Semiparametric Bounds On Variances And Covariance Of Functions Of Unimodal Distributions

This paper studies the semi-parametric boundary problem of expectations,variance and covariance of the functions under the unimodal distribution. Thepurpose of the paper provides semi-parametric bounds of numerical characteris--tic for the functions of unimodal variable under the several given momentinformation.In this paper, the problem of moment belongs to probability theory.Studying the moment bounds of random variable naturally appears in theeconomics, operational research, probability, statistics and other fields. Momentproblem is motivated by practical problems. Moment inequalities can be appliedin financial pricing that can be used to ensure the balance of property. It also hasmany applications in stochastic programming which can be used in estimatingEuropean option, stock price and other financial values.The chosen functions under unimodal distribution are European call optionsmax(S-K,0)and European gap optionsSI(S≥K)in the paper, here S is stock pricethat satisfies unimodal distribution. K is exercise price. We use the dualityprinciple and then introduce a new measure. The estimates for the upper andlower bounds of the functions’ expectation are established by measuretransformation. The inequality is achievable. These results enrich the pre--decessor’s research. We present the equivalent formula for the function’scovariance under unimodal distribution in the third chapter. It can be regard asthe generalization of Khinchine transform. Using the method of estimatingsemi-parametric bounds, we find upper bound and lower bound control functionby using equivalent formula and measure transform. Then we obtain the supperand lower bounds for European call options max(S-K,0)and European gapoptionsSI(S≥K), all these results is completely new. In the final chapter, we givethe supper and lower bounds for variance as the extension of the function’scovariance semi-parametric bounds. The semi-parametric bounds estimate’sresult of European call options max(S-K,0)and European gap optionsSI(S≥K)are finally completed.