Modeling Bid–Ask Spreads in Mean–Variance Portfolio Optimization
Summary
The document proposes a mean–variance model for a static portfolio of assets quoted at bid and ask prices. It represents each asset’s long and short exposure as separate nonnegative variables, calculates expected returns and covariances for those positions, and minimizes portfolio variance subject to a target return and a budget constraint based on the quoted prices. The holding period is specified as 90 days, and the example considers ten tradable assets.
The author asks whether this formulation handles transaction costs and remains convex, and whether including covariance between an asset’s long and short returns causes a problem. The document presents the proposed equations but gives no answers, validation, or empirical evidence. Its assumptions therefore need scrutiny before use, including how long and short returns are defined, how cash flows and initial spending are represented, and whether simultaneous long and short positions in the same asset are prevented. The piece is useful as a modeling question, rather than a settled optimization method.
Key ideas
- The proposal represents long and short exposures as separate nonnegative portfolio variables.
- It seeks to minimize return variance while meeting a target expected return and budget constraint.
- The author asks whether bid–ask prices and the proposed constraints correctly model trading costs.
- Covariance between long and short returns of the same asset is identified as a modeling concern.
- The document does not resolve these questions or validate the formulation.
Tags
Full text
# Optimisation problem with bid-ask spread
# 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:
- 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?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.