tailieunhanh - Báo cáo " On the martingale representation theorem and approximate hedging a contingent claim in the minimum mean square deviation criterion "

In this work we consider the problem of the approximate hedging of a contingent claim in minimum mean square deviation criterion. A theorem on martingale representation in the case of discrete time and an application of obtained result for semi-continous market model are given. | VNU Journal of Science Mathematics - Physics 23 2007 143-154 On the martingale representation theorem and approximate hedging a contingent claim in the minimum mean square deviation criterion Nguyen Van I luu1 Vuong Quan Hoang2 1 Department of Mathematics Mechanics Informatics College of Science VNU 334 Nguyen Trai Hanoi Vietnam 2 ULB Belgium Received 15 November 2006 received in revised form 12 September 2007 Abstract. In this work we consider the problem of the approximate hedging of a contingent claim in minimum mean square deviation criterion. A theorem on martingale representation in the case of discrete time and an application of obtained result for semi-continous market model are given. Keywords Hedging contingent claim risk neutral martingale measure martingale representation. 1. Introduction The activity of a stock market takes place usually in discrete time. Unfortunately such markets with discrete time are in general incomplete and so super-hedging a contingent claim requires usually an initial price two great which is not acceptable in practice. The purpose of this work is to propose a simple method for approximate hedging a contingent claim or an option in minimum mean square deviation criterion. Financial market model with discrete time Without loss of generality let us consider a market model described by a sequence of random vectors Sn n 0 1 . N Sn G Rd which are discounted stock prices defined on the same probability space Q 3 P with Fn n 0 1 . . N being a sequence of increasing sigmaalgebras of information available up to the time n whereas risk free asset chosen as a numeraire S0 1. A Fn -measurable random variable H is called a contingent claim in the case of a standard call option H max Sn K 0 K is strike price. Corresponding author. Tel. 84-4-8542515. E-mail huunv@ 143 144 Huu . Hoang VNU Journal of Science Mathematics - Physics 23 2007 143-154 Trading strategy A sequence of random vectors of d-dimension Y Yn n 1 2 . N with Yn