Delta jump at activation of a down-and-in call

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.