On The Problem Of Pricing And Utility Maximization With Random Interval Payoffs In One-period Setting

This article introduced a market with random interval payoffs where all securities have random interval payoffs.Under one-period setting,the payoffs of securities are given by interval matrix.Both pricing a contingent claim and utility maximization are the two important topics.This paper works on the two problems in the market with random interval payoffs under one-period setting.In section 2,we recall classical results and methods on contingent claim pricing theory under one-period setting.Then we introduce the market with random interval payoffs.After proposing a concept of acceptable market based on no-arbitrage principle,we give the concept of acceptable risk-neutral probability measure.We give the price interval for contingent claim with random interval payoffs.After that,we introduced the conception of replicate strategy for contingent claim.Based on it,we find that the result is the same as that based on acceptable risk-neutral probability measure.In section 3,we discuss the problem of utility maximization in a market with random-interval payoffs.We introduced a investor by three characters:utility function u,the initial endowment e and the subjective attitude to the interval uncertaintyλ. We introduce the synthetic utility model under short-selling constraints using these three indexes and study the existence and propositions of utility maximization.We find that we can get an acceptable risk-neutral probability measure by using best portfolio which proves the acceptability of the market.Also the fair price of an interval-valued contingent claim is proposed from the utility maximization and is given by an expression which using the propositions of the utility maximization.The result shows that the fair price is in the acceptable price interval.In section 4,we introduce the market with fuzzy stochastic payoffs.In every confidence degree market of the fuzzy stochastic market,we get the acceptable price of every confidence degree by using the results of section 2.At last,we get the fuzzy price of contingent claim in the acceptable market by using fuzzy representation theorem.This article also gives the conception of the replicate strategy under the fuzzy stochastic market and gets the fuzzy price of the contingent claim.Noticing that the market with random interval payoffs is the extension of the classical market,we discuss the propositions based on no-arbitrage principle.We get many meaningful conclusions.We also find that the results in the classical market are included by the results in the market with random interval payoffs.So it is obviously that the market with random interval payoffs is the meaningful extension of the classical market,but it includes more information and can help investor make better decision.