A Class of Strict Local Martingales

Martin Herdegen and Sebastian Herrmann
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Martin Herdegen: ETH Zurich
Sebastian Herrmann: ETH Zurich

No 14-18, Swiss Finance Institute Research Paper Series from Swiss Finance Institute

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 ? 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 ? 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 (search for similar items in EconPapers)
JEL-codes: Y80 (search for similar items in EconPapers)
Pages: 35 pages
Date: 2014-02, Revised 2014-10
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:chf:rpseri:rp1418

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