Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble

Mathematical Finance, Forthcoming

Swiss Finance Institute Research Paper No. 13-58

39 Pages Posted: 17 Nov 2013 Last revised: 18 Nov 2017

See all articles by Martin Herdegen

Martin Herdegen

University of Warwick - Department of Statistics

Sebastian Herrmann

University of Manchester

Multiple version iconThere are 2 versions of this paper

Date Written: November 17, 2017

Abstract

There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen-Ledoit-Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations which can be strict local martingales and which preserve the key assumption of a log-periodic power law for the hazard rate of the time of the crash. We then study the optimal investment problem for an investor with constant relative risk aversion in this model. We show that for positive instantaneous expected returns, investors with relative risk aversion above one always ride the bubble.

Keywords: Bubbles; Strict local martingales; JLS model; Optimal investent; Utility maximisation; Power utility

JEL Classification: G11, C61

Suggested Citation

Herdegen, Martin and Herrmann, Sebastian, Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble (November 17, 2017). Mathematical Finance, Forthcoming; Swiss Finance Institute Research Paper No. 13-58. Available at SSRN: https://ssrn.com/abstract=2354997 or http://dx.doi.org/10.2139/ssrn.2354997

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, M13 9PL
United Kingdom

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

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