A consistent pricing process (CPP) is any representation of (
frictionless) "prices" of assets in a market. It is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
in a
filtered probability space
In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in ...
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 the study of stochastic processes, a stochastic process is adapted (also referred to as a non-anticipating or non-anticipative process) if information about the value of the process at a given time is available at that same time. An informal int ...
in
if ''Z'' is a
martingale with respect to the physical
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
, and if
at all times
such that
is the
solvency cone for the market at time
.
The CPP plays the role of an
equivalent martingale measure in markets with
transaction costs
In economics, a transaction cost is a cost incurred when making an economic trade when participating in a market.
The idea that transactions form the basis of economic thinking was introduced by the institutional economist John R. Commons in 1 ...
.
In particular, there exists a
1-to-1 correspondence between the CPP
and the EMM
.
References
Financial risk modeling
Mathematical finance
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