Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

Area of Science:

• Financial Mathematics
• Quantitative Finance
• Risk Management

Background:

• Systemic risk in banking poses a significant threat to financial stability.
• Central banks act as lenders of last resort to distressed financial institutions.
• Modeling the optimal intervention strategy for a central bank is complex due to mutual obligations and potential defaults.

Purpose of the Study:

• To determine the minimum cash injection required by a central bank to limit defaults to a predefined proportion.
• To analyze the behavior of the central agent's control problem as the number of institutions grows infinitely large.
• To derive optimal bailout strategies for financial institutions to mitigate systemic risk.

Main Methods:

• Structural default model with mutual obligations.
• Convergence analysis of the central agent's control problem.
• Mean-field control problem formulation and numerical solution using a policy gradient method.
• Identification of a drift-controlled supercooled Stefan problem.

Main Results:

• The value of the central agent's control problem converges as the number of institutions approaches infinity.
• The problem's solution is shown to satisfy a drift-controlled supercooled Stefan problem.
• Optimal strategies are computed in feedback form.
• Simulations reveal that subsidizing banks with equity values in a specific time-dependent region is optimal.

Conclusions:

• Central bank interventions can effectively manage systemic risk in the banking sector.
• The optimal bailout strategy is dynamic and depends on the equity values of financial institutions.
• Mathematical modeling provides insights into optimal policy design for financial stability.