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How to take the expectation of an exponential martingale? And an exponential with a random v...

Given that $exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)$ is a martingale, you just need to substitute $m = 0$ into it to find the value of its expectation, as for any martingale $Z_t$ we have that $E[Z_t] …
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