Bid–ask matrix

The bid–ask matrix is a matrix with elements corresponding with exchange rates between the assets. These rates are in physical units (e.g. number of stocks) and not with respect to any numeraire. The $(i,j)$ element of the matrix is the number of units of asset $i$ which can be exchanged for 1 unit of asset $j$.

Mathematical Definition

A $d \times d$ matrix $\Pi = \left[\pi_{ij}\right]_{1 \leq i,j \leq d}$ is a bid-ask matrix, if

1. $\pi_{ij} > 0$ for $1 \leq i,j \leq d$. Any trade has a positive exchange rate.
2. $\pi_{ii} = 1$ for $1 \leq i \leq d$. Can always trade 1 unit with itself.
3. $\pi_{ij} \leq \pi_{ik}\pi_{kj}$ for $1 \leq i,j,k \leq d$. A direct exchange is always at most as expensive as a chain of exchanges.^[1]

Example

Assume a market with 2 assets (A and B), such that $x$ units of A can be exchanged for 1 unit of B, and $y$ units of B can be exchanged for 1 unit of A. Then the bid–ask matrix $\Pi$ is:

$\Pi = \begin{bmatrix} 1 & x \\ y & 1 \end{bmatrix}$

Relation to solvency cone

If given a bid–ask matrix $\Pi$ for $d$ assets such that $\Pi = \left(\pi^{ij}\right)_{1 \leq i,j \leq d}$ and $m \leq d$ is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally $m = d$). Then the solvency cone $K(\Pi) \subset \mathbb{R}^d$ is the convex cone spanned by the unit vectors $e^i, 1 \leq i \leq m$ and the vectors $\pi^{ij}e^i-e^j, 1 \leq i,j \leq d$.^[1]

Similarly given a (constant) solvency cone it is possible to extract the bid–ask matrix from the bounding vectors.

Notes

• The bid–ask spread for pair $(i,j)$ is $\left\{\frac{1}{\pi_{ji}},\pi_{ij}\right\}$.
• If $\pi_{ij} = \frac{1}{\pi_{ji}}$ then that pair is frictionless.

References

1. Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".