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Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$ $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ can be rewritten in a risk-neutral way so that the drift is the risk free interest rate $r$:

$$ dS_t = r S_t dt + \sigma S_t d \tilde{W}_t. $$

My question is this:

if I am an investor who is only buying/selling $S_t$ (no bonds, no derivatives, etc.), why should I care about this at all? How would this have any impact on my investment decisions?

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    $\begingroup$ The risk-neutral one is only for the valuation of derivatives by arbitrage. So no, if you are not working with derivatives you don't need to know about that one. $\endgroup$ – noob2 May 11 at 20:23
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Risk-neutral pricing is to help with relative value type questions: If I know the value of this what should the value of that be if it depends in some way on this. It doesn't help with absolute value type questions: Should I buy this or that, is the implied volatility too low or high etc. Those are generally "real world measure" type questions.

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