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 ( Ω , F , { F t } t = 0 T , P ) {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)} such that at time t {\displaystyle t} the i t h {\displaystyle i^{th}} component can be thought of as a price for the i t h {\displaystyle i^{th}} asset. Mathematically, a CPP Z = ( Z t ) t = 0 T {\displaystyle Z=(Z_{t})_{t=0}^{T}} in a market with d-assets is an adapted process in R d {\displaystyle \mathbb {R} ^{d}} if Z is a martingale with respect to the physical probability measure P {\displaystyle P} , and if Z t ∈ K t + ∖ { 0 } {\displaystyle Z_{t}\in K_{t}^{+}\backslash \{0\}} at all times t {\displaystyle t} such that K t {\displaystyle K_{t}} is the solvency cone for the market at time t {\displaystyle t} . The CPP plays the role of an equivalent martingale measure in markets with transaction costs. In particular, there exists a 1-to-1 correspondence between the CPP Z {\displaystyle Z} and the EMM Q {\displaystyle Q} .

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 ( Ω , F , { F t } t = 0 T , P ) {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)} such that at time t {\displaystyle t} the i t h {\displaystyle i^{th}} component can be thought of as a price for the i t h {\displaystyle i^{th}} asset.

Mathematically, a CPP Z = ( Z t ) t = 0 T {\displaystyle Z=(Z_{t})_{t=0}^{T}} in a market with d-assets is an adapted process in R d {\displaystyle \mathbb {R} ^{d}} if Z is a martingale with respect to the physical probability measure P {\displaystyle P} , and if Z t ∈ K t + ∖ { 0 } {\displaystyle Z_{t}\in K_{t}^{+}\backslash \{0\}} at all times t {\displaystyle t} such that K t {\displaystyle K_{t}} is the solvency cone for the market at time t {\displaystyle t} .^[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 Z {\displaystyle Z} and the EMM Q {\displaystyle Q} .^{[citation needed]}

References

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1. ^ Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time". {{cite journal}}: Cite journal requires |journal= (help)
2. ^ Yuri M. Kabanov; Mher Safarian (2010). Markets with Transaction Costs: Mathematical Theory. Springer. p. 114. ISBN 978-3-540-68120-5.
3. ^ Jacka, Saul; Berkaoui, Abdelkarem; Warren, Jon (2008). "No arbitrage and closure results for trading cones with transaction costs". Finance and Stochastics. 12 (4): 583–600. arXiv:math/0602178. doi:10.1007/s00780-008-0075-7. S2CID 17136711.

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