Consistent pricing process (original) (raw)

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

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

^ Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".
^ Yuri M. Kabanov; Mher Safarian (2010). Markets with Transaction Costs: Mathematical Theory. Springer. p. 114. ISBN 978-3-540-68120-5.
^ 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.