Weighted norm inequalities and hedging in incomplete markets
type: Item
instance of: scholarly article
title: Weighted norm inequalities and hedging in incomplete markets
zbMATH Open document ID: 0916.90016
author: Pascale Monat
author: Martin Schweizer
author: Ch. Stricker
author: Freddy Delbaen
author: Walter Schachermayer
published in: Finance and Stochastics
publication date: 1999-07-06T00:00:00Z
review text: This paper is concerned with questions which arise from a number of optimization problems in financial applications. That's why let us explain the backgrounds and financial interpretation of obtained results before we turn to describe them very briefly. The authors start with a \(d\)-dimensional stochastic process \(X= (X_i)_{0\leq t\leq T}\), defined on a probability space \((\Omega,{\mathfrak F},\mathbb{F}, P)\) with filtration \(\mathbb{F}=({\mathfrak F}_t)\) with filtration \(\mathbb{F}= ({\mathfrak F}_t)\), which describes the discounted price evolution of risky assets in a financial market containing also some riskless asset with discounted price \(Y\equiv 1\). One of the central problems in financial mathematics in such a framework is the pricing and hedging of contingent claims (like European call option on some asset \(i\) with expiration data \(T\) and strike price \(R\), say) by means of dynamic trading strategies based on \(X\). More generally, a contingent claim will here simply be an \({\mathfrak F}_t\)-measured random variable \(H\) describing the net payoff at \(T\) of the financial instrument we want to consider. The problems of pricing and hedging \(H\) can then be formulated as follows: What price should the seller \(S\) of \(H\) charge the buyer \(B\) at time \(0\)? And having sold \(H\), how can the seller \(S\) insure himself against the upcoming random loss at time \(T\)? A natural way to approach these questions is to consider dynamic portfolio strategies of the form \((\theta,\eta)= (\theta_t, \eta_t)_{0\leq t\leq T}\), where \(\theta_t\) describes the number of units of \(d\) assets and \(\eta_t\) is the amount invested in the riskless asset, helt at time \(t\). The value of portfolio \((\theta_t,\eta_t)\) is then given by \(V_t= \theta_t' X_t+ \eta_t\), the cumulative gains from trade up to time \(t\) are \(G_t(\theta)= \int^t_0\theta_s dX_s=: (\theta\cdot X)_t\) (it is here assumed that \(X\) is a semimartingale and \(G_t(\theta)\) is the…
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 91G10
zbMATH DE Number: 1210555
zbMATH Keywords: semimartingales
zbMATH Keywords: stochastic integrals
zbMATH Keywords: reverse Hölder inequalities
zbMATH Keywords: stochastic process
zbMATH Keywords: contingent claim
zbMATH Keywords: dynamic portfolio strategies
zbMATH Keywords: self-financing strategy
zbMATH Keywords: mean-variance optimal hedging strategies
reviewed by: Serguey Victorovich Zhulenev
MaRDI profile type: Publication
full work available at URL: s007800050021
OpenAlex ID: W2021434857
DOI: 10.1007/S007800050021
description: scientific article; zbMATH DE number 1210555