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 Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...

(\Omega,\mathcal,\_^T,P)

such that at time

t

the

i^

component can be thought of as a price for the

i^

asset. Mathematically, a CPP

Z = (Z_t)_^T

in a market with d-assets is an

adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...

\mathbb^d

if ''Z'' is a

martingale Martingale may refer to: * Martingale (probability theory), a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value * Martingale (tack) for horses * Martingale (coll ...

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 probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

P

, and if

Z_t \in K_t^+ \backslash \

at all times

t

such that

K_t

is the

solvency cone The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of portfolios that can be ...

for the market at time

t

. The CPP plays the role of an

equivalent martingale measure In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price und ...

in markets with

transaction costs In economics and related disciplines, a transaction cost is a cost in making any economic trade when participating in a market. Oliver E. Williamson defines transaction costs as the costs of running an economic system of companies, and unlike pro ...

. In particular, there exists a 1-to-1 correspondence between the CPP

Z

and the EMM

Q

References

Financial risk modeling Mathematical finance {{econometrics-stub