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
element of the matrix is the number of units of asset
which can be exchanged for 1 unit of asset
.
Mathematical definition
A
matrix
is a bid-ask matrix, if
for
. Any trade has a positive exchange rate.
for
. Can always trade 1 unit with itself.
for
. 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
units of A can be exchanged for 1 unit of B, and
units of B can be exchanged for 1 unit of A. Then the bid–ask matrix
is:
![{\displaystyle \Pi ={\begin{bmatrix}1&x\\y&1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1eacfe23d9b3a6861f3756adcff5c4f743aeac0)
It is required that
by rule 3.
With 3 assets, let
be the number of units of i traded for 1 unit of j. The bid–ask matrix is:
![{\displaystyle \Pi ={\begin{bmatrix}1&a_{12}&a_{13}\\a_{21}&1&a_{23}\\a_{31}&a_{32}&1\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4ec22f0fac26bf66a89240af6603b50a318ba3)
Rule 3 applies the following inequalities:
![{\displaystyle a_{12}a_{21}\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d648d6c3371e51f332531d48ae15b97890c8d19)
![{\displaystyle a_{13}a_{31}\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916867627844a5b1881cec85bab9a29bbf920f57)
![{\displaystyle a_{23}a_{32}\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a02f336f30b7e43f7e6c862f9113280b4ed9841)
![{\displaystyle a_{13}a_{32}\geq a_{12}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/204dc9d914a39d9733d1e785f11a4f5bceb03bd9)
![{\displaystyle a_{23}a_{31}\geq a_{21}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a6d3ea0fe1b589541c7a0bab5ddee25b30b18b)
![{\displaystyle a_{12}a_{23}\geq a_{13}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125c367ca4b9cee2c0e298752869894cf64f72c0)
![{\displaystyle a_{32}a_{21}\geq a_{31}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e1e3824a28420d9af8f158d3a5abbf77c73307)
![{\displaystyle a_{21}a_{13}\geq a_{23}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d194445902f9a88ad42353d9043a2d53531719a5)
![{\displaystyle a_{31}a_{12}\geq a_{32}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53011bbdd37af1fb07a22afd6736c41aab576547)
For higher values of d, note that 3-way trading satisfies Rule 3 as
![{\displaystyle x_{ik}x_{kl}x_{lj}\geq x_{il}x_{lj}\geq x_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0805171ee519277c9edef5fa3dddf2176a8002be)
Relation to solvency cone
If given a bid–ask matrix
for
assets such that
and
is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally
). Then the solvency cone
is the convex cone spanned by the unit vectors
and the vectors
.[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
is
.
- If
then that pair is frictionless.
- If a subset
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
and a portfolio
. Then
![{\displaystyle P_{n}\pi _{n}=V_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/301fd6e2c33c3448485388e320acd34a97efba35)
where the i-th entry of
is the value of
in terms of asset i.
Then the tensor product defined by
![{\displaystyle V_{n}\square V_{n}={\frac {v_{i}}{v_{j}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608a3b416f90012a90758a7b4094090af81858f9)
should resemble
.
References
- ^ a b Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".