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In the 5th page of Black and Scholes' original paper on option pricing formulas, they write the following assumption:

b) The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is log- normal. The variance rate of the return on the stock is constant.

I'm a bit confused about the first sentence and how it relates to the GBM assumption $$dS_t = S_t[ \mu dt + \sigma dW_t],$$ especially since the last sentence says that the "variance rate of return" (which I assume is $\sigma^2$) is constant over time. In particular, I don't see where $S_t^2$ comes into play.

Also, I read somewhere that Merton cleaned up the derivation and put it in more mathematically rigorous language. Where can I find that paper?

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Answer: if $dX_t = \mu(t,X_t,\dots) dt + \sigma(t,X_t, \dots) dW_t$, then $\mu$ is called the "drift rate" and $\sigma^2$ is called the variance rate. The connection to quadratic variation is that $d\langle X \rangle_t = \sigma(t,X_t,\dots)^2 dt$, i.e. the drift rate of the quadratic variation of $X_t$ is in fact the variance rate of $X_t$.

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  • $\begingroup$ What about the factor $S_t$ in your formula? Also variance is the square of the standard deviation. On the other hand, the return is the log of $S_t$. $\endgroup$ – Raskolnikov Dec 14 '17 at 23:05
  • $\begingroup$ The SDE implies that the log of the stock prices is a Brownian motion with mean $(\mu - \sigma^2/2)$ and variance parameter $\sigma^2$, but this still doesn't clear up my question of what a "variance rate" of a stock price is. $\endgroup$ – user217285 Dec 14 '17 at 23:11
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    $\begingroup$ "variance rate" refers to the fact that the variance scales with time: if $\sigma^2$ is the variance rate per year, then the variance over a 6 month period (half a year) is $\frac{1}{2}\sigma^2$ and so forth. R.C. Merton does not discuss this, it is just standard terminology: "variance rate" = variance divided by units of time. Your "Pay rate" = how much money you are paid per unit of time (hour, year, whatever). $\endgroup$ – Alex C Dec 15 '17 at 0:49
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    $\begingroup$ Variance rate = $\frac{\textbf{Var}[\ln S_T-\ln S_0]}{T}$ $\endgroup$ – Alex C Dec 15 '17 at 1:53
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    $\begingroup$ If memory serves merton's paper is called "theory of rational option pricing" $\endgroup$ – berkorbay Dec 15 '17 at 16:15

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