Finance: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 [math]\displaystyle{ (i,j) }[/math] element of the matrix is the number of units of asset [math]\displaystyle{ i }[/math] which can be exchanged for 1 unit of asset [math]\displaystyle{ j }[/math].

Mathematical definition

A [math]\displaystyle{ d \times d }[/math] matrix [math]\displaystyle{ \Pi = \left[\pi_{ij}\right]_{1 \leq i,j \leq d} }[/math] is a bid-ask matrix, if

1. [math]\displaystyle{ \pi_{ij} \gt 0 }[/math] for [math]\displaystyle{ 1 \leq i,j \leq d }[/math]. Any trade has a positive exchange rate.
2. [math]\displaystyle{ \pi_{ii} = 1 }[/math] for [math]\displaystyle{ 1 \leq i \leq d }[/math]. Can always trade 1 unit with itself.
3. [math]\displaystyle{ \pi_{ij} \leq \pi_{ik}\pi_{kj} }[/math] for [math]\displaystyle{ 1 \leq i,j,k \leq d }[/math]. 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 [math]\displaystyle{ x }[/math] units of A can be exchanged for 1 unit of B, and [math]\displaystyle{ y }[/math] units of B can be exchanged for 1 unit of A. Then the bid–ask matrix [math]\displaystyle{ \Pi }[/math] is:

[math]\displaystyle{ \Pi = \begin{bmatrix} 1 & x \\ y & 1 \end{bmatrix} }[/math]

It is required that [math]\displaystyle{ xy\ge1 }[/math] by rule 3.

With 3 assets, let [math]\displaystyle{ a_{ij} }[/math] be the number of units of i traded for 1 unit of j. The bid–ask matrix is:

[math]\displaystyle{ \Pi = \begin{bmatrix} 1 & a_{12} & a_{13}\\ a_{21} & 1 & a_{23}\\ a_{31}& a_{32}& 1 \end{bmatrix} }[/math]

Rule 3 applies the following inequalities:

• [math]\displaystyle{ a_{12}a_{21}\ge1 }[/math]
• [math]\displaystyle{ a_{13}a_{31}\ge1 }[/math]
• [math]\displaystyle{ a_{23}a_{32}\ge1 }[/math]

• [math]\displaystyle{ a_{13}a_{32}\ge a_{12} }[/math]
• [math]\displaystyle{ a_{23}a_{31}\ge a_{21} }[/math]

• [math]\displaystyle{ a_{12}a_{23}\ge a_{13} }[/math]
• [math]\displaystyle{ a_{32}a_{21}\ge a_{31} }[/math]

• [math]\displaystyle{ a_{21}a_{13}\ge a_{23} }[/math]
• [math]\displaystyle{ a_{31}a_{12}\ge a_{32} }[/math]

For higher values of d, note that 3-way trading satisfies Rule 3 as

[math]\displaystyle{ x_{ik}x_{kl}x_{lj}\ge x_{il}x_{lj}\ge x_{ij} }[/math]

Relation to solvency cone

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

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 [math]\displaystyle{ (i,j) }[/math] is [math]\displaystyle{ \left\{\frac{1}{\pi_{ji}},\pi_{ij}\right\} }[/math].
• If [math]\displaystyle{ \pi_{ij} = \frac{1}{\pi_{ji}} }[/math] then that pair is frictionless.
• If a subset [math]\displaystyle{ \prod_s\pi_{ij} = \frac{1}{\prod_s\pi_{ji}} }[/math] then that subset is frictionless.

Arbitrage in bid-ask matrices

Arbitrage is where a profit is guaranteed. A method to determine if a BAM is arbitrage-free is as follows.

Consider n assets, with a BAM [math]\displaystyle{ \pi_n }[/math] and a portfolio [math]\displaystyle{ P_n }[/math]. Then

[math]\displaystyle{ P_n\pi_n = V_n }[/math]

where the i-th entry of [math]\displaystyle{ V_n }[/math] is the value of [math]\displaystyle{ P_n }[/math] in terms of asset i.

Then the tensor product defined by

[math]\displaystyle{ V_n \square V_n = \frac{v_i}{v_j} }[/math]

should resemble [math]\displaystyle{ \pi_n }[/math].