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.

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}$$

It is required that $$xy\ge1$$ by rule $3$.

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


 * $$\Pi = \begin{bmatrix}

1 & a_{12} & a_{13}\\ a_{21} & 1 & a_{23}\\ a_{31}& a_{32}& 1 \end{bmatrix}$$

Rule $3$ applies the following inequalities:


 * $$a_{12}a_{21}\ge1$$
 * $$a_{13}a_{31}\ge1$$
 * $$a_{23}a_{32}\ge1$$


 * $$a_{13}a_{32}\ge a_{12}$$
 * $$a_{23}a_{31}\ge a_{21}$$


 * $$a_{12}a_{23}\ge a_{13}$$
 * $$a_{32}a_{21}\ge a_{31}$$


 * $$a_{21}a_{13}\ge a_{23}$$
 * $$a_{31}a_{12}\ge a_{32}$$

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


 * $$x_{ik}x_{kl}x_{lj}\ge x_{il}x_{lj}\ge x_{ij}$$

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 $$\pi^{ij}e^i-e^j, 1 \leq i,j \leq d$$.

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

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 $$\pi_n$$ and a portfolio $$P_n$$. Then


 * $$P_n\pi_n = V_n$$

where the i-th entry of $$V_n$$ is the value of $$P_n$$ in terms of asset i.

Then the tensor product defined by


 * $$V_n \square V_n = \frac{v_i}{v_j}$$

should resemble $$\pi_n$$.