Expected Call Payout Minimization Is a Linear Weighting Problem
Summary
The question asks whether Markowitz minimum-variance optimization can choose weights for a basket of call options on different stocks, given fixed strikes, a covariance matrix, and weights summing to one. The answer distinguishes minimizing expected payout from minimizing variance: each option has an expected payout given by its Black–Scholes value, and the basket’s expected payout is the weighted sum of those individual expectations.
With nonnegative weights and no other constraints, this objective is minimized by assigning all weight to the option with the lowest expected payout. That generally differs from a minimum-variance portfolio, which optimizes dispersion rather than the mean payout. The conclusion depends on the stated simplified setup; it does not account for constraints such as required diversification or minimum allocations, and the response does not discuss transaction costs or other real-world frictions.
Key ideas
- The expected payout of a weighted option basket is the weighted sum of the individual expected payouts.
- Under nonnegative weights that sum to one, the lowest-payout option alone minimizes expected payout.
- Minimum-variance optimization targets a different objective from minimizing expected payout.
- Additional portfolio constraints could change the optimal allocation.
Tags
Full text
# Minimum variance in a portfolio of call options
# 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.
## Answer by Brian B (score 3)
https://quant.stackexchange.com/a/33304
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.