I'm trying to find the optimal portfolio of options and stock which minimizes the standard deviation of the portfolio returns but also taking into consideration the bid and ask prices of the assets. I am given a snapshot of the stock and related options (all with same maturity but different strikes) with their bid and ask prices.

I am making the assumption that the stock price follows a geometric brownian motion and that infinite amounts of each asset can be sold/bought at their bid/ask price.

My approach, if there was only a mid price would be to simulate returns for each asset, calculate the average returns and the covariance matrix then pass this into a quadprog function in MATLAB to minimize the variance of the portfolio with no upper or lower bounds on the quantity of each asset (i.e. short selling is allowed).

However, with the bid/ask spread, the return on selling an asset is slightly different to the return of buying that same asset and so passing the returns and covariances the same way as above would not be accurate.

Can anyone suggest a plausible method to account for this bid/ask spread? I thought about treating the buy and the sell of an asset as two separate assets, calculating their respective returns and altering the constraints to accommodate the buying and selling direction but it gets very messy and my results don't look correct. Is there an easier way to do this?

Thanks, Michael.

Depending on how big is bid-ask spread, it may not matter at all. To start with I'd suggest that for each security you randomly pick either it's bid or ask price, and see whether the optimal weights differ much depending on your choice. If not, you can simply disregard such asymmetry. If not, I am not sure whether MPT will work nicely with your setting since it is a very linear theory, whereas using ask (for buying) and bid (for selling) introduces non-linearity.

I can suggest a simple way that is not perfect but is reasonable and not too difficult.

### Transaction Costs Will Change the Optimal Portfolio

What you want to do is account for the transactions costs of trading. Note that Engle and Ferstenberg (2007) have shown that including transactions costs shifts the efficient frontier down and to the right (because there is an added mean and variance of transactions costs).

This will lead to a different optimal portfolio than if you ignore transactions costs. A common complaint about Markowitz portfolio optimization (without transactions costs) is that the portfolios underperform "out-of-sample." Including transactions costs (mean and variance) may reduce the underperformance problem.

For "small" trade amounts, you can approximate the mean and variance of trading as being affected only by the percentage spread $$S_i$$ for asset $$i$$ $$S_i = \log(A_i/B_i)$$ where $$A_i$$ is the ask price and $$B_i$$ is the bid price.

Also, you must account for the cost of entry and exit -- so we multiply the single-trade mean $$E(c_i)$$ and variance $$\textrm{var}(c_i)$$ of cost by 2.

If you are patient and wait for a possibly better price before giving up and crossing the spread (buying at the offer or selling at the bid), this would be:

• mean: half the mean bid-ask spread per trade; total = $$2\times\bar{S}_i/2=\bar{S}_i$$, and
• variance: cf Rademacher random variable; total = $$2\times\bar{S}_i^2/4=\bar{S}_i^2/2$$.

If you are impatient, your execution price is much more certain (and worse):

• mean: the full spread per trade; total = $$2\times\bar{S}_i$$, and
• variance: 0.

You then subtract each of these mean costs from your positive expected returns and add it to the negative expected returns and add the cost variances to your return variances. Or, equivalently: $$E(r_{i,new}) = E(r_i) - 2\textrm{sign}(r_i)E(c_i), \\ \textrm{var}(r_{i,new}) = \textrm{var}(r_i) + 2\textrm{var}(c_i).$$ Note that you should not let the expectations cross through 0. If your expected return is 5% and the expected entry+exit cost is 6%, you are better off treating this like it has a 0% expected return rather than a -1% expected return (which might lead to shorting the stock).

Should you propagate the cost variances through to covariances, i.e. create covariances from correlations and (new) variances? That depends on whether you think execution costs are correlated or not. If you want execution costs to be correlated, propagate the uncertainty; if not, only alter your variances and not the covariances.

### Approximation for Larger Trades: Using a Price Impact Model

You mention wanting to use bid-ask spreads; however, I will also hint at a better approximation. Basically, we can assume your transactions costs have a few parts and use a price impact model to get a better sense of costs.

While there are lots of price impact models, you are not doing trade scheduling at this step -- so you can use a model that might be unsuitable for trade scheduling but an OK approximation for portfolio optimization.

For trading a quantity $$Q_i$$ of asset $$i$$, I would use something like the Torre and Ferrari (1997) model: $$E(c_i) = \alpha_i + \beta_{iS} \bar{S}_i + \beta_{iQ} \sigma_i\sqrt{\frac{|Q_i|}{\bar{V}}}$$ where

• $$\alpha_i$$ captures fixed costs (like a commission fee),
• $$\beta_{iS}$$ gets at how much of the log-spread you typically pay,
• $$\beta_{iQ}$$ estimates the permanent impact (and cumulated decaying impact as per Obizhaeva and Wang (2013)) paid, and
• $$\bar{V}$$ is the average volume per trading period.

The variance calculation is tricky: you want to incorporate the uncertainty of your model coefficients with the uncertainty of the model terms. If I had to guess at this, I would expect something like $$\textrm{var}(c_i) = \sigma_\pi^2 + \hat\beta_{iS}^2 \frac{\bar{S}^2}{4} + \hat\beta_{iQ}^2 \left[ \sigma_i^2\frac{n-1}{2}\left(\frac{\Gamma(n/2)}{\Gamma(\frac{n-1}{2})}\right)^2 \frac{Q_i^2}{\bar{V}^2} + \sigma_i^2 Q_i^2\frac{\textrm{var}(\bar{V})}{\bar{V}^4}\right]$$ where $$\sigma_\pi$$ is the standard deviation from a fitted model prediction interval and $$n$$ is the number of observations used to estimate $$\sigma_i$$. The last term comes from the variance of $$\sigma_i$$, a one-term delta method approximation for $$\textrm{var}(1/\bar{V}))$$, and the variance of a product of random variables.

With these (hopefully better) estimates of $$E(c_i)$$ and $$\textrm{var}(c_i)$$ you can then double them (cost of entry and cost of exit) and add them in as done for the simpler small-trade case.