Optimisation problem with bid-ask spread

I want to optimise a static portfolio with a holding period of 90 days given 10 tradable assets. The assets are quoted in bid and ask prices. I want to minimise the risk measured by standard deviation given a certain level of expected return.

My Idea: I treat the bid-ask- prices as two separate assets. Therefore I will have 20 assets and using $$\begin{pmatrix} x_1^+\\ x_1^-\\ x_2^+ \\ x_2^- \\ .\\ .\\ \end{pmatrix}=\begin{pmatrix} x^+\\ x^-\\ \end{pmatrix}$$ I can denote the portfolio with $x^+$ for long position and $x^-$ for short with the constraint $x^+\geq0$ and $x^-\geq0$. The total position is thus equivalent to $x^+-x^-$. Based on the 10 assets, treating short and long separately, I can calculate the mean and the covariance of the return in 90 days based on the data. This will gives me a 20x1 mean vector $\mu=\begin{pmatrix} \mu^+\\ \mu^-\\ \end{pmatrix}$ and a 20x20 covariance matrix $\Sigma$. Then the optimisation problem can be written as: $$\text{minimalise} \begin{pmatrix} x^+\\ x^-\\ \end{pmatrix}^T\Sigma \begin{pmatrix} x^+\\ x^-\\ \end{pmatrix} \\\text{subject to: } x^+S_0^+-x^-S_0^-\leq w\\\begin{pmatrix} x^+\\ x^-\\ \end{pmatrix}^T\ \begin{pmatrix} \mu^+\\ \mu^-\\ \end{pmatrix}=r\\x^+\geq0\\x^-\geq0$$

Here, $w$ is the initial budget for constructing a portfolio and $S_0^+$ bid price and $S_0^-$ the ask price of the asset at the beginning of the holding period.

Questions:

1. Does this model make sense? I have tried to fix the nonlinearity of transaction cost by restricting $x^+\geq0$ $x^-\geq0$ and treat them separately. Is the reformulated problem indeed a linear convex problem?
2. In the covariance matrix, I am also considering the covariance of the return in long position of an asset with itself's short position. Could this be a problem?
• Short answer so a comment: I would say the answers are 1. yes and 2. yes. They will be perfectly correlated which makes the problem unstable. Maybe it would be a good idea to add the constraint $x^+ - x^- \geq 0$. Further more, you will want to have some current holdings for the transaction costs to make sense and then adjust the constraints for the difference. – vanguard2k Jun 8 '17 at 11:50
• @vanguard2k Thank you. I didn't quite get why adding the constraint $x^+-x^-\geq0$ This would impose that I will buy more than sell. Could you emphasis more on the motivation of this constraint and how to fix the unstability – quallenjäger Jun 8 '17 at 11:59
• It was really a first intuition. I just assumed that you dont want to sell assets short. The thing that happens with a variance covariance matrix that is ill-conditioned (has eigenvalues close to $0$) is, that the optimization zooms in on the estimation errors. You can see this in results like $x_1^+ \approx 10^{12}$, $x_2^- \approx 10^{12}$. Now, from the work of Jagannathan and Ma, we know that constraints can be equivalent to stabilizing the matrix. I just figured that my constraint would avoid the scenario above for every single asset. – vanguard2k Jun 8 '17 at 12:30
• whether its helpful or not depends on how you estimate $\Sigma$. of course, if you want to allow short selling, then you cant use it. – vanguard2k Jun 8 '17 at 12:31
• @vanguard2k Thank you! Very intuitive answer. Unfortunately I need to allow short selling. Is there another way to stabilise the matrix? Or is there any resources I can read about this kind of methods? – quallenjäger Jun 8 '17 at 12:34