# Call options and portfolio of the same options worth less?

A portfolio of long positions in call options with the same maturity and strikes on different assets is worth more than a call option on a portfolio of the same assets with the same weight; i.e. $$\sum_{i=1}^{n}\lambda_i C(T,K,S^{(t)},t) \geq C(T,K,\hat{S},t),$$ where $\lambda_i\geq 0$ and $S^{(i)}$ for $i = 1,\ldots,n$ are assets and $\hat{S} = \sum_{i=1}^{n}\lambda_i S^{(i)}$ is a value of a portfolio which has $\lambda_i$ units of asset $S^{(i)}$ for each $i = 1,\ldots,n$.

This seems like an application of the triangle inequality, but I am not sure how to formally write it. Any suggestions is greatly appreciated.

• For the multiple calls, you get to take the sum of the per-component increased values and ignore any decreased values since calls can't be worth less than 0. For the portfolio, you have to take the positive and negative sums and look at the delta for the entire portfolio. – barrycarter Jan 16 '16 at 4:57
• This theorem is an illustration of the effect of diversification in a portfolio. It is easiest to see if all stocks have the same volatilit $\sigma$, then the portfolio volatility will be lower ($\sigma_p <= \sigma$), so the call on the pfolio will be less valuable. – Alex C Jan 16 '16 at 16:22
• Could you provide a solution or example so I can understand it – Wolfy Jan 18 '16 at 1:47
• You might get some insight by doing the derivation for the case where the correlation = 1, then see what happens when you relax that. – experquisite Jan 18 '16 at 21:25
• @experquisite I do not understand what you mean – Wolfy Jan 19 '16 at 2:16

For the case where $\sum_{i=1}^n \lambda_i =1$, you need only note that the payoff \begin{align*} (x-K)^+ \end{align*} is a convex function in $x$. That is, \begin{align*} \Big(\sum_{i=1}^n \lambda_i S_i -K\Big)^+ \le \sum_{i=1}^n\lambda_i(S_i-K)^+. \end{align*} Then \begin{align*} e^{-rT}E\left(\Big(\sum_{i=1}^n \lambda_i S_i -K\Big)^+\right) \le \sum_{i=1}^n\lambda_ie^{-rT}E\left((S_i-K)^+\right). \end{align*}