The fundamental theorem of asset pricing for unbounded stochastic processes
type: Item
instance of: scholarly article
title: The fundamental theorem of asset pricing for unbounded stochastic processes
zbMATH Open document ID: 0917.60048
published in: Mathematische Annalen
publication date: 1999-07-18T00:00:00Z
full work available at URL: document.pdf
review text: The fundamental theorem of asset pricing is a basic result in mathematical finance. Using the concept of martingale measures it characterizes the class of semimartingale models for financial assets which exclude arbitrage, i.e., riskless profits from dynamic trading. Beginning with the seminal work by \textit{J. M. Harrison} and \textit{D. M. Kreps} [J. Econ. Theory 20, 381--408 (1979; Zbl 0431.90019)] and \textit{J. M. Harrison} and \textit{S. R. Pliska} [Stochastic Processes Appl. 11, 215--260 (1981; Zbl 0482.60097)] several formulations of this theorem have been established. The article reviewed here states and proves this theorem in the following very general form: Let \(S=(S_t)_{t \geq 0}\) be an \(R^d\)-valued semimartingale defined on the stochastic base \((\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)\). Then \(S\) satisfies the condition of no free lunch with vanishing risk (NFLVR) if and only if there exists a probability measure \(Q \sim P\) such that \(S\) is a sigma-martingale with respect to \(Q\). The concept of NFVLR is a mild strengthening of the no-arbitrage property (see e.g., [the authors, Math. Ann. 300, No. 3, 463--520 (1994; Zbl 0865.90014)]). A sigma-martingale is defined as a semimartingale \(S=(S_t)_{t \geq 0}\) which can be written as a stochastic integral \(S=\varphi \cdot M\) where \(M\) is a martingale and \(\varphi\) an \(M\)-integrable predictable process with nonnegative values. The concept of a sigma-martingale is new in the context of mathematical finance. It is introduced to deal with possibly unbounded jumps of the asset price process \(S\). Indeed, under the assumption that \(S\) is locally bounded, previous versions of the fundamental theorem, e.g., Delbaen-Schachermayer [loc. cit.], yield equivalence of NFLVR to the existence of a local martingale measure. The authors provide an example of a sigma-martingale which does not admit an equivalent local martingale measure, illustrating that sigma-martingales are in fact unavoid…
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 46N30
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 91B25
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 1221884
zbMATH Keywords: no-arbitrage
zbMATH Keywords: martingale measure
zbMATH Keywords: sigma-martingale
zbMATH Keywords: asset pricing
author: Freddy Delbaen
author: Walter Schachermayer
reviewed by: Peter Bank
MaRDI profile type: Publication
OpenAlex ID: W2095487918
DOI: 10.1007/S002080050220
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description: scientific article; zbMATH DE number 1221884