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_\right is a ''bid-ask matrix'', if
# $\pi_ > 0$ for $1 \leq i,j \leq d$. Any trade has a positive exchange rate.
# $\pi_ = 1$ for $1 \leq i \leq d$. Can always trade 1 unit with itself.
# $\pi_ \leq \pi_\pi_$ 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:

: $\Pi = \begin 1 & x \\ y & 1 \end$

Relation to solvency cone

If given a bid–ask matrix $\Pi$ for $d$ assets such that $\Pi = \left(\pi^\right)_$ 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 $K(\Pi) \subset \mathbb^d$ is the convex cone spanned by the unit vectors $e^i, 1 \leq i \leq m$ and the vectors $\pi^e^i-e^j, 1 \leq i,j \leq d$.

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\$.
* If $\pi_ = \frac$ then that pair is frictionless.

References

{{DEFAULTSORT:Bid-ask matrix
Mathematical finance