Thread: Partial Differential Equations PDEs and their numerical approximation

[Daniel Duffy]

Black Scholes pde 101

 

[John TIAN]

Hi Dr. Duffy, I am a final year student looking for learning some numerical methods. Recently, I saw a MIT course 18.085 Computational Science and Engineering taught by Prof. Strang. And He has a book about it. What do you think of it? Thanks a lot

 

[Daniel Duffy]

Gil Strang is probably the best maths educator around. His focus is mostly linear algebra and his videos are excellent, especially his linear algebra videos.
Actually, he was PhD supervisor of my PhD supervisor, so he is my academic grandfather

I usually write all my own stuff. My forthcoming (self-contained) book on PDE/FDM (for finance) (story A-Z) is

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach

This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners, novices...

www.wiley.com

I actually discuss Strang FDM splitting in the book.


And online courses

Online Courses :: Datasim

www.datasim.nl

 

[Daniel Duffy]

A bit of a preview of some topics

Several chapters are devoted to FDM for 2-factor European/American options We tested the relatrve performance on spread options for the same level of accuracy:

1 Saulyev ADE : 1 second
2 Barakat and Clark 2.2
3 Yanenko 4.2
4 "Classic" ADI 4.0 (4 times slower!)
5 Predictor-Corrector 3.8
6 Marchuk-Strang 7.6

. Methods 1+2 do not involve solving matrix systems. 3-6 solve tridiagonal systems.
. It is possible to compute greeks by writing them as PDEs and using CSE approach (essentially tbe Cauchy-Kowalewski (Lax-Wendroff) technique). We did it for 1-factor and CSE for spreads should also be fast as well.
. Sauly'ev ADE is also the easiest to program.
. We mapped PDEs to unit square making boundary conditions easy (Fichera theory).
. Schemes 1 and 2 can easily be applied to 3-factor problems.,
. Code used is C++20, of course.
. The methods are also applicable to fixed income (Hull-White, CIR-style, mean-reverting).