It is well known that a european call option with strike price $C(K)=(S_T-k)^+$ coul be hedge using the Black-Scholes formula $BS(t,T,r,K,S_0)$. I would like to find a hedge (or sub-hedge) of the the next portfolio (or find some strategy) \begin{equation} \chi_{[S_T\geq B]}+\frac{1}{B-K}\left((S_t-B)^+-(K-S_T)^+(S_{H_B}-S_T)\chi {[H_B\leq T]}\right) \end{equation} where $0<K<B$, $\chi$ is the indicator function and $H_B=\inf_{t\geq 0}{S_t\geq B }$. It is because I am trying to make make a sub-hedge of the one-touch option $\chi_{[H_B\leq t]}$. If anyone has some idea I really appreciate it. Thank you!
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$\begingroup$ First of all you want a price from which you calculate its delta. That delta is the amount of stock you need to hold in the hedge. This applies to every model and every payoff. Esp. for such an exotic payoff I would however be careful with the assumption that a BS price (and its delta) correctly price and hedge that option. $\endgroup$– Kurt G.Sep 1 at 12:34
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