For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much?
Avoid math heavy answers please.
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You can compute expectation of drifted processes as well and derive same pricing formulas,but usually its more complicated (compare derivation of Black Scholes using martinglaes and through PDE. PDE proof ,where drift is explicit, is much longer)
With martingale representations you have more analytical mathematical tools/formulas available (e.g. barrier hitting times,easy expectation calculation ,etc).