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Difficulty: Hard
Category: Finance
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Topics: brownian-motion, probability, reflection-principle, stochastic-calculus
Consider a standard Brownian motion $W_t$, starting at 0. You are tasked with pricing a barrier option that pays out if the maximum value of the Brownian motion before time $t$ exceeds a level $a$. Given $a > 0$, what is the probability that the maximum value of the Brownian motion from time 0 to $t$ exceeds $a$; that is, what is $P(\max_{0 \le s \le t} W_s \ge a)$? Recall that $\Phi(x)$ represents the cumulative distribution function (CDF) of the standard normal distribution.
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