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

$\pi _{{ij}}>0$ for $1\leq i,j\leq d$ . Any trade has a positive exchange rate.
$\pi _{{ii}}=1$ for $1\leq i\leq d$ . Can always trade 1 unit with itself.
$\pi _{{ij}}\leq \pi _{{ik}}\pi _{{kj}}$ 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 2 Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.