Deriving Black Scholes using CAPM

Question

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf

There are several things that are not clear to me, and I would like to ask someone to explain them:

(1) on page 7 we read "Drop dt from both sides and take the covariance of rV and rM, noting that only the second term on the right-hand side of Equation (10) is stochastic" : how on Earth do we see this? the first term also contains Vt and St just like the second one, so what would make it less stochastic?

(2) first we find E[dV_t/V_t] and then we multiply the equation by V_t... Without arguing at all we assume that E[dV_t/V_t]=E[dV_t]/V_t. Why is that so? Isn't Vt random as well, it's not a constant, is it?

(3) after all, what is the precise mathematical meaning of the expression E[dV_t]?

Hope someone can clarify this. Thanks.

Answer 1

It should be noted that the original paper by Black & Scholes is not mathematically rigorous. There is no rigorous meaning to the expression $\mathrm{d}t$. Nor is there any rigorous meaning to the expression $\mathrm{d}V_t$. Actions such as 'dropping $\mathrm{d}t$' have no mathematical meaning. Nevertheless they hold because beneath this heuristic garbage lies a deep mathematical theory pinning what you are allowed to do. If you are looking for a decent, non rigorous derivation to the Black Scholes equation, then Wilmott - The Mathematics of Financial Derivatives is a good book to look at.

That paper has no meaning in a mathematical sense. It is impossible to answer your question.

(too long to be a comment)

Answer 2

The CAPM derivations that you link to use shorthand notation to describe stochastic integrals, which is not very rigorous indeed and can lead to a lot of confusion. This question on QSE shows that as a result of using this notation, the CAPM derivation by Black and Scholes (and also the one by Rouah that you link to) contains two mistakes. As a result, it seems that the PDE of Black and Scholes cannot be directly derived from the CAPM.

Answer 3

Just seen this - probably too late, but if not..:

re. (1): You are in $t$, i.e. $t$ is now, and $V_t$ and $S_t$ are real market prices observable right in this moment ($t$).

re (2): same answer as to (1). $S_t$ and $V_t$ are here not symbols for random variables, but current realizations which you see now (in time $t$). Of course, you don't know future prices, so from your current perspective, $V_{t+1}$ and $S_{t+1}$ (and hence $dV_t$ and $dS_t$) are random (note that I use infinitesimal small units here for $t$...)

re (3) $E$ is the expectation operator, $d$ indicates change in an infinitesimal small time interval, $V_t$ is the option value. $E(dV_t)$ is the expected change of the option value over the next infinitesimal small time interval.