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.

element of the matrix is the number of units of asset

which can be exchanged for 1 unit of asset

{\displaystyle \Pi =\left[\pi _{ij}\right]_{1\leq i,j\leq d}}

is a bid-ask matrix, if Assume a market with 2 assets (A and B), such that

units of A can be exchanged for 1 unit of B, and

units of B can be exchanged for 1 unit of A.

be the number of units of i traded for 1 unit of j.

The bid–ask matrix is: Rule 3 applies the following inequalities: For higher values of d, note that 3-way trading satisfies Rule 3 as If given a bid–ask matrix

is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally

Then the solvency cone

is the convex cone spanned by the unit vectors

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

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.

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

Consider n assets, with a BAM

Then where the i-th entry of

in terms of asset i.

Then the tensor product defined by should resemble