Abstract
This paper is a generalization of the results of the previous papers. Using these results a class of evolutions of risk assets based on the geometric Brownian motion is constructed. Among these evolutions of risk assets, the important class of the random processes is the random processes with parameters built on the basis of the discrete geometric Brownian motion. For this class of random processes the interval of non-arbitrage prices are found for the wide class of contingent liabilities. In particular, for the payoff functions of standard options call and put of the European type the fair prices of super-hedge are obtained. Analogous results are obtained for the put and call of arithmetical options of Asian type. For the parameters entering in the definition of random process the description of all statistical estimates is presented. Statistical estimate for which the fair price of super-hedge for the payoff functions of standard call and put options of European type is minimal is indicated. From the formulas found it follows that the fair price of super-hedge can be less than the price of the underlying asset. In terms of estimates the simple formula for the fair price of super-hedge is found. Every estimates can be realized in the reality. This depends on the distribution function of the observed dates in the financial market.
Highlights
In reality, all financial markets are incomplete and the evolution of risky assets is discrete
In this work, which is a continuation of the paper [1], we construct the random processes based on the discrete geometric Brownian motion which can describe the evolution of risky assets
We generalize the class of evolutions of risk assets proposed in [1] and which contains a class of evolutions built on the discrete geometric Brownian motion
Summary
All financial markets are incomplete and the evolution of risky assets is discrete. It is proved that every martingale measure can be represented as an integral on some measure on the set of extreme points of the set of martingale measures This crucial fact is a base for the estimation of contingent liabilities in the incomplete financial markets with the evolution of risk assets introduced in [1]. For this class of evolutions of risk assets the set of martingale measures is described and the representation for every martingale measure as integral over the set of extreme points is obtained Having this representation the formulas for the lower and upper bounds of non arbitrage prices are found. Optional decomposition Theorems see in [13] [14] [15] [16]
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