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I'm currently studying the Black&Scholes model and I'm not sure about the following thing: the log return, say r, doesn't evolve in time? I mean, dr/dt = 0, its derivative is zero? Does only its average evolve in time, that is, d average(r)/dt = something? Thank you.

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  • $\begingroup$ In the classic model, r is assumed to be constant, as well explained in the answer below. In terms of dynamics, it features as a a constant coefficient in the evolution of the bank account: $dB_t=r B_t$, and through this (combined with the magic of the valuation formula) it makes its way to the drift of the stock price evolution equation. $\endgroup$ – Magic is in the chain Jun 7 at 14:44
  • $\begingroup$ Ismael, if you like Bob's (or mine) answer below, could you pls click on the "tick mark" next to one of the answers so that this question can be marked as "complete"? $\endgroup$ – Jan Stuller Jun 9 at 16:09
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Let me try to answer. In the Black-Scholes model, we have the following dynamics for a stock Price $S_t$:

$$S(t)=S(0)+\int^{t}_{0}r S(h)dh+\int^{t}_{0}\sigma S(h)dW(h)$$

The short-hand notation for the above would be:

$$dS_t= r S_t dt+\sigma S_tdW_t$$

The two equations are the same thing (just two different notations) and the solution to both is the log-normal process:

$$S_t = S_0exp{(rt+0.5\sigma^2t+ \sigma W(t)})$$

The log-return is defined as $ln\left(\frac{S_t}{S_0}\right)$, so we can easily see that:

$$ln\left(\frac{S_t}{S_0}\right)=rt+0.5\sigma^2t+ \sigma W(t)$$

You can see that the log-return is Normally distributed with mean $=rt+0.5\sigma^2t$ and standard deviation $=\sigma \sqrt(t)$ (why? Because by definition $\sigma W(t)$ is normally distributed with mean zero and standard deviation equal to $\sigma \sqrt(t)$) .

So the log-return itself evolves in time: it is a stochastic process that is normally distributed around its (time-dependent) mean and it has a (time-dependent) standard deviation. If you plot the log-return on x-y axes, with y being time and x being the log-return, you can picture it as a straight line with slope $rt$ where the Normal distribution of the log-return is tangentially centered on this straight line. As time goes on, the standard deviation of this Normal distribution around the line gets wider and wider.

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The risk-free rate is assumed to be constant, see for example Wikipedia:

(riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate.

Under the risk-neutral measure (where pricing using the Black-Scholes formula happens), the return on the stock is equal to the risk-free rate.

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