= Reflected European payoff for an up-and-in option
{title2=$g(x)=f(x)\mathbf1_{\{x>b\}}+\kappa^{-\nu/\sigma}f(x/\kappa)\mathbf1_{\{x\leq\kappa b\}}$}
For an <up-and-in claim> in the dividend-free <Black-Scholes model>, set $\nu=(\rho-\sigma^2/2)/\sigma$, $\kappa=(S_0/b)^2$ and assume finite absolute payoff expectations. Its initial price equals the <risk-neutral pricing> value of the displayed <European option>. To prove this, express the log-price as $\sigma(W_t+\nu t)$ and apply the <joint endpoint and maximum law for drifted Brownian motion>. Above the logarithmic barrier $a=\log(b/S_0)/\sigma$, use the unrestricted endpoint density. Below it, the density of paths which hit the barrier is $e^{2a\nu}\phi_T(y-2a-\nu T)$. Substitution $z=y-2a$ gives the scaled payoff, the threshold $\kappa b$, and the factor $\kappa^{-\nu/\sigma}$. This is an initial-price identity, not a pathwise equality of terminal payoffs.
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