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

44 Pages Posted: 11 Jan 2019

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: January 2019

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 that can be strict local martingales and that 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, JLS model, optimal investment, power utility, strict local martingales, utility maximization

Suggested Citation

Herdegen, Martin and Herrmann, Sebastian, Strict Local Martingales and Optimal Investment in a Black–Scholes Model with a Bubble (January 2019). Mathematical Finance, Vol. 29, Issue 1, pp. 285-328, 2019, Available at SSRN: https://ssrn.com/abstract=3313653 or http://dx.doi.org/10.1111/mafi.12175

Martin Herdegen (Contact Author)

University of Warwick - Department of Statistics ( email )

Coventry CV4 7AL
United Kingdom

Sebastian Herrmann

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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