Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

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

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

Ingredients required:

  • Ito’s Lemma: Given stochastic process for the stock price dS = \mu S dt + \sigma S dX, Ito’s lemma gives stochastic process for a derivative F(t, S) as:

\displaystyle dF = \Big( \frac{\partial F}{\partial t} + \mu S \frac{\partial F}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \sigma S \frac{\partial F}{\partial S} dX(t)

  • CAPM: Expected return from a stock is sum of the reward for waiting (the risk-free rate r), and reward for bearing risk over and above the risk free rate (\beta_s (E[r_M] - r_f)), i.e.:

E[r_S] = r + \beta_S (E[r_M] - r)

Given CAPM, the instantaneous return r_S dt on the underlying follows:

\begin{aligned} \displaystyle E[r_S dt] = E[\frac{dS}{S}] &= r dt + \beta_S (E[r_M] - r) dt \\ \Rightarrow E[dS] &= rS dt + \beta_S (E[r_M] - r) S dt \end{aligned}

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

\begin{aligned} \displaystyle E[r_F dt] = E[\frac{dF}{F}] &= r dt + \beta_F (E[r_M] - r) dt \\ \Rightarrow E[dF] &= rF dt + \beta_F (E[r_M] - r) F dt \end{aligned}

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

\displaystyle \frac{dF}{F} = \frac{1}{F} \Big( \frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \frac{1}{F} \frac{\partial F}{\partial S} dS

Dividing and multiplying by S in the last term, and writing \frac{dS}{S} and \frac{dF}{F} respectively as r_S dt and r_F dt implies:

\begin{aligned} r_F dt &=\frac{1}{F}\Big( \frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \frac{\partial F}{\partial S} \frac{S}{F} r_S dt\end{aligned}

Canceling dt on both sides and noting that the only random term on the RHS is r_S, plus the fact that for any three random variables x, y and z, y = a + bx implies \mbox{Cov}(y, z) = b \mbox{Cov}(x, z) allows us to write:

\displaystyle \mbox{Cov}(r_F, r_M) = \frac{\partial F}{\partial S} \frac{S}{F} \mbox{Cov}(r_S, r_M)

Finally, dividing both sides by variance of market returns \sigma_M^2 gives the following relationship between the option beta and the stock beta:

\displaystyle \beta_F = \frac{\partial F}{\partial S} \frac{S}{F} \beta_S

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

\displaystyle E[dF] = \Big( \frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \frac{\partial F}{\partial S} E[dS]

Using CAPM expressions for E[dF] and E[dS] in the above gives:

\displaystyle rF dt + (r_M - r) \beta_F F dt = \Big( \frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \frac{\partial F}{\partial S} \big(rS dt + (r_M - r) \beta_S S dt \big)

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

\displaystyle \frac{\partial F}{\partial t} + rS\frac{\partial F}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2}= rF

which is the Black-Scholes PDE.

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Written by Vineet

July 12, 2014 at 2:24 pm

2 Responses

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  1. @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. 🙂

    Quasar

    May 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.

      Vineet

      August 12, 2016 at 10:33 pm


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