I was wondering how to derive the price of a cash-or-nothing and asset-or-nothing option by trying to work out the expectation under the risk-neutral measure, while assuming that the underlying follows a Geometric Brownian motion.

I know that the value of the Asset-or-nothing call is supposed to be $Value = S_0\Phi(d_1)$

Furthermore the value of the Cash-or-nothing call should be $Value = e^{-rT}A\Phi(d2)$, if we assume that it pays out A if $S_T > K$.

Yet I don't know how to derive these results myself, and I haven't been able to find a book that does it


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