A Class of Strict Local Martingales

35 Pages Posted: 7 Mar 2014 Last revised: 22 Oct 2014

See all articles by Martin Herdegen

Martin Herdegen

University of Warwick - Department of Statistics

Sebastian Herrmann

University of Manchester

Date Written: October 8, 2014

Abstract

Many results in stochastic analysis and mathematical finance involve local martingales. However, specific examples of strict local martingales are rare and analytically often rather unhandy. We study local martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local martingales, uniformly integrable martingales that are not in H¹, etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H • M is a strict local martingale. Moreover, we characterise all local martingale deflators and all equivalent local martingale measures for a given special semimartingale with respect to the smallest filtration that turns γ into a stopping time. Two new counter-examples show, using direct arguments only, that neither of the no-arbitrage conditions NA and NUPBR implies the other. The structural simplicity of these examples allows to understand the difference between NA and NUPBR on an intuitive level.

Keywords: Single jump; Strict local martingales; Stochastic integrals; Local martingale deflators; No arbitrage; No unbounded profit with bounded risk

JEL Classification: Y80

Suggested Citation

Herdegen, Martin and Herrmann, Sebastian, A Class of Strict Local Martingales (October 8, 2014). Swiss Finance Institute Research Paper No. 14-18, Available at SSRN: https://ssrn.com/abstract=2402248 or http://dx.doi.org/10.2139/ssrn.2402248

Martin Herdegen

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

Sebastian Herrmann (Contact Author)

University of Manchester ( email )

Oxford Road
Manchester, N/A M13 9PL
United Kingdom

HOME PAGE: http://personalpages.manchester.ac.uk/staff/sebastian.herrmann

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