Non-Arbitrage Models of Financial Markets
In the first part of the paper, we construct the models of the complete non-arbitrage financial markets for a wide class of evolutions of risky assets. This construction is based on the observation that for a certain class of risky asset evolutions the martingale measure is invariant with respect to these evolutions. For such a financial market model the only martingale measure being equivalent to an initial measure is built. On such a financial market, formulas for the fair price of contingent liabilities are presented. A multiparameter model of the financial market is proposed, the martingale measure of which does not depend on the parameters of the model of the evolution of risky assets and is the only one. In the second part of the paper, a model of an incomplete non-arbitrage financial market is proposed. As in the first part of the paper, we use the fact that the family of spot martingale measures is invariant with respect to a certain class of evolutions of risky assets. The set of all martingale measures being equivalent to an initial measure is completely described. Each martingale measure is a linear convex combination of the finite number of spot measures whose structure is completely described. For a wide class of models for the evolution of risky assets, a formula is found for the fair price of a super-hedge, as well as an interval of non-arbitrage prices for any contingent liability. A multi-parameter model of the incomplete financial market is proposed, the martingale measures of which do not depend on the parameters of the model of the evolution of risky assets. For the parameters of the models of the evolution of risky assets, statistical estimates are found for both complete and incomplete non-arbitrage markets.
