Finite Differences for option pricing

[d.koutsomito]

Hello there,

I would like to know how efficient and how popular is to price options using finite difference methods. If I am not wrong, finite differences are used for their PDEs. If it is not the best way to do so, can you suggest any?

I am looking forward to your opinion.

Thanks.

 

[Weiyi CHEN]

Yes, FD methods are generally used to approximate the solutions of certain PDEs, numerically. I'm not sure it's the most efficient one when solving PDEs but certainly it's widely used in industry at least currently. The reason why FD methods are a popular choice for pricing options is that all options will satisfy the Black-Scholes PDE. Finite Difference methods can be applied to American Options and they can also be used for many exotic contracts.

Most popular FD methods in computational finance includes Explicit Euler, Implicit Euler, CN method and so on. Each FD method has its pros and cons, so they are used in different situations. For example, Explicit Euler is unstable for certain choices of domain discretisation, while Implicit Euler and CN are unconditionally stable with respect to the domain discretisation though solving linear systems of equations. So in short, FD methods has powerful abilities to solve PDEs when you're able to adjust your explicit strategy according to given conditions, I can't see it would become less populars in recent future, personally speaking.

 

[Daniel Duffy]

http://www.datasimfinancial.com/forum/viewforum.php?f=24&sid=5e9c4d0ea7a210558cab1fc30caee0bf

http://www.datasimfinancial.com/articles.php

I'm not sure it's the most efficient one when solving PDEs but certainly it's widely used in industry at least currently.

FDM is probably the fastest numerical method out there.

 

[d.koutsomito]

I have now a guide and enough material to study further.

Thank you both very much.

 

[Daniel Duffy]

FDM is also used to approximate SDEs, but the theoretical foundations are trailing behind those for PDEs.

 

[d.koutsomito]

FDM is also used to approximate SDEs, but the theoretical foundations are trailing behind those for PDEs.

Do you mean by solving the PDE of the drift of the SDE?

If not, could you propose any material on this topic too?

 

[Daniel Duffy]

Do you mean by solving the PDE of the drift of the SDE?

If not, could you propose any material on this topic too?

See the above links where there is more.

 

[Bastian Gross]

In large dimension PDE-solving could be really expensive.
So be tricky and use reduced basis methods.

 

[Daniel Duffy]

In large dimension PDE-solving could be really expensive.
So be tricky and use reduced basis methods.

in 32-bit a 4d pde won't even fit into the memory!

 

[ExSan]

in 32-bit a 4d pde won't even fit into the memory!

could you please explain why ? -sample-

 

[MEqnet]

I've seen people recommend using eigendecomposition to knock out the cross terms for BS multi-asset options, then apply a standard ADI method after making some change-of-variables. I haven't seen this implemented for a stochastic vol term, although there is a paper floating out there the applies extended ADI methods to the Heston model.

Out of curiosity, what type of option are you trying to value? 'Efficiency' beyond that of your time marching scheme may start with how you discretize your domain.

 

[Daniel Duffy]

could you please explain why ? -sample-

make a nXmXpXq 4-d array

 

[ExSan]

make a nXmXpXq 4-d array

m = ? n = ? p = ? q = ?
I have tested a 4-D and n-D arrays for not so small values, my own abstract data structure

 

[Daniel Duffy]

m = ? n = ? p = ? q = ?
I have tested a 4-D and n-D arrays for not so small values, my own abstract data structure

n >= 50 etc.

Using Boost::multi_array in the past with a few objects brought my PC to a halt!

 

[Daniel Duffy]

Here is an overview article on PDE/FDM. It was written for a mathematically savvy audience in academia, not necessarily in computational finance.

 

[PepeQuant]

To solve multi-dim PDEs, you can either reduce the dimension...most of the case X2/X1 or...sometimes X/P(t, T)... if none of them works, try to use operator splitting method

 

[Daniel Duffy]

To solve multi-dim PDEs, you can either reduce the dimension...most of the case X2/X1 or...sometimes X/P(t, T)... if none of them works, try to use operator splitting method

When n >=3 life gets difficult for PDE/FDM scheme.

BTW have you used ADE?

Datasim Financial Forums • View topic - The Alternating Direction Explicit (ADE) Thread

 

[PepeQuant]

When n >=3 life gets difficult for PDE/FDM scheme.

BTW have you used ADE?

Datasim Financial Forums • View topic - The Alternating Direction Explicit (ADE) Thread

lol cant agree more on n >= 3. I havnt used ADE but I will definitely take a look. It seems interesting

update: I just took a quick look and it is very similar to operator splitting method

 

[Daniel Duffy]

update: I just took a quick look and it is very similar to operator splitting method

Until you get to know it; it's like learning the names of trees or birds

Soviet Splitting originated in USSR 1960s; ADE originated there in the 1950s!

 

[Daniel Duffy]

lol cant agree more on n >= 3. I havnt used ADE but I will definitely take a look. It seems interesting

update: I just took a quick look and it is very similar to operator splitting method

Actually, ADE can handle all n >=3. The only(?) issue is memory
See Part 4
https://www.quantnet.com/threads/boost-c-libraries-and-applications-to-computational-finance.9961/