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

1. ${\pi }_{ij}>0$ for $1\le i,j\le d$. Any trade has a positive exchange rate.
2. ${\pi }_{ii}=1$ for $1\le i\le d$. Can always trade 1 unit with itself.
3. ${\pi }_{ij}\le {\pi }_{ik}{\pi }_{kj}$ for $1\le i,j,k\le d$. 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 $\mathrm{\Pi }$ is:

$\mathrm{\Pi }=\left[\begin{array}{cc}1& x\\ y& 1\end{array}\right]$

It is required that $xy\ge 1$ by rule Template:Mvar.

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

$\mathrm{\Pi }=\left[\begin{array}{ccc}1& a_{12}& a_{13}\\ a_{21}& 1& a_{23}\\ a_{31}& a_{32}& 1\end{array}\right]$

Rule Template:Mvar applies the following inequalities:

• $a_{12}{a}_{21}\ge 1$
• $a_{13}{a}_{31}\ge 1$
• $a_{23}{a}_{32}\ge 1$

• $a_{13}{a}_{32}\ge a_{12}$
• $a_{23}{a}_{31}\ge a_{21}$

• $a_{12}{a}_{23}\ge a_{13}$
• $a_{32}{a}_{21}\ge a_{31}$

• $a_{21}{a}_{13}\ge a_{23}$
• $a_{31}{a}_{12}\ge a_{32}$

For higher values of Template:Mvar, note that 3-way trading satisfies Rule Template:Mvar as

$x_{ik}{x}_{kl}{x}_{lj}\ge x_{il}{x}_{lj}\ge x_{ij}$

Relation to solvency cone

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

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 $\{\frac{1}{{\pi }_{ji}},{\pi }_{ij}\}$.
• 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.

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 ${\pi }_n$ 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

$V_n◻V_n=\frac{v_i}{v_j}$

should resemble ${\pi }_n$.