= 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.
Back to article page
