Consider a basket of correlated assets $(S_1(t),\ldots, S_N(t))$, as well as a vector of strike prices $(K_1,\ldots,K_N)$, and let's look at the following European payoff types:

  1. An option that pays 1€ if $S_i(T)\geq K_i$ for all $i\in \{1,\ldots,N\}$.
  2. Pick a number $n \in \{1,\ldots,N\}$. The option pays 1€ if (at least/exactly) $n$ assets satisfy $S_i(T)\geq K_i$. (This is basically case 1. with $n=N$).

Does one or both of these these options exist and have a name? Moreover, are they possibly traded?

  • $\begingroup$ It is called a digital rainbow option. $\endgroup$ – Gordon Sep 16 '19 at 14:27
  • $\begingroup$ @Gordon: thank you for your answer. A rainbow option often involves the best asset, e.g. $(\max(S_1,\ldots,S_N) - K)^+$. The payoff of this one here is rather $1_{S_1>K_1}\cdots 1_{S_N>K_N}$, so it's not directly a rainbow option imo. $\endgroup$ – davidhigh Sep 16 '19 at 14:36
  • $\begingroup$ A rainbow option involves the weighted average of the sorted asset returns, that is, $\sum_{i=1}^n w_i X_{(i)}$, where $X_i = S_i -K_i$. $\endgroup$ – Gordon Sep 16 '19 at 14:47
  • $\begingroup$ @Gordon: ok, I see this general definition for a rainbow option incorporates the options above. Still, it seems quite exotic for a rather common type of option (for example in sports games, where one simply stacks the individual bets -- if both team 1 and team 2 win their games, then ...). Do you have a reference for that? $\endgroup$ – davidhigh Sep 16 '19 at 14:57
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    $\begingroup$ @davidhigh what you're describing is an accumulator, where it is just a set of many trades/bets where you must win at least some number of them. in this case, you would be talking about an accumulator on a load of digital options. It's not called this in the finance world (that i'm aware of) though, and in fact an accumulator refers to another, unrelated, structure. $\endgroup$ – will Sep 17 '19 at 20:21

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