I am currently working on a research project involving the numerical solution of the Black-Scholes partial differential equation (PDE) for option pricing. To simplify the problem, I am interested in converting the Black-Scholes PDE into an ordinary differential equation (ODE) form.
I would like to inquire about the techniques and methods used for this conversion process. Specifically, I am seeking guidance on how to transform the Black-Scholes PDE into an ODE and the subsequent steps involved in solving the resulting ODE.
Any insights or suggestions regarding the conversion process, mathematical techniques, or numerical methods would be greatly appreciated. Additionally, if you could recommend any relevant research papers, textbooks, or online resources that discuss this topic in detail, it would be of great help.
Thank you in advance for your assistance!
Conversion of BSM PDE to an ODE is not possible unless you are interested in an approximate solution. The following book has a detailed derivation of a closed-form solution using the substitution of variables.
Kallianpur, Gopinath and Karandikar Rajeeva L. Introduction to Option Pricing Theory
To solve this equation it's necessary to impose boundary conditions. This means that the solution can be expanded in a series of functions that are a complete set and satisfy the boundary conditions. This is an exact representation-the numerical methods enter the picture when approximating the infinite series by a finite sum.
There are many choices and any one is as good as any other. A typical choice is Chebyshev polynomials in the ``spatial'' variable, which transforms the Black-Scholes equation to a system of ODEs in the ``time" variable.
Another choice could be plane waves, depending on the boundary conditions.
This is the so-called ``(pseudo-)spectral'' approach. It would be the ``spectral'' approach, were it easy to find the eigenfunctions and eigenvalues of the ``spatial'' differential operator, given the boundary conditions.
A typical reference is: Article A brief introduction to pseudo-spectral methods: application...
Cf. also https://master-tma.gricad-pages.univ-grenoble-alpes.fr/num-methods-turb/part2/chapter1_pseudospectral_methods.html
It is possible to express the solution in closed form: https://digitalcommons.liu.edu/cgi/viewcontent.cgi?article=1074&context=post_honors_theses so it's possible to test the quality of the numerical approach that implements the pseudo-spectral method, that, once more, is exact.
Dear Fawad,
You will find the exact solution of how to solve the Black-Scholes PDE equation by transforming it and eventually reducing it down to the Heat-Diffusion equation in "Introduction to Partial Differential Equations, by Peter J. Olver", on pages 299 to 301. Olver then proceeds to solve this resulting Heat-Diffusion equation for its Fundamental Solution and this is explained further on pages 301 to 302 of the book. Alternatively, the Heat-Diffusion equation may also be solved by substituting z(Tau, y) = F(Tau)*G(y) and its derivatives into the Heat-Diffusion equation, where upon solving the two resulting ODEs (one in terms of time, Tau, and the other in terms of the spatial coordinate, y), you will obtain an eigenfunction solution, for example, of the form Sum(exp(-k*Tau)*Cos(ky)). This latter solution is often of a damped sinusoidal nature, but I do not know if it is used in this area of financial mathematics as I have never really studied the area myself; I only know how to solve the initial Black-Scholes PDE by each of these methods. Here are some other details about Peter J. Olver's book, which I repeat once again as "Introduction to Partial Differential Equations". I highly recommend this text, and you can find the book using the following numbers: ISSN 0172-6056, ISSN 2197-5604 (electronic), ISBN 978-3-319-02098-3, ISBN 978-3-319-/02099-0 (eBook), DOI 10.1007/978-3-319-2099-0, Library of Congress Control Number 2013954394, Springer Publication
I hope that this helps.
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