  This paper discusses the connection between mathematical finance and
statistical modelling which turns out to be more than a formal mathematical
correspondence. We like to figure out how common results and notions in
statistics and their meaning can be translated to the world of mathematical
finance and vice versa. A lot of similarities can be expressed in terms of
LeCam's theory for statistical experiments which is the theory of the behaviour
of likelihood processes. For positive prices the arbitrage free financial
assets fit into filtered experiments. It is shown that they are given by
filtered likelihood ratio processes. From the statistical point of view,
martingale measures, completeness and pricing formulas are revisited. The
pricing formulas for various options are connected with the power functions of
tests. For instance the Black-Scholes price of a European option has an
interpretation as Bayes risk of a Neyman Pearson test. Under contiguity the
convergence of financial experiments and option prices are obtained. In
particular, the approximation of Ito type price processes by discrete models
and the convergence of associated option prices is studied. The result relies
on the central limit theorem for statistical experiments, which is well known
in statistics in connection with local asymptotic normal (LAN) families. As
application certain continuous time option prices can be approximated by
related discrete time pricing formulas.
