Solvency cone
Encyclopedia
The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market
. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone
of portfolios that can be exchanged to portfolios of non-negative components (including paying of any transaction costs).
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 
.
is any closed convex cone such that 
and 
.
is a model of a financial market. This is sometimes called a market process.
The negative of a solvency cone is the set of portfolios that can be obtained starting from the zero portfolio. This is intimately related to self-financing portfolio
s.
The dual cone of the solvency cone (
) are the set of prices which would define a friction-less pricing system for the assets that is consistent with the market. This is also called a consistent pricing system.
, we can obviously make (1A,-1M) and (-1A,1M) into non-negative portfolios, therefore
. Note that (1,1) is the "price vector."
.
The dual cone of prices is thus easiest to see in terms of prices of A in terms of M (and similarly done for price of M in terms of A):
:
Market
A market is one of many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services in exchange for money from buyers...
. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone
Convex cone
In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...
of portfolios that can be exchanged to portfolios of non-negative components (including paying of any transaction costs).
Mathematically
If given a bid-ask matrixBid-ask matrix
The bid–ask matrix is a matrix with elements corresponding with exchange rates between the assets. These rates are in physical units and not with respect to any numeraire...
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 .Definition
A solvency cone is any closed convex cone such that and .Uses
A process of (random) solvency cones is a model of a financial market. This is sometimes called a market process.The negative of a solvency cone is the set of portfolios that can be obtained starting from the zero portfolio. This is intimately related to self-financing portfolio
Self-financing portfolio
Self-financing portfolio, an important concept in financial mathematics.A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one.- Mathematical definition :...
s.
The dual cone of the solvency cone (
) are the set of prices which would define a friction-less pricing system for the assets that is consistent with the market. This is also called a consistent pricing system.Examples
Assume there are 2 assets, A and M with 1 to 1 exchange possible.Frictionless Market
In a frictionless marketFrictionless market
A Frictionless market is a financial market without transaction costs. Friction is a type of market incompleteness. Every complete market is frictionless, but the converse does not hold. In a frictionless market the solvency cone is the halfspace normal to the unique price vector. The...
, we can obviously make (1A,-1M) and (-1A,1M) into non-negative portfolios, therefore
. Note that (1,1) is the "price vector."With Transaction Costs
Assume further that there is 50% transaction costs for each deal. This means that (1A,-1M) and (-1A,1M) cannot be exchanged into non-negative portfolios. But, (2A,-1M) and (-1A,2M) can be traded into non-negative portfolios. It can be seen that.The dual cone of prices is thus easiest to see in terms of prices of A in terms of M (and similarly done for price of M in terms of A):
- someone offers 1A for tM:
therefore there is arbitrage if 
- someone offers tM for 1A:
therefore there is arbitrage if 
Note
If a solvency cone:
- contains a line, then there is an exchange possible without transaction costs.
-
, then there is no possible exchange.