Thermal lag: Difference between revisions

Latest revision as of 01:51, 8 July 2014

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-{{too technical|date=December 2013}}
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-The '''solvency cone''' is a concept used in [[financial mathematics]] which models the possible trades in the [[market|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).
-==Mathematically==
-If given a [[bid-ask matrix]] <math>\Pi</math> for <math>d</math> assets such that <math>\Pi = \left(\pi^{ij}\right)_{1 \leq i,j \leq d}</math> and <math>m \leq d</math> is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally <math>m = d</math>).  Then the solvency cone <math>K(\Pi) \subset \mathbb{R}^d</math> is the convex cone spanned by the unit vectors <math>e^i, 1 \leq i \leq m</math> and the vectors <math>\pi^{ij}e^i-e^j, 1 \leq i,j \leq d</math>.<ref name=WS02>{{cite journal|last=Schachermayer|first=Walter|date=November 15, 2002|title=The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time}}</ref>
-===Definition===
-A solvency cone <math>K</math> is any closed convex cone such that <math>K \subseteq \mathbb{R}^d</math> and <math>K \supseteq \mathbb{R}^d_+</math>.<ref>{{cite doi|10.1137/080743494}}</ref>
-==Uses==
-A process of (random) solvency cones <math>\left\{K_t(\omega)\right\}_{t=0}^T</math> 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.{{Citation needed|date=February 2011}}
-The [[dual cone]] of the solvency cone (<math>K^+ = \left\{w \in \mathbb{R}^d: \forall v \in K: 0 \leq w^Tv\right\}</math>) 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]].<ref name="WS02" />
-==Examples==
-[[Image:SolvencyCone noTransactionCosts.png|thumb|alt=Solvency cone with no transaction costs|Sample solvency cone with no transaction costs]]
-[[Image:SolvencyCone withTransactionCosts.png|thumb|alt=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 <math>K = \{x \in \mathbb{R}^2: (1,1)x \geq 0\}</math>.  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 <math>K = \{x \in \mathbb{R}^2: (2,1)x \geq 0, (1,2)x \geq 0\}</math>.
-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: <math>(0,t) \rightarrow (1,0) \rightarrow (0,\frac{1}{2})</math> therefore there is arbitrage if <math>t < \frac{1}{2}</math>
-* someone offers tM for 1A: <math>(1,0) \rightarrow (0,t) \rightarrow (\frac{t}{2},0)</math> therefore there is arbitrage if <math>t > 2</math>
-==Notes==
-If a solvency cone <math>K</math>:
-* contains a line, then there is an exchange possible without transaction costs.
-* <math>K = \mathbb{R}^d_+</math>, then there is no possible exchange, i.e. the market is completely [[illiquid]].
-==References==
-{{Reflist}}
-[[Category:Mathematical finance]]
-[[Category:Financial risk]]

Thermal lag: Difference between revisions

Latest revision as of 01:51, 8 July 2014

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