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:
- 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?
- 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?