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Can the drift rate μ depend on specific stock ? If not what is the rationale for the discounted Stock price to be a martingale ?

\begin{align} & dS_t/S_t = \mu dt + \sigma dW_t \end{align}

Thanks

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Your SDE describes the evolution of the stock price under the physical measure $\mathbb{P}$. However, the BSM model is developed under the risk neutral measure $\mathbb{Q}$. Without digging into results of measure theory and the Girsanov theorem of change of measure, you intuitively "change the weight" of the trajectories such that on average the rate of change is $r dt$. In fact, you define a new probability measure. In the risk neutral world, investors are not compensated for the excess risk and all risk premiums diminish. Back in your question, under the $\mathbb{P}$-Measure the stock can have its own idiosyncratic drift parameter $\mu$, but under the $\mathbb{Q}$-Measure, the drift parameter should be $r$. Hence, under this measure the discounted process with numaire the money market account is a martingale. Then, steps are straightforward.

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  • $\begingroup$ The question is fundamental and does not need any change in measure. If I have two different stocks and construct an instantaneous portfolio of <S1> of S2 and -<S2> of S1, I should be able to cancel the drift. Is that still a truly drift free GBM ? Doesn't look so. $\endgroup$ – Chris Aug 6 at 4:22
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An easy way to think about it:

In the risk-neutral world ($\mathbb{Q}$), investors don’t care about risk and don’t pay you more or less than they pay for a risk-free investment. They don’t really see a difference. Hence, all assets have the same return, the risk-free rate $r$.

Of course, in the real world ($\mathbb{P}$), people do care and a stock has a different return $\mu$. There is a risk premium investors pay. And of course, different stocks may have different drift rates.

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  • $\begingroup$ Yes, that is the question. What is the justification for RN ? $\endgroup$ – Chris Aug 6 at 4:16
  • $\begingroup$ Under RN measure $\mathbb{Q}$, discounted stock prices are martingales (we choose $\mathbb{Q}$ such that this holds). Why do we do this? Then, we can price options as discounted conditional expectation which may be easier than solving PDEs and MC Simulation etc. The martingale approach is one of the easiest ways of finding option prices and relies on discounted stock prices to be martingales. If you ask why can we do that, the first fundamental theorem of asset pricing states that the absence of arbitrage is equivalent to the existence of (at least) one equivalent martingale measure. $\endgroup$ – KeSchn Aug 6 at 4:28

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