Modified: December 04, 2023
basket of options
This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.Matt Levine points out a basic fact of derivatives theory: a basket of options is worth more than an option on a basket.
If five stocks go up and five stocks go down, so that we are net zero, the option on the basket is worthless. But call options on the stocks that went up would be valuable.
This is 'just' a special case of Jensen's inequality! The price of a stock is a random variable (think of the randomness as selecting a member of the basket, so the 'expectation' is the average value of the basket), and the value of an option is a function of the underlying stock value . In the simplest case (an at-the-money call option) this function is just a ReLu f(x) = max(x, 0), which is convex! So this is saying
which is indeed a special case of Jensen's inequality.