Risk Hedging in Financial Markets

Abstract

A recursive method of martingale measures construction for a wide class of evolutions of risky asset is proposed. An integral representation for each equivalent martingale measure is obtained. A complete description of all martingale measures is established. The formulas for both infimum and supremum for the average values of payment functions of call and put options with respect to all equivalent martingale measures are established. The invariance of the set of all martingales with respect to a certain class of evolutions of risky assets is proved. A parametric class of evolutions of risky asset is introduced, which includes ARCH and GARCH models and their generalizations. A parameter estimation method for the introduced parametric models is proposed.
Necessary and sufficient conditions are obtained under which the martingale measure is unique. A significant number of examples of the discounted evolution of risky assets are presented for which the existence of a single martingale measure is established. An explicit construction of a single martingale measure in these cases is given. Formulas for fair price of options contracts and investor hedging strategies are provided. A parametric model of evolution of risky asset is introduced so that the single martingale measure does not depend on the entered parameters. A complete description of the family of martingale measures is given for multinomial models of the evolution of risky asset. Each martingale measure is a finite sum of the introduced spot measures. The attractive side of such models is that the lower and upper price of the interval non arbitrage prices are, respectively, the minimum and maximum of the average values of the payment functions on a set of spot measures. A class of parametric models is introduced that describe the multinomial evolution of risky asset such that the family of martingale measures does not depend on the entered parameters.