Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble
Mathematical Finance, Forthcoming
39 Pages Posted: 17 Nov 2013 Last revised: 18 Nov 2017
Date Written: November 17, 2017
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: Suggested Citation