## Black-Scholes PDE – I: 1st (Original) Derivation

**The original CAPM-based derivation of the Black-Scholes PDE**

*Ingredients required:*

- Ito’s Lemma: Given stochastic process for the stock price , Ito’s lemma gives stochastic process for a derivative as:

- CAPM: Expected return from a stock is sum of the reward for waiting (the risk-free rate ), and reward for bearing risk over and above the risk free rate , i.e.:

…

Given CAPM, the instantaneous return on the underlying follows:

And, similarly, the instantaneous return on the derivative follows:

Re-writing Ito’s Lemma in terms of and dividing by gives:

Dividing and multiplying by in the last term, and writing and respectively as and implies:

Canceling on both sides and noting that the only random term on the RHS is , plus the fact that for any three random variables and , implies allows us to write:

Finally, dividing both sides by variance of market returns gives the following relationship between the option beta and the stock beta:

Coming back to Ito’s Lemma, we can take expectation on both sides of the expression for to write:

Using CAPM expressions for and in the above gives:

The last step now is substituting expression for in terms of , and cancelling terms to show that:

which is the Black-Scholes PDE.

…

@Vineet, I plan to apply at NTU, MFE in ’17. Decent math, fin. and programming skills.

I want to have a firm grip on stochal. Plan to systematically work(slog) through MJ’s(Mark Joshi’s) reading list – http://www.markjoshi.com/RecommendedBooks.html – the suggested readings on prob, stochastic processes and basic mathematical fin in the next 6-7 months. I would be glad to have any tips or suggestions. Thanks a tonne in advance! Cheers. 🙂

QuasarMay 30, 2016 at 6:28 am

It’s been a bit off time on the blog, so maybe it’s already too late, but what you need most is a good math and programming background. Not knowing finance well can be addressed during the program, but if you are behind relevant math, then it could quite an uphill task to compete. I would not recommend any one book, but see your background and your comfort level with advanced calculus / probability and accordingly shortlists a few books, and then decide based on the time that you can afford to study all of that (and time is almost always the biggest constraint). And while I am not fond of general purpose lists, I would second Joshi’s recommendation about Wiersema’s book; as for many others, well, if you could read them, you probably don’t need a math-fin degree 🙂

And it would definitely help if you knew Python well.

VineetAugust 12, 2016 at 10:33 pm