Optimal Bubble Riding: A Mean Field Game with Varying Entry Times

TL;DR

This paper develops a mean field game model for optimal liquidation during financial bubbles, accounting for varying entry times of traders and both endogenous and exogenous market crashes, with theoretical and numerical analysis.

Contribution

It introduces a novel mean field game framework with varying entry times and incorporates exogenous crash times using progressive enlargement of filtrations.

Findings

Equilibrium strategies decompose into pre- and post-burst segments.

Existence of MFG equilibria proved in a generalized setting.

Numerical simulations reveal relationships between bubble bursts and trader strategies.

Abstract

Recent financial bubbles such as the emergence of cryptocurrencies and "meme stocks" have gained increasing attention from both retail and institutional investors. In this paper, we propose a game-theoretic model on optimal liquidation in the presence of an asset bubble. Our setup allows the influx of players to fuel the price of the asset. Moreover, traders will enter the market at possibly different times and take advantage of the uptrend at the risk of an inevitable crash. In particular, we consider two types of crashes: an endogenous burst which results from excessive selling, and an exogenous burst which cannot be anticipated and is independent from the actions of the traders. The popularity of asset bubbles suggests a large-population setting, which naturally leads to a mean field game (MFG) formulation. We introduce a class of MFGs with varying entry times. In particular, an equilibrium will depend on the entry-weighted average of conditional optimal strategies. To incorporate the exogenous burst time, we adopt the method of progressive enlargement of filtrations. We prove existence of MFG equilibria using the weak formulation in a generalized setup, and we show that the equilibrium strategy can be decomposed into before-and-after-burst segments, each part containing only the market information. We also perform numerical simulations of the solution, which allow us to provide some intriguing results on the relationship between the bubble burst and equilibrium strategies.