Puneet Varma (Editor)

Bid–ask matrix

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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 .

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Mathematical definition

A d × d matrix Π = [ π i j ] 1 ≤ i , j ≤ d is a bid-ask matrix, if

  1. π i j > 0 for 1 ≤ i , j ≤ d . Any trade has a positive exchange rate.
  2. π i i = 1 for 1 ≤ i ≤ d . Can always trade 1 unit with itself.
  3. π i j ≤ π i k π k j for 1 ≤ i , j , k ≤ 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 Π is:

Π = [ 1 x y 1 ]

Relation to solvency cone

If given a bid–ask matrix Π for d assets such that Π = ( π i j ) 1 ≤ i , j ≤ d and m ≤ 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 ( Π ) ⊂ R d is the convex cone spanned by the unit vectors e i , 1 ≤ i ≤ m and the vectors π i j e i − e j , 1 ≤ i , j ≤ d .

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

References

Bid–ask matrix Wikipedia
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