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".