# Is there a closed form formula for the value of a European Put KO/KI?

Was able to find closed form formula for single barrier options KO OR KI. However I haven't found that for a double barrier option.

I am looking for a put down & in KI, up and out KO, where:

H(KI) < K < H(KO) && H(KI) < S < H(KO) where H(KI) is KI barrier, H(KO) is KO barrier, S is stock price, K is strike

Thank you very much in advance

Writing $$M_T=\max_{0\le t\le T}S_t\,,\,\, m_T=\min_{0\le t\le T}S_t$$ the option payoff is \begin{align} (K-S_T)^+\underbrace{1_{\{m_T\le L\}}}_{\text{KI}}\underbrace{1_{\{M_T< U\}}}_{\text{KO}}=(K-S_T)^+(1-1_{\{m_T>L\}})1_{\{M_T In other words, the KI-KO-option is a portfolio of a long positon in a single barrier KO option and a short position in a double barrier KO option.
• once the option is knocked in the lower barrier $$L$$ has done its job and becomes irrelevant;
• likewise, once the option is knocked out by $$U$$, it cannot subsequently by knocked in by $$L$$ anymore.