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I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price process is performed (Ito lemma part) in the first place. Price process is already a function of time and Wiener process, so I wonder why do we need to apply another function (ln). I do not think it has to do with log normality of prices or normality of returns. I have seen such a transformation taking place in solution of other problems that were not related to GBM - BS framework.

Thanks,

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closed as off-topic by Matthew Gunn, noob2, LocalVolatility, vonjd, JejeBelfort Aug 25 '17 at 7:27

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    $\begingroup$ Small log differences can be approximated by the percent change.. If prices followed Brownian motion, then they could go negative (which is entirely non-sensical). On the other hand, log prices following Brownian motion is equivalent to saying that log returns follow the normal distribution which is similar to saying that over small periods (where returns are small) that returns follow the normal distribution. $\endgroup$ – Matthew Gunn Aug 24 '17 at 18:51
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    $\begingroup$ Remember that $dW$ is normally distributed it, so we have to transform it (or the price) if we want some other distribution. $\endgroup$ – noob2 Aug 24 '17 at 18:56
  • $\begingroup$ @noob2: Thanks for your reply. I really appreciate it, even though I disagree with it. One does not need to transform price process under GBM to arrive to non-normality. The dynamics of price process, under GBM, by definition is log normally distributed (even before one mathematically applies Ito lemma) $\endgroup$ – Larevue Aug 24 '17 at 20:03
  • $\begingroup$ @Matthew Gunn: Thanks Matthew for your contibution. As indicated in my post, this is the answer I expected not to receive. I would like to complete your comment, because it might be misleading. If a price process follows arithmetic Brownian motion, price can take negative value. If a price process follows geometrical Brownian motion, only positive values are enabled. In both cases I assume classical Brownian motion where the error term is standard normal variable. $\endgroup$ – Larevue Aug 24 '17 at 20:08
  • $\begingroup$ If $X_t$ follows geometric Brownian motion, then $Y_t = \log X_t$ follows Brownian motion with drift. Geometric brownian motion is still an imperfect model of stock prices, it underestimates tail events. Anyway, a broader aesthetic point is that anytime you're dealing with exponential growth, you may want to think about the model and the problem in terms of log differences. Under the log transform, multiplication because a sum. Repeated multiplication becomes a summation or integral. $\endgroup$ – Matthew Gunn Aug 24 '17 at 20:14
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This is merely a mathematical trick.

You cannot easily integrate $dS_t = S_t(\mu dt + \sigma dW_t)$ over time because the RHS depends on $S_t$.

Using Ito's lemma on the log price gets you: $d\ln(S_t) = \left(\mu-\frac{1}{2}\sigma^2\right) dt + \sigma dW_t$ which is straightforward to integrate.

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  • $\begingroup$ I see that logarithmical transformation nicely cancels price / price square terms under SDE. But is there no way one could apply directly the second order approximation using Taylor expansion (with the first and second derivative taken w.r.t. Winer process)? Or can another transformation be applied? Thanks. $\endgroup$ – Larevue Aug 25 '17 at 8:37
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    $\begingroup$ You cannot use standard calculus for functions of stochastic processes. Instead you should use Itô calculus for instance. In that case, taylor approximation is basically Itô's lemma. $\endgroup$ – Quantuple Aug 25 '17 at 10:10
  • $\begingroup$ thanks again. Agree re Ito's lemma being a "counterparty" of Taylor's expansion. Of course, non-differentiability of Winer process requires use of stochastic calculus. $\endgroup$ – Larevue Aug 26 '17 at 11:40

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