Most financial models introduced for the purpose of pricing and hedging derivatives concentrate
on the dynamics of the underlying stocks, or underlying instruments on which the derivatives
are written. However, as certain types of derivatives became liquid, it appeared reasonable to model
their prices directly and use these market models to price or hedge exotic derivatives. This framework
was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets.
We discuss the characterization of arbitrage free dynamic stochastic models for the markets with
infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the
presence of jumps in the underlying, the option prices are represented either through local volatility or
through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic
process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage
in this framework and address the issue of construction of the arbitrage-free models.