Minimization with respect to entropy in the problem of finding a martingale measure
type: Item
instance of: scholarly article
title: Minimization with respect to entropy in the problem of finding a martingale measure
zbMATH Open document ID: 0999.60046
author: O. V. Shataev
published in: Russian Mathematical Surveys
publication date: 2002-12-02T00:00:00Z
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 49N15
zbMATH DE Number: 1729035
zbMATH Keywords: martingale measure
zbMATH Keywords: convex functional
MaRDI profile type: Publication
DOI: 10.1070/RM2000V055N05ABEH000330
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review text: Let \((\Omega, F,P)\) be the probability space and \(h_1,h_2,\dots, h_N\) be real-valued random variables having a continuous joint distribution with everywhere non-zero density \(f_0(x_1,\dots,x_N)> 0\). Let \(S_n= S_0 e^{h_1+\cdots+ h_n}\) be the price of a stock at time \(n= 1,2,\dots, N\), \(B_0\equiv 1\) be the price of a bond, \({\mathcal I}_n= \sigma(h_1,\dots, h_n)\), \(n=1,\dots, N\), and denote by \({\mathcal P}(P)\) the set of all probability measures equivalent to the measure \(P\) and such that \((S_n/B_n)\) is a martingale with respect to the filtration \(({\mathcal I}_n)_n\). The author determines the density \(f\) of the joint distribution of \(h_1,\dots, h_N\) corresponding to a measure \(\widetilde P\in{\mathcal P}(P)\) that satisfies the conditions \(E_{\widetilde P}G_i\) \((S_1,\dots, S_N)= \int G_i f dx_1\cdots dx_N= C_i\), \(i=1,\dots,M\), where the \(G_i\) are given functions to be interpreted as discounted payment functions of certain options and the \(C_i\) are the given values of their prices at time \(0\).
reviewed by: Ion C. Vladimirescu
description: scientific article; zbMATH DE number 1729035