  The general method is proposed for constructing a family of martingale
measures for a wide class of evolution of risky assets. The sufficient
conditions are formulated for the evolution of risky assets under which the
family of equivalent martingale measures to the original measure is a non-empty
set. The set of martingale measures is constructed from a set of strictly
nonneg ative random variables, satisfying certain conditions. The inequalities
are obtained for the non-negative random variables satisfying certain
conditions. Using these inequalities, a new simple proof of optional
decomposition theorem for the nonnegative super-martingale is proposed. The
family of spot measures is introduced and the representation is found for them.
The conditions are found under which each martingale measure is an integral
over the set of spot measures. On the basis of nonlinear processes such as ARCH
and GARCH, the parametric family of random processes is introduced for which
the interval of non-arbitrage prices are found. The formula is obtained for the
fair price of the contract with option of European type for the considered
parametric processes. The parameters of the introduced random processes are
estimated and the estimate is found at which the fair price of contract with
option is the least.
