# What is the name of these digital basket options?

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?

• It is called a digital rainbow option. – Gordon Sep 16 '19 at 14:27
• @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. – davidhigh Sep 16 '19 at 14:36
• A rainbow option involves the weighted average of the sorted asset returns, that is, $\sum_{i=1}^n w_i X_{(i)}$, where $X_i = S_i -K_i$. – Gordon Sep 16 '19 at 14:47
• @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? – davidhigh Sep 16 '19 at 14:57
• @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. – will Sep 17 '19 at 20:21