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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).
A market is one of the 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. It can be said that a market is the process by which the prices of goods and services are established. Markets facilitate trade and enable the distribution and resource allocation in a society. Markets allow any trade-able item to be evaluated and priced. A market emerges more or less spontaneously or may be constructed deliberately by human interaction in order to enable the exchange of rights of services and goods. Markets generally supplant gift economies and are often held in place through rules and customs, such as a booth fee, competitive pricing, and source of goods for sale.
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.
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]
A solvency cone is any closed convex cone such that and . [2]
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 portfolios.[ citation needed ]
In financial mathematics, a self-financing portfolio is a portfolio having the feature that, 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.
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. [1] [3]
Assume there are 2 assets, A and M with 1 to 1 exchange possible.
In a frictionless market, we can obviously make (1A,-1M) and (-1A,1M) into non-negative portfolios, therefore . Note that (1,1) is the "price vector."
In economic theory 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 Black-Scholes model assumes a frictionless market.
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):
If a solvency cone :
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