Nash equilibria in market impact models : differential game, transient price impact and transaction costs

Market impact is the effect caused by transactions that can move asset prices. Nash equilibria describe an optimal state for the players in a non-cooperative game. In this thesis, we combine these two concepts to analyze the competing behavior of two or more large traders in a financial market. We first consider n risk-averse agents who compete for liquidity in an Almgren-Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear-quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed-form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations. Motivated by the observation that high-frequency traders may use oscillatory trading strategies, which result in a ``hot-potato game", we propose quadratic transaction costs in order to make the market more stable and efficient. We identify a critical value for the size of the transaction costs, above which all oscillations disappear and strategies become buy-only or sell-only. Numerical simulations show that for both traders the expected costs can be lower with transaction costs than without. The liquidation costs can increase with trading frequency when there are no transaction costs, but decrease with trading frequency when transaction costs are sufficiently high. Moreover, we extend this model in several aspects including incorporation of permanent impact, unequal splitting of combined liquidation costs, optimal closed-loop strategies, and we introduce a continuous-time version of the model. In particular, we prove that a Nash equilibrium for continuous-time strategies exists only if the transaction costs are exactly equal to a critical value. For the nonexistence of optimal strategies we give intuitive and mathematical explanations.