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

TL;DR

This paper explores models that combine the Johansen-Ledoit-Sornette bubble framework with strict local martingale behavior, analyzing their implications for optimal investment strategies in a Black-Scholes setting.

Contribution

It clarifies the connection between the JLS bubble model and strict local martingales, and studies optimal investment strategies in such models with risk aversion.

Findings

Investors with risk aversion above one tend to ride the bubble when expected returns are positive.

Relaxed JLS models can exhibit strict local martingale behavior while maintaining key bubble characteristics.

The paper links two major approaches to modeling financial bubbles within a unified framework.

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.