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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

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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<U\}}\,. \end{align} 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.

I made the assumption that there is no complicated interaction between KI and KO. That is:

  • 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.

Closed formulas for double barrier KO options have been known for a long time. See for example the book by E.G.Haug, Complete Guide to Option Pricing Formulas.

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  • $\begingroup$ Thank you very much. That makes a lot of sense. Am I correct in that as per Haug, the double barrier options have a closed form solution but it is an infinite sum which must be approximated linearly? Apologies if trivial, just making sure I am not missing something. I guess that makes it a quasi closed form formula in that you still need an iterative process to estimate it iteratively! Best and thank you again! $\endgroup$
    – Count
    Jan 13, 2022 at 19:39
  • $\begingroup$ @Count, yes double barrier option pricing formulas come as infinite sums but they converge very quickly. Haug's version should converge slower when the barriers are very close. You will have no problems to check that. $\endgroup$
    – Kurt G.
    Jan 14, 2022 at 8:52

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