Bid–ask matrix

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title: "Bid–ask matrix" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["mathematical-finance"] topic_path: "economics" source: "https://en.wikipedia.org/wiki/Bid–ask_matrix" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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.

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.
  • If a subset \prod_s\pi_{ij} = \frac{1}{\prod_s\pi_{ji}} then that subset is frictionless.

Arbitrage in bid-ask matrices

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.

Iterative computation

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.

References

References

  1. Schachermayer, Walter. (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".

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