Magrabe Exchange Option: not equal drifts

Question

I need to calculate the price of exchange option between 2 assets $S_1$ and $S_2$ The formula is given here Wiki: Magrabe formula or here Quant Stack Exchange. In the derivation of the formula it is assumed, that price is the discounted expecation of future claim under risk-neutral measure $Q$: $$p=e^{-r_{f}}\mathbb{E}_{Q}(S_1-S_2)^{+}$$. and under risk-neutral measure it is assumed, that: $$dS_{i,t}=\mu_{i}dt+\sigma_{i}dW^{i}_{t}$$ $$\mu_{i}=r_{f}\text{, for }i=1,2$$.

My question is:

Why do we calculate expectation under risk-neutral measure $Q$ and not real-world measure $P$ ? In real world return on each asset can be different, i.e. $\mu_1\neq\mu_2\neq r_f$. Then it would make sence to use real-world measure, under which this condition is fullfilled.

Answer 1

You can essentially get your answer from any quantitative finance book. I recommend Shreve's book. Note that this has nothing to do with pricing an exchange option.

You could price in the real-world measure, but that's very difficult if impossible because you'd need to know the risk premium for an instrument. By changing the probability measure to risk-neutral, you can assume all investors demand only the risk-free rate and thus you could discount your expected price with the risk-free rate.

Please read risk neutral measure on Wikipedia. In particular, focus on "Unfortunately, the discount rates would vary between investors and an individual's risk preference is difficult to quantify."