Games on Portfolio Optimization and Bitcoin Mining

Abstract

In this thesis, we propose two game models for portfolio optimization and Bitcoin mining respectively, to understand agents strategic decisions with interaction and the system evolution in the equilibrium.

In the first part, we study the feedback effect of portfolio optimization on market prices, returns and volatility.

In a model where a significant proportion of trading in a stock is conducted by utility maximizers who perturb the supply and demand, we characterize an equilibrium, including the optimal investment strategy and the dynamics of the market price, by a system of coupled nonlinear PDEs.

Furthermore, we prove the existence of the equilibrium and discuss its uniqueness. We also use asymptotic analysis to study the feedback effect in the equilibrium. This helps us deal with coupled equations and gives us analytical approximations. We find that the drift and volatility of the market price will be smaller than those of the classical Merton problem, if investors hold the positions without any leverage. We also examine the model from other perspectives. For instance, the mean field game and Stage-k approach are explored. In addition, we connect the continuous equilibrium model in Section 2.2 with oil Exchanged Traded Funds (ETFs). Our model can explain the inefficiency in the tracking performance of those ETFs.

In the second part, we are interested in the centralization of the reward and computational power that occurs in Bitcoin mining. We propose a mean field game model to study this phenomenon. Miners compete against each other for the reward by increasing their computational power.

We show that the heterogeneity of their wealth leads to the imbalance of the reward distribution or a ``the rich get richer'' effect.

This phenomenon is aggravated by a higher price of bitcoin and reduced by the competition.

Additionally, an advanced miner with cost advantages, contributes a significant amount of computational power.

Hence, cost efficiency can also result in the centralization.