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

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.