The interval market model in mathematical finance. Game-theoretic methods
type: Item
instance of: scholarly article
title: The interval market model in mathematical finance. Game-theoretic methods
zbMATH Open document ID: 1279.91005
DOI: 10.1007/978-0-8176-8388-7
author: Pierre Bernhard
author: Berend Roorda
author: Johannes M. Schumacher
author: Patrick Saint-Pierre
author: Jean-Pierre Aubin
author: J. C. Engwerda
author: Vassili Kolokoltsov
published in: Static \& Dynamic Game Theory: Foundations \& Applications
publication date: 2012-11-15T00:00:00Z
review text: The interval market model, introduced independently by the authors of this book, postulates that for a market consisting of \(n\) assets, there is a compact convex set in \(\mathbb R^{n}\) which contains the one-step relative price changes (in a discrete-time setting) or the relative asset price velocities (in a continuous-time setting). E.g.\ for a single asset in discrete time, this means there are constants \(d < 1 < u\) such that \(S(t + \Delta t) \in [dS(t), uS(t)]\) for all possible price trajectories; compare this to the \(S(t + \Delta t) \in \{dS(t), uS(t)\}\) specification of the binary tree model. In contrast with classic mathematical finance, the interval market model is not of probabilistic-stochastic nature. Instead, as the authors state it, ``\dots most developments to be reported here belong to the realm of robust control, i.e.\ minimax approaches to decision making in the presence of uncertainty. These take several forms: the discrete Isaacs equation, Isaacs' and Breakwell's geometric analysis of extremal fields, Aubin's viability approach, Crandall and Lions' viscosity solutions as extended to differential games by Evans and Souganidis, Bardi, and others, Frankowska's non-smooth analysis approach to viscosity solutions of positively complete sets. As a consequence, we will not attempt to give here a general introduction to dynamic game theory, as different parts of the book use different approaches. We will, however, strive to make each part self-contained. Nor will we try to uniform the notation, although some of these works deal with closely related topics. As a matter of fact, the developments we report here evolved, relatively independently, over more than a decade. As a result, they have developed independent, consistent notation systems. Merging them at this late stage would have been close to impossible. We will provide a concise `dictionary' between the notations of Parts II--V.'' Part I of the book is written by P.\ Bernhard. Chapter 1 revi…
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 91-02
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 91G10
Mathematics Subject Classification ID: 91G20
Mathematics Subject Classification ID: 91G60
Mathematics Subject Classification ID: 91B26
Mathematics Subject Classification ID: 49N90
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 91A40
Mathematics Subject Classification ID: 91A80
zbMATH DE Number: 6106769
zbMATH Keywords: interval market model
zbMATH Keywords: option pricing
zbMATH Keywords: Black-Scholes model
zbMATH Keywords: robust control
zbMATH Keywords: game theory
zbMATH Keywords: minimax formula
zbMATH Keywords: differential games
zbMATH Keywords: viscosity solution
zbMATH Keywords: discrete-time and continuous-time hedging
zbMATH Keywords: fair price interval
zbMATH Keywords: rainbow options
zbMATH Keywords: real options
zbMATH Keywords: American options
zbMATH Keywords: Bermudan options
zbMATH Keywords: digital options
zbMATH Keywords: incomplete market
zbMATH Keywords: credit default swaps
zbMATH Keywords: Guaranteed Capture Basin Algorithm
zbMATH Keywords: Constant Proportion Portfolio Insurance
zbMATH Keywords: Viabilist Portfolio Performance and Insurance
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-8176-8388-7
OpenAlex ID: W2157557807
description: scientific article; zbMATH DE number 6106769