Solvency cone

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).


Contents

Mathematically

If given a bid-ask matrix Π 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.[1]

Definition

A solvency cone K is any closed convex cone such that K \subseteq \mathbb{R}^d and K \supseteq \mathbb{R}^d_+.[2]

Uses

A process of (random) solvency cones \left\{K_t(\omega)\right\}_{t=0}^T 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]

The dual cone of the solvency cone (K^+ = \left\{w \in \mathbb{R}^d: \forall v \in K: 0 \leq w^Tv\right\}) 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]

Examples

Solvency cone with no transaction costs
Sample solvency cone with no transaction costs
Solvency cone with transaction costs
Sample solvency cone with transaction costs

Assume there are 2 assets, A and M with 1 to 1 exchange possible.

Frictionless Market

In a frictionless market, we can obviously make (1A,-1M) and (-1A,1M) into non-negative portfolios, therefore K = \{x \in \mathbb{R}^2: (1,1)x \geq 0\}. 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 K = \{x \in \mathbb{R}^2: (2,1)x \geq 0, (1,2)x \geq 0\}.

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: (0,t) \rightarrow (1,0) \rightarrow (0,\frac{1}{2}) therefore there is arbitrage if t < \frac{1}{2}
  • someone offers tM for 1A: (1,0) \rightarrow (0,t) \rightarrow (\frac{t}{2},0) therefore there is arbitrage if t > 2

Note

If a solvency cone K:

  • contains a line, then there is an exchange possible without transaction costs.
  • K = \mathbb{R}^d_+, then there is no possible exchange.

References

  1. ^ a b Schachermayer, Walter (November 15, 2002). The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time. 
  2. ^ Hamel, Andreas; Heyde, Frank (December 11, 2008) (pdf). Duality for Set-Valued Risk Measures. http://www.princeton.edu/~ahamel/SetRiskHamHey.pdf. Retrieved July 22, 2010. [dead link]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Coherent risk measure — In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function …   Wikipedia

  • Consistent pricing process — A consistent pricing process (CPP) is any representation of (frictionless) prices of assets in a market. It is a stochastic process in a filtered probability space such that at time t the ith component can be thought of as a price for the ith… …   Wikipedia

  • Frictionless market — A Frictionless market is a financial market without transaction costs.[1] 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… …   Wikipedia

  • Social Protection — ▪ 2006 Introduction With medical costs skyrocketing and government programs scaled back, citizens bore more responsibility for their health care costs; irregular migration, human trafficking, and migrant smuggling posed challenges for… …   Universalium

  • Germany — /jerr meuh nee/, n. a republic in central Europe: after World War II divided into four zones, British, French, U.S., and Soviet, and in 1949 into East Germany and West Germany; East and West Germany were reunited in 1990. 84,068,216; 137,852 sq.… …   Universalium

  • international relations — a branch of political science dealing with the relations between nations. [1970 75] * * * Study of the relations of states with each other and with international organizations and certain subnational entities (e.g., bureaucracies and political… …   Universalium

  • K — Кадастровая стоимость (Cadastre value) Кадастровый учет, кадастр (Cadastre) Кадастровый номер земельного участка (Cadastre number of a plot of land) Кадровый резерв (Personnel reserve) Кадры (Personnel) Казенное предприятие (state owned unitary… …   Экономико-математический словарь

  • Foreign relations of Mexico — Mexico This article is part of the series: Politics and government of Mexico …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.