Drifted Brownian first-passage density ($h_c(t)=a{e}^{-(a-ct{)}^2/(2t)}/\sqrt{2\pi t^3}$)

For a positive level $a$, this density describes the first-passage time of $B_t+ct$. The Girsanov theorem multiplies the zero-drift density by the stopped likelihood $e^{ca-c^{2}t/2}$. Its total mass is one for nonnegative drift and $e^{2ac}$ for negative drift; the latter has remaining mass at infinite hitting time.

For $a>0$, real $b$ and standard Brownian motion, the first-passage time to a linear boundary hasHere $\Phi$ is the standard normal distribution function. Differentiation gives $a\exp (-(a+bt{)}^2/(2t))/\sqrt{2\pi t^3}$, the drifted Brownian first-passage density. Its Laplace transform is $\exp [-a(b+\sqrt{b^2+2\theta })]$ for $\theta >0$. There is mass $1-e^{-2ab}$ at infinity when $b>0$; for $b\le 0$ the hitting time is finite almost surely.

When $d=\sqrt{c^2+2r}$ is real and nonnegative, the discounted drifted Brownian first-passage density is $e^{a(c-d)}{h}_d(t)$. Integrating yields $e^{a(c-d)}\Phi ((dT-a)/\sqrt{T})+e^{a(c+d)}\Phi ((-a-dT)/\sqrt{T})$. In a Black-Scholes model with $c=(r-{\sigma }^2/2)/\sigma$, the identity $c^2+2r=(r+{\sigma }^2/2{)}^2/{\sigma }^2$ ensures this formula covers every real interest rate.