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knowing that the log of the prices in a GBM follows the following normal distribution: you can create a normal distribution with these values and then check the CDF: here is the python code: from scipy.stats import norm; mu=0.16; sigma=0.24;S_0=95;T=1 my_var=sigma**2*T my_norm=norm(np.log(S_0) + (mu-sigma**2/2*T),np.sqrt(my_var)) my_norm.cdf(np.log(93)) ...

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Anyone knows how to derive this formula or knows the reference paper ? The answer requires reflected Brownin motion $w=(\ln (p/b)+rt+ \frac{\sigma^2}{2} t)/(\sqrt{t}\sigma)$ $z=(\ln (p/b)-rt- \frac{\sigma^2}{2} t)/(\sqrt{t}\sigma)$ $g=2(\frac{\sigma^2}{2}+r)/\sigma^2$ probability of hitting= $N(w)+(b/p)^gN(z)$ for $b>p$ $N$ cumulative normal ...

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For in-the-money options the probability of ever getting in-the-money (hitting the strike) before maturity naturally equals unity. The risk neutral probability for an out-of-the-money option ever getting in-the-money is equal to the barrier hit probability used for computing the value of a rebate, developed by Reiner/Rubinstein (1991): p_c = (X/S)^{\mu+\...

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You got to be careful with $\mathbb{P}$ and $\mathbb{Q}$. Indeed, $N(d_2)$ is the probability of the event $\{S_T\geq K\}$ in the risk-neutral world. Note that $r$ (or $r-q$) is the drift in the risk-neutral world and hence this variable occurs in $d_2$. Since time to maturity and volatility are typically small numbers, i.e. \$d_1=d_2+\sigma\sqrt{T-t}\approx ...

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