# About the log return in the Black&Scholes model

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.

• 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. – Magic is in the chain Jun 7 at 14:44
• 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"? – Jan Stuller Jun 9 at 16:09

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.

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.