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.Comment: 54 pages, 3 figure