= Up-and-out power claim
{title2=$S_T^p\mathbf1_{\{\sup_{t\leq T}S_t<c\}}$}
An <up-and-out power claim> pays a power of the terminal <stock> price provided an upper barrier has never been reached. In the <Black-Scholes model>, set $A=\log(c/S_0)/\sigma$ and $m_p=(\rho+(p-\tfrac12)\sigma^2)/\sigma$, where $\sigma>0$, $c>S_0$ and $T>0$. The <risk-neutral pricing> value is
$$
S_0^p e^{[(p-1)\rho+p(p-1)\sigma^2/2]T}\left[\Phi\left(\frac{A-m_pT}{\sqrt T}\right)-e^{2m_pA}\Phi\left(\frac{-A-m_pT}{\sqrt T}\right)\right].
$$
Here $\Phi$ is the <standard normal distribution function>. Weighting the <Brownian motion> endpoint by $e^{p\sigma W_T-p^2\sigma^2T/2}$ changes the logarithmic drift to $m_p$; the <finite-horizon maximum of Brownian motion with drift> then supplies the survival factor.
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