Trionomial Trees

mucki · December 9th, 2003, 5:52 pm

Hi there,I have implemented a trinomial tree for the stock price movements (BS-world) to price barriers. I found a very good approximation of the theoretical prices. However, when applying the implementation to binaries things do not work out so well: Is there a problem with trinomial trees when applying them to this problem? Or have I just made a mistake?MuckiPS.: I used the follwowing model: S_ 0= 100, X = 120, Cashpayment = 100, r = 10%, sigma = 30%, T = 4 years==> Analytical price: 35.19==> Discrete price: 34.85 (step width = 0.001 years)

daveangel · December 9th, 2003, 7:44 pm

U havent stated how many steps u r using ... the error should decrease as u increase the number of steps. But this of course gets quite tricky. The crux of th issue is that a trinomial tree is an explicit finite difference scheme and hence it is not very accurate. If you want to improve things, use a Crnk-Nicholson method (implicit finite difference). Much better and faster.

Alan · December 9th, 2003, 7:46 pm

if both S=100 and X=120 are not exactly on nodes, you might try putting them there.

mucki · December 9th, 2003, 7:52 pm

Thanks for the replys. I already put 120 on the nodes but that didn't help either.I used 4 divided by 0.001 = 4000 time steps. For barrier options (standard and two strike) as well as standard bs calls I obtained excellent convergence.Do you have any further ideas?

daveangel · December 9th, 2003, 8:03 pm

4000 time steps is far too many... it simply wont work as a front office app.You need to look at implict crank-nicholson type approaches.

mucki · December 9th, 2003, 8:14 pm

I just use the tree for backtesting, it's not meant for front office app. I find it more intuitive than cn. That's why I am so curious about wether I made a mistake. But I can't find any. Maybe the trinomial tree has a general problem with binaries.

Fermion · December 9th, 2003, 8:32 pm

QuoteOriginally posted by: muckiI used 4 divided by 0.001 = 4000 time steps. For barrier options (standard and two strike) as well as standard bs calls I obtained excellent convergence.Do you have any further ideas?Have you checked error convergence using different time-steps? Once you have a pattern you can extrapolate. If you don't see any sort of convergence, then you have a numerical issue: probably a rounding error due either to the number of steps or a critical part of the code, such as a mutual cancellation of two similar numbers.

mucki · December 10th, 2003, 9:17 am

Funny! For very even smaller time steps the discrete option prices move arroud the true price. Seems that the tree is not suitable for binaries?Thanks a lot for the advice

daveangel · December 10th, 2003, 10:54 am

A trick u can use is to compute the avergae of the price for n and n+1 steps... this improces the stability of the result

FDAXHunter · December 10th, 2003, 11:03 am

A much better trick is to center the tree nodes on the actual exercise level. This is where the oscillation comes from, as the exercise price dips above and below a node, so you need a huge number of steps.This tree (called a LR tree) also converges much faster than quadrature's trick.Regards.

mucki · December 11th, 2003, 12:55 pm

Thanks a lot that really helps

filthy · December 11th, 2003, 1:25 pm

if you are using the leisen reimer for american options makesure you use richardson extrapolation to take advantageof its monotonic convergence. otherwise is isn'tan awful lot faster.