I am trying to come up with a static hedge for a Digital Call with strike K that knocks out when price > barrier H. I know it will involve non-knockout digital calls with strike K and strike H but I am not sure in what proportion and what other digitals I will need to include in the hedge.
Appreciate any advice and suggestions. Thank you in advance
\begin{equation*} \begin{split} \mathbb{1}_{S_T > K, \max_{[0,T]} S_t < H} &\approx \frac{(S_T - (K-\varepsilon))^+ - (S_T - (K+\varepsilon))^+}{2 \varepsilon} \mathbb{1}_{\max_{[0,T]} S_t < H} \\ &= \frac{(S_T - (K-\varepsilon))^+\mathbb{1}_{\max_{[0,T]} S_t < H} - (S_T - (K+\varepsilon))^+\mathbb{1}_{\max_{[0,T]} S_t < H} }{2 \varepsilon} \end{split} \end{equation*} so you can start with the static hedge of standard knock out calls $(S_T - (K-\varepsilon))^+\mathbb{1}_{\max_{[0,T]} S_t < H}$ and $(S_T - (K+\varepsilon))^+\mathbb{1}_{\max_{[0,T]} S_t < H}$ if you're already familiar with that.
