Down-and-in claim

A down-and-in claim with lower barrier $b$ pays a specified terminal payoff only if the stock reaches $b$ before maturity. Before activation its value solves a killed-domain pricing problem; at activation its value equals the ordinary terminal-payoff claim value.

For strike $c>b$, write $C(s,c,\tau )$ for the ordinary European call option value. Before activation the down-and-in claim is $I(s,\tau )=(b/s{)}^{\alpha }C(b^2/s,c,\tau )$, where $\alpha =2\rho /{\sigma }^2-1$. Consequently its left-limit option delta at the barrier is $-\alpha C(b,c,\tau )/b-C_s(b,c,\tau )$, whereas after activation it is $C_s(b,c,\tau )$. Their difference is strictly positive for $\tau >0$: differentiating the killed normal transition density at its boundary gives the integral of a strictly positive function against the positive call payoff. The value is continuous, so changing the bank holding finances the stock rebalance.

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 $g(x)=f(x)1_{x\le b}+{\kappa }^{-\nu /\sigma }f(x/\kappa )1_{x>\kappa b}$. Reflection of the log-price endpoint above the lower barrier changes its normal probability density to $e^{2\nu \ell }{\varphi }_T(y-2\ell -\nu T)$, where $\ell =\log (b/s)/\sigma$. Substitution $z=y-2\ell$ gives the payoff identity. This is price equality, not pathwise equality of payments.