Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints.
type: Item
instance of: scholarly article
title: Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints.
zbMATH Open document ID: 1048.91072
DOI: 10.1016/S0304-4149(00)00007-7
author: Nizar Touzi
published in: Stochastic Processes and their Applications
publication date: 2004-09-07T00:00:00Z
review text: The paper studies the existence of a super-replicating investment strategy for a European (i.e. dependent only on prices at a fixed execution date) contingent claim in a market with one riskless and one risky asset with stochastic volatility. Both the risky asset price and its volatility are defined by Markovian stochastic differential equations. There are borrowing and short-selling constraints. The existence of super-replication follows from the existence of a viscosity solution to the Hamilton-Jacobi-Bellman equation for the value function of the minimal replication cost problem. The proof is based on a dynamic programming principle developed by \textit{M. Soner} and the author in their earlier work. The lower semicontinuous envelope of the value function turns out to be independent of the driving process of the volatility parameter. Under a set of regularity conditions on the payoff function, the author characterizes the value function for both the unbounded and bounded stock volatility cases. The results cover the examples of classic European calls and puts as well as digital options.
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 2098336
zbMATH Keywords: stochastic control
zbMATH Keywords: viscosity solution
reviewed by: Alexis Derviz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2098336