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I read that return is normal and stock price is log normal. But I also read that return is log normal. So I am confused about which it is.

In the 14th Chapter of Options, Futures, and other Derivatives, John C Hull says

... the expected percentage return required by investors from a stock is independent of stock's price...

A reasonable assumption is that the variability of the return in a short period of time, $\Delta t$, is the same regardless of the stock price.

Based on these two observations, he got the model : $$\frac{\Delta S}{S}=\mu\Delta t + \sigma\epsilon\sqrt{\Delta t}$$

where $S$ is price, $\epsilon$ has a standard normal distribution. From this model, it seems like return has a normal distribution. This model also says $S$ follows a geometric Brownian motion and so we can apply Ito's lemma on some function of $S, t$. Applying Ito's Lemma to $lnS$ he arrived at the equation $$lnS_T - lnS_0\sim N((\mu-\frac{\sigma^2}{2})T, \sigma^2 T)$$ His conclusion from this is that price follows a log normal distribution. But it seems that return $r$ is also normal since $lnS_T-lnS_0=ln\frac{S_T}{S_0}=lnr$ is normal.

My question is, assuming the proposed model is right, does return have a normal or log normal distribution? Or is it a matter of time span, for small $\Delta t$ return is normal, and for longer periods $T$ return is log normal?

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  • $\begingroup$ If returns had a lognormal distribution, prices would never go down. $\endgroup$
    – Chris Haug
    Commented May 19, 2022 at 10:32

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The actual model here is the one described by the equation with the tilde (~). In that model:

  • $S_t$ has a lognormal distribution for any $t$
  • $S_t/S_0$ has a lognormal distribution for any $t$
  • $S_t/S_0$ is approximately normal for any small $t$.

The equation with $\Delta$'s is less a definition of the model, and more a guideline to its numerical implementation.

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  • $\begingroup$ But the delta equation has clear relation with the physical assumptions. If instead the tilde equation is the model definition, what sort of market or investor behaviours is the model based on? $\endgroup$
    – Sara
    Commented May 5, 2022 at 3:02
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    $\begingroup$ The model is still based on expected return being independent of stock price. The equation with deltas is one way of writing out the intuition, but it is not a consistent model on its own, because carrying it out for two steps with $\Delta t =1$ gives a different result from one step with $\Delta t =2$. $\endgroup$
    – user225256
    Commented May 5, 2022 at 12:47

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