In an incomplete financial market, the axiomatic of Time Consistent Pricing
Procedure (TCPP), recently introduced, is used to assign to any financial asset
a dynamic limit order book, taking into account both the dynamics of basic
assets and the limit order books for options.
Kreps-Yan fundamental theorem is extended to that context. A characterization
of TCPP calibrated on options is given in terms of their dual representation.
In case of perfectly liquid options, these options can be used as the basic
assets to hedge dynamically. A generic family of TCPP calibrated on option
prices is constructed, from cadlag BMO martingales.
We give characterizations of asymptotic arbitrage of the first and second
kind and of strong asymptotic arbitrage for large financial markets with small
proportional transaction costs $\la_n$ on market $n$ in terms of contiguity
properties of sequences of equivalent probability measures induced by
$\la_n$--consistent price systems. These results are analogous to the
frictionless case. Our setting is simple, each market $n$ contains two assets
with continuous price processes. The proofs use quantitative versions of the
Halmos--Savage Theorem and a monotone convergence result of nonnegative local
martingales. Moreover, we present an example admitting a strong asymptotic
arbitrage without transaction costs; but with transaction costs $\la_n>0$ on
market $n$ ($\la_n\to0$ not too fast) there does not exist any form of
In this paper a simple model for the evolution of the forward density of the
future value of an asset is proposed. The model allows for a straightforward
initial calibration to option prices and has dynamics that are consistent with
empirical findings from option price data. The model is constructed with the
aim of being both simple and realistic, and avoid the need for frequent
re-calibration. The model prices of $n$ options and a forward contract are
expressed as time-varying functions of an $(n+1)$-dimensional Brownian motion
and it is investigated how the Brownian trajectory can be determined from the
trajectories of the price processes. An approach based on particle filtering is
presented for determining the location of the driving Brownian motion from
option prices observed in discrete time. A simulation study and an empirical
study of call options on the S&P 500 index illustrates that the model provides
a good fit to option price data.