Superreplication in stochastic volatility models and optimal stopping
type: Item
instance of: scholarly article
title: Superreplication in stochastic volatility models and optimal stopping
zbMATH Open document ID: 0951.91029
author: Rüdiger Frey
published in: Finance and Stochastics
publication date: 2000-11-01T00:00:00Z
review text: The pricing and hedging of derivative securities is nowadays well-understood in the context of the classical Black-Scholes model of geometric Brownian motion. However, recent empirical research has produced a lot of statistical evidence that is difficult to reconcile with the assumption of independent and normally distributed asset returns. Researches have therefore attempted to build models for asset price fluctuations that are flexible enough to cope with these empirical deficiencies of the Black-Scholes model. In particular, a lot of work has been devoted to relaxing the assumption of constant volatility in the Black- Scholes model and there is a growing literature on Stochastic Volatility models (SV-models). In this class of models the Stochastic Differential Equation (SDE) that governs the asset price process is driven by a Brownian motion, but the diffusion coefficient of this SDE is modelled as a stochastic process which is only imperfectly correlated to the Brownian motion driving the asset price process. It is well-known that SV-models are incomplete, i.e. one cannot replicate the payoff of a typical derivative by dynamic trading in the underlying risky asset (``the stock'') and in some riskless money market account. The author restricts himselves to SV-models where the range of the volatility is bounded. Under this additional assumption ``nontrivial'' superhedging strategies for a large class of derivatives whose payoff may even be path-dependent are obtained. These strategies are universal in the sense that they depend only on the bounds imposed on the volatility and not on a particular parametric model for the volatility dynamics. The value process of the superhedging strategy is characterized by an optimal-stopping problem in the context of the Black-Scholes model. Roughly speaking the result of the paper can be phrased as follows: the value of a superhedging strategy for a European type derivative under stochastic volatility equals the value of a corre…
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 91B24
zbMATH DE Number: 1524265
zbMATH Keywords: stochastic volatility
zbMATH Keywords: optimal stopping
zbMATH Keywords: incomplete markets
zbMATH Keywords: superreplication
MaRDI profile type: Publication
full work available at URL: s007800050010
OpenAlex ID: W1995951988
DOI: 10.1007/S007800050010
description: scientific article; zbMATH DE number 1524265