Replicating a Double-Barrier Put Knock-In Knock-Out with Single and Double Knock-Outs

Summary

The document explains how to value a European put that knocks in at a lower barrier and knocks out at an upper barrier. It expresses the payoff using the terminal put payoff and indicators for whether the underlying has crossed the lower barrier and remained below the upper barrier. This decomposition represents the position as a long single-barrier knock-out put combined with a short double-barrier knock-out put.

The construction relies on the specified interaction convention: after lower-barrier activation, that barrier no longer matters, and an upper-barrier knock-out prevents later activation. The answer notes that closed-form formulas for double-barrier knock-out options are available and points to established option-pricing references. It does not derive those formulas or discuss model assumptions such as the underlying price process, so applying the decomposition requires matching the contract’s barrier rules and valuation model.

Key ideas

• The payoff can be written as a long single-barrier knock-out put minus a double-barrier knock-out put.
• The lower barrier activates the option, while the upper barrier can terminate it.
• The decomposition assumes activation and knock-out have no subsequent interaction.
• Valuation relies on a suitable formula for the double-barrier knock-out component.

Tags

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


# 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

## Answer by Kurt G. (score 2)

https://quant.stackexchange.com/a/69453

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.