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Date
2014-09Type
- Working Paper
ETH Bibliography
yes
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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. Show more
Publication status
publishedExternal links
Journal / series
SSRNPages / Article No.
Publisher
Social Science Research NetworkSubject
Single jump; Strict local martingales; Stochastic integrals; Local martingale deflators; No arbitrage; No unbounded profit with bounded riskOrganisational unit
03658 - Schweizer, Martin / Schweizer, Martin
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ETH Bibliography
yes
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