wilmott.com

Basic concept: Please see my notes on arriving at the Black-Scholes PDE using Ito's Lemma and Geometric Brownian Motion. I have checked online and there are several references that detail on how to solve this PDE using boundary conditions in order to arrive at the famous Black-Scholes formula for option values. Can anyone here cite a source that shows the solution in a succinct and easy-to-understand manner? Thanks in advance.

Examples of some online sources that I have found so far, but looking for better ones:
Finding the Solution to the Black-Scholes Equation (liu.edu)
L26-27-Nov5.pdf (uregina.ca)

Thanks and Kind Regards.

How did you manage to find such longwinded solutions?!

What is your maths background?

Thanks Paul. I cobbled the above from a couple of online sources. And it only arrives at the B-S PDE. From your response, it sounds like you might know of a [relatively] simple way to solve the B-S PDE. I would rate myself 2/10 from a maths background perspective. Keen to learn how to solve the B-S PDE. Thanks in advance.

Cuch, on this forum, will probably have some suggestions, but I'd go with:

1. Change variables (S, t and V) to turn BS PDE into simplest diffusion eqn
2. Use Green's function approach to write down the solution 
3. Manipulate this solution so it's in terms of the normal CDF 
4. Change back to the original variables

But maths is not something you can pick and choose from, there's a well-trodden path that you really ought to be taking. Anyway...

How did you manage to find such longwinded solutions?!

I see it in other contexts as well. People google anonymous online sources, a  bit like Forest Gump “online gazing is like a box of chocolates, you never know what you're going to get.”

//  BTW that solution is horrendous 

Paul (PWOQF Volume I, chapter 6) answers OP's question step-by-step (steps 1 to 4 above). You need to fill in the (simple) algebraic details..

// A follow-on question: what is the rational for this question... student exercise, write a BS PDE solver?

Cuch, on this forum, will probably have some suggestions, but I'd go with:

1. Change variables (S, t and V) to turn BS PDE into simplest diffusion eqn
2. Use Green's function approach to write down the solution 
3. Manipulate this solution so it's in terms of the normal CDF 
4. Change back to the original variables

But maths is not something you can pick and choose from, there's a well-trodden path that you really ought to be taking. Anyway...

PWOQF Volume I, chapter 6!
And Green's function is essential e.g. see Stakgold's (1998) book on this.

 solve this PDE using boundary conditions

BS PDE does not have BCs; you probably mean initial conditions @ t = 0 (or t = T in your case).

But numerically, you need numerical BC at S = 0, S = SMax when the time comes.

Thanks all. Thanks for the reference to chapters in Dr. Wilmott's book. I will go through chapters 5, 6 and 7 as #5 teaches basics of stochastic calculus and builds up to #6 and #7.

I think you deleted your other question about the integral. I wrote down another solution (alternative to noting that the question is about the expected value of a Gaussian with mean m), which I haven't managed to post, so I'm adding it here:

If you want to solve it using calculus rather than probability theory, note that you can split your integral in two terms:
$$\int_{-\infty}^{\infty} c x f(x) dx = c \int_{-\infty}^{\infty} (x-m) f(x) dx + c \int_{-\infty}^{\infty} c m f(x) dx$$. The first integral is over a product of an odd function and an even function with respect to m, which gives an odd function, and its limits are symmetric, so it will give 0. The secong integral, from the conditions of the task is $$m \int_{-\infty}^{\infty} c f(x) dx = m$$

Thanks @kat. This is super helpful.

P.S. I did realize that the function in question resembled the probability density of a Gaussian with mean m, and that the sum product (integral) of all values times their probabilities would result in the expected value m, but instead wanted to solve it using calculus rather than probability theory, and I failed Thanks again.

Thank you, Kat, for that. Exactly what I wrote. But yours is so much more pretty! As I kept trying to explain, nothing to do with probability, means, or c.

Sigh.

No, ska, probability would have been overkill. The observation about odd/even is all you need.

FYI other post removed by request.

Thanks all. 

@Kat - Am I misunderstanding this or does your solution have an extra c (inadvertently) in the second term? Please see highlighted in below screenshot.

That said, love the way the solution uses odd and even function concepts. Makes sense.

A good way to go is to seek a solution of heat equation on infinite interval using well-known separation of variables

[$]u(x,t) = X(x) T(t) [$]

It's nitty-gritty.

Easier would be Fourier Transform.

Thanks all. 

@Kat - Am I misunderstanding this or does your solution have an extra c (inadvertently) in the second term? Please see highlighted in below screenshot.
cm.png

No extra c. Sorry!