# Difference between revisions of "Consistent pricing process"

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A '''consistent pricing process (CPP)''' is any representation of ([[frictionless market|frictionless]]) "prices" of assets in a market. It is a [[stochastic process]] in a [[filtered probability space]] <math>(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t=0}^T,P)</math> such that at time <math>t</math> the <math>i^{th}</math> component can be thought of as a price for the <math>i^{th}</math> asset. | A '''consistent pricing process (CPP)''' is any representation of ([[frictionless market|frictionless]]) "prices" of assets in a market. It is a [[stochastic process]] in a [[filtered probability space]] <math>(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t=0}^T,P)</math> such that at time <math>t</math> the <math>i^{th}</math> component can be thought of as a price for the <math>i^{th}</math> asset. | ||

− | Mathematically, a CPP <math>Z = (Z_t)_{t=0}^T</math> in a market with d-assets is an [[adapted process]] in <math>\mathbb{R}^d</math> if ''Z'' is a [[martingale]] with respect to the physical [[probability measure]] <math>P</math>, and if <math>Z_t \in K_t^+ \backslash \{0\}</math> at all times <math>t</math> such that <math>K_t</math> is the [[solvency cone]] for the market at time <math>t</math>.<ref>{{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><ref>{{cite book|title=Markets with Transaction Costs: Mathematical Theory|author1=Yuri M. Kabanov|author2=Mher Safarian|publisher=Springer|year=2010|isbn=978-3-540-68120-5|page=114}}</ref> | + | Mathematically, a CPP <math>Z = (Z_t)_{t=0}^T</math> in a market with d-assets is an [[adapted process]] in <math>\mathbb{R}^d</math> if ''Z'' is a [[martingale (probability theory)|martingale]] with respect to the physical [[probability measure]] <math>P</math>, and if <math>Z_t \in K_t^+ \backslash \{0\}</math> at all times <math>t</math> such that <math>K_t</math> is the [[solvency cone]] for the market at time <math>t</math>.<ref>{{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><ref>{{cite book|title=Markets with Transaction Costs: Mathematical Theory|author1=Yuri M. Kabanov|author2=Mher Safarian|publisher=Springer|year=2010|isbn=978-3-540-68120-5|page=114}}</ref> |

The CPP plays the role of an [[equivalent martingale measure]] in markets with [[transaction costs]].<ref>{{cite journal|title=No arbitrage and closure results for trading cones with transaction costs|first1=Saul|last1=Jacka|first2=Abdelkarem|last2=Berkaoui|first3=Jon|last3=Warren|journal=Finance and Stochastics|volume=12|number=4|pages=583–600|doi=10.1007/s00780-008-0075-7}}</ref> In particular, there exists a [[1-to-1]] correspondence between the CPP <math>Z</math> and the EMM <math>Q</math>.{{Citation needed|date=February 2011}} | The CPP plays the role of an [[equivalent martingale measure]] in markets with [[transaction costs]].<ref>{{cite journal|title=No arbitrage and closure results for trading cones with transaction costs|first1=Saul|last1=Jacka|first2=Abdelkarem|last2=Berkaoui|first3=Jon|last3=Warren|journal=Finance and Stochastics|volume=12|number=4|pages=583–600|doi=10.1007/s00780-008-0075-7}}</ref> In particular, there exists a [[1-to-1]] correspondence between the CPP <math>Z</math> and the EMM <math>Q</math>.{{Citation needed|date=February 2011}} |

## Revision as of 16:42, 11 October 2013

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 the component can be thought of as a price for the asset.

Mathematically, a CPP in a market with d-assets is an adapted process in if *Z* is a martingale with respect to the physical probability measure , and if at all times such that is the solvency cone for the market at time .^{[1]}^{[2]}

The CPP plays the role of an equivalent martingale measure in markets with transaction costs.^{[3]} In particular, there exists a 1-to-1 correspondence between the CPP and the EMM .{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]}
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