. 23 in the paper). However, the running time was ridiculous and as a result I could not increase the resolution which in turn meant that the output was not at the desired accuracy. I am quite desperate to get this project over as I have been on it for way too long!

I never thought to use FDM on the diff eq (eq. 22). I was too focused on the U(x,t) "solution". I would like to try use FDMs, although I have little experience using them. In the Wilmott forum were you and "aedallan" talking about FDM used on Zhu's problem or were you talking about FDM on american options in general?

I have just got your book "Financial instrument pricing using C++" , one of the few sources I could find that speak about exponentially FDMs.

How do you suggest I go about trying to solve this problem?

I really appreciate any help you can throw my way!

Cuchulainn · Posted: Wed Sep 19, 2007 10:44 am Post subject: American Options

I discuss it in my FDM book (Duffy)

main techniques are

- front fixing
- penalty
- variational (PSOR)
- others (treat as plain option FDM and correct at level n+1)

An article by Nielson et al JCF 5, 4, 2002 discusses a number of techniques.
Khaliq has some articles on these problems with penalty methods. Also G.H. Meyer.

And all these methods are based on the classic Free and Moving Boundary Value Problems by John Crank.