= Static terminal-payoff representation of a down-and-in claim
{title2=$\kappa=(s/b)^2$}
For initial spot $s>b$ in the <Black-Scholes model>, put $\nu=(\rho-\sigma^2/2)/\sigma$ and $\kappa=(s/b)^2$. A <down-and-in claim> with terminal payoff $f$ has the same initial price as the terminal claim $g(x)=f(x)\mathbf1_{x\le b}+\kappa^{-\nu/\sigma}f(x/\kappa)\mathbf1_{x>\kappa b}$. Reflection of the log-price endpoint above the lower barrier changes its <normal probability density> to $e^{2\nu\ell}\phi_T(y-2\ell-\nu T)$, where $\ell=\log(b/s)/\sigma$. Substitution $z=y-2\ell$ gives the payoff identity. This is price equality, not pathwise equality of payments.
Back to article page
