Call options and portfolio of the same options worth less?

Question

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.

Answer 1

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*}