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.

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:

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

It is required that $xy\geq 1$ by rule 3.

With 3 assets, let $a_{ij}$ be the number of units of i traded for 1 unit of j. The bid–ask matrix is:

$\Pi ={\begin{bmatrix}1&a_{12}&a_{13}\\a_{21}&1&a_{23}\\a_{31}&a_{32}&1\end{bmatrix}}$

Rule 3 applies the following inequalities:

• $a_{12}a_{21}\geq 1$
• $a_{13}a_{31}\geq 1$
• $a_{23}a_{32}\geq 1$

• $a_{13}a_{32}\geq a_{12}$
• $a_{23}a_{31}\geq a_{21}$

• $a_{12}a_{23}\geq a_{13}$
• $a_{32}a_{21}\geq a_{31}$

• $a_{21}a_{13}\geq a_{23}$
• $a_{31}a_{12}\geq a_{32}$

For higher values of d, note that 3-way trading satisfies Rule 3 as

$x_{ik}x_{kl}x_{lj}\geq x_{il}x_{lj}\geq x_{ij}$

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 $\pi ^{ij}e^{i}-e^{j},1\leq i,j\leq d$.

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.
• If a subset $\prod _{s}\pi _{ij}={\frac {1}{\prod _{s}\pi _{ji}}}$ then that subset is frictionless.

Arbitrage in bid-ask matrices

Arbitrage is where a profit is guaranteed. A method to determine if a BAM is arbitrage-free is as follows.

Consider n assets, with a BAM $\pi _{n}$ and a portfolio $P_{n}$. Then

$P_{n}\pi _{n}=V_{n}$

where the i-th entry of $V_{n}$ is the value of $P_{n}$ in terms of asset i.

Then the tensor product defined by

$V_{n}\square V_{n}={\frac {v_{i}}{v_{j}}}$

should resemble $\pi _{n}$.