Bid–ask matrix

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

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

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

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

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

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

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

Rule 3 applies the following inequalities:

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

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

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

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

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

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

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

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

• The bid–ask spread for pair ${\displaystyle (i,j)}$ is ${\displaystyle \left\{{\frac {1}{\pi _{ji}}},\pi _{ij}\right\}}$.
• If ${\displaystyle \pi _{ij}={\frac {1}{\pi _{ji}}}}$ then that pair is frictionless.
• If a subset ${\displaystyle \prod _{s}\pi _{ij}={\frac {1}{\prod _{s}\pi _{ji}}}}$ then that subset is frictionless.

Arbitrage is where a profit is guaranteed.

If Rule 3 from above is true, then a bid-ask matrix (BAM) is arbitrage-free, otherwise arbitrage is present via buying from a middle vendor and then selling back to source.

Iterative computation

A method to determine if a BAM is arbitrage-free is as follows.

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

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

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

Then the tensor product defined by

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

should resemble ${\displaystyle \pi _{n}}$.