Complete markets with discontinuous security price
type: Item
instance of: scholarly article
title: Complete markets with discontinuous security price
zbMATH Open document ID: 0930.91014
published in: Finance and Stochastics
publication date: 1999-09-14T00:00:00Z
review text: A classical model for continuous time asset pricing models is a geometric Brownian motion, namely, the standard model. An important feature of the standard model is that markets are ``complete'': That is, there will always exist a self-financing trading strategy that replicates the contingent claim. However as has been long known, empirical studies show that stock prices often have jumps (that is, are not continuous). Various models incorporating discontinuities have been proposed. The most common is the ``jump diffusion'' model, where Poisson jumps have been added to Brownian (or ``white'') noise, modeled by a Wiener process. These models do have equivalent martingale measures, and they do have market completeness. In this paper authors are interested in models that have strict market completeness and jumps that are intrinsic to the stock price with a unique equivalent martingale measure, and no arbitrage. For this framework the authors produce a family of distinct semimartingales, indexed by a parameter \(\beta\) (\(-2\leq\beta\leq 0\)), that gives rise to strict market completeness. For each value of the parameter \(\beta\) the model is just as rich as the standard model using white noise and a drift; as \(\beta\) increases to zero the model converges weakly to the standard model. A hedging result analogous to the one of \textit{I. Karatzas, D. Okone} and \textit{Jinju Li} [Stochastics Stochastics Rep. 37, No. 3, 127-131 (1991; Zbl 0745.60056)] is presented.
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1336683
zbMATH Keywords: market completeness
zbMATH Keywords: arbitrage
zbMATH Keywords: stochastic calculus
zbMATH Keywords: Azema martingales
zbMATH Keywords: equivalent martingale measure
zbMATH Keywords: weak convergence
zbMATH Keywords: hedging strategies
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: option pricing
zbMATH Keywords: Black-Scholes model
zbMATH Keywords: contingent claims
zbMATH Keywords: martingale limit theorem
author: Michael A. Dritschel
author: Philip Protter
MaRDI profile type: Publication
full work available at URL: s007800050058
OpenAlex ID: W2064341413
Wikidata QID: Q57451944
DBLP publication ID: journals/fs/DritschelP99
DOI: 10.1007/S007800050058
description: scientific article; zbMATH DE number 1336683