# Minimum variance in a portfolio of call options

My apologies if this question might be better suited elsewhere, however it regards probability and mathematical finance, so I thought I would post it here.

The question is:

Assume a universe where Black-Scholes is valid and Alice wants to sell a basket of $X$ call options to Bob on $Y$ different stocks with weights given by the vector $W$, subject to $X>Y$ and that the sum of $W=1$. She is given a vector of strike prices $K$ which she is unable to change, additionally she is given a co-variance matrix $\Sigma$.

If she wants to minimize the expected amount she has to pay at expiration to Bob by only changing the weights in the vector $W$ then how would you go about calculating this?

My own take is that I would just use the Markowitz minimum variance portfolio, but I am actually unsure whether this would yield a valid result.

• Maybe I'm missing something but your question does not really make sense to me. Could you maybe provide a reference? Indeed if the problem is "she wants to minimize her expected payout by only changing the weights in the vector W", the trivial solution is to set all weights to $-\infty$ (since the expected payout is a weighted sum of call prices (hence positive value instruments)). – Quantuple Mar 28 '17 at 14:29
• Hmm sorry for the confusion. I'll edit my answer above, let me know if it makes more sense now. – no nein Mar 28 '17 at 14:36

You probably meant to specify that all $w_i>0$ and that

$$\sum w_i=1$$

otherwise it is tough to make sense of the question.

The expected payout of any one of these options is given by the Black-Scholes formula. The expectation of a sum is the sum of expectations, so we have that this expected payout is

$$\sum w_i BS(k_i, x_i, \Sigma)$$

where I am taking $x_i$ to be the $i^{th}$ stock chosen from $y_i$ and $k_i$ to be the strike.

It is easy to see that this sum is minimized when we take the cheapest option at index $i_{\mathrm{min}}$, set

$$w_{i_{\mathrm{min}}}=1$$

and all other $w_i=0$.

The Markowitz minimum variance portfolio will be quite different.

• I think you can do better by considering correlations. For example if you have another stock $j$ with $\rho_{i_{\textrm{min}}j}=-1$, you could actually do better by selling both these options. – msitt Mar 28 '17 at 15:19
• Thanks a bunch! But yeah that was my consideration as well, that the correlation between the assets should factor into the weights. Anyway you are entirely right regarding the specifications I missed. – no nein Mar 28 '17 at 15:58
• Selling some of both will reduce variance but increase her expected payout. Generally speaking in these portfolio scenarios, max expected P&L specifications lead to trivial weights. – Brian B Mar 28 '17 at 16:14
• Why? If the strike price vector K is constructed such that all of them have the same expected payout. Then a correlation of -1, and a weight of 0.5 in each stock would lead to an expected payout of 0. – no nein Mar 28 '17 at 16:15
• Sorry, they will not have a payout of 0. But the expected payout would be lower unless I am misunderstanding something? Or would the varians of the expected payout merely be lower? In which case my actual question then is, how we we minimize the variance of her payout. – no nein Mar 28 '17 at 16:22