A Class of Strict Local Martingales

Martin Herdegen, Sebastian Herrmann

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Article number14-18
Pages (from-to)1-35
Number of pages35
JournalSSRN Electronic Journal
DOIs
Publication statusPublished - 8 Oct 2014

Keywords

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

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