If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock price will breach $H$ before time $T (T>0)$?
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$\begingroup$ See Section 3.1 of the book "Mathematical Methods for Financial Markets", amazon.com/Mathematical-Methods-Financial-Markets-Springer/dp/…, for discussions. $\endgroup$ – Gordon Sep 15 '15 at 17:16
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You can use the reflection principle to solve this problem. First you calculate the probability that the Stock will be above the barrier at time T. i.e. $P(S_T > H) = blabla$ (which is easy to calculate since it is just normally distributed). Then via the reflection principle you know that the probability of crossing the barrier between time 0 and T is 2x as big, i.e. $2 blabla$.
https://en.wikipedia.org/wiki/Reflection_principle_(Wiener_process)
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$\begingroup$ Thanks. But isn't it $P(S_t > H) = blabla$ and $P(S_T > H) = 2 * blabla$ as the equation seems to state that $P(S_t > H) = 2 * P(S_T > H) $? $\endgroup$ – Jojo Sep 17 '15 at 12:16
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$\begingroup$ Let's use the following notation: $P_{[0,T]}$ is the probability that the stock price will breach H before time T. Let $P_T$ be the probability that the stock breaches H AT time T (i.e. like a european option). Then the reflection principle states $P_{[0,T]} = 2* P_T$. $\endgroup$ – mbison Sep 17 '15 at 13:27
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