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I am currently looking at an exotic option that allows the holder, at some time $\tau$, to receive either a call or put — the choice of which is decided by the option writer — of which both have the same expiry $T>\tau$ and strike $K$.

Is there a name for this kind of option (I searched up chooser options but it doesn't match the definition)?

Also, why would anyone want to purchase such a option? I guess at the end of the day this option should be a linear combination of European options, but I am not sure how to prove this rigorously, any guidance is appreciated!

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    $\begingroup$ Indeed ramurapt.files.wordpress.com/2009/10/chooseroptions.doc Mark Rubinstein "Options for the Undecided," Risk Magazine, 4(4) discusses "chooser" options where the option holder, rather than the option writer, decides whether the option will become a put or a call. $\endgroup$
    – Dimitri Vulis
    Dec 24, 2020 at 3:53
  • $\begingroup$ @DimitriVulis hi, I’ve just read through the doc and it definitely makes a lot more sense when the option holder makes the choice, but under what circumstances would it make sense to purchase one where the option writer makes the decision? $\endgroup$
    – user107224
    Dec 24, 2020 at 19:06
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    $\begingroup$ Under the assumption that the writer will always choose the least valuable option to the holder, which is likely to be tail risk hedge. And perhaps it is cheaper for the holder than buying a tail risk hedge outright at a given point in time? Off the top of my head, otherwise I agree it is somewhat unusual. $\endgroup$
    – Attack68 ♦
    Dec 24, 2020 at 19:09

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