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Delegated portfolio management with random default

Alberto Gennaro, Thibaut Mastrolia

Abstract

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

Delegated portfolio management with random default

Abstract

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

Paper Structure

This paper contains 17 sections, 5 theorems, 103 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The model and the optimization problem
  3. Risky assets and portfolio dynamics
  4. Default time and enlargement of filtration
  5. Admissible strategy and contracts
  6. Economical interpretation.
  7. Delegated portfolio management and bi-level stochastic programming
  8. Optimal contract and investment strategy
  9. Optimal investment with random horizon: solving the agent problem
  10. The optimal contract and verification results
  11. Bounded default time
  12. Viscosity solution and dynamic programming.
  13. Unbounded default time
  14. Numerical solutions
  15. Main numerical challenges and remedies.
  16. ...and 2 more sections

Key Result

Lemma 1

Let $\xi\in \Xi$ and let $(R^{\pi}_{t\wedge \tau})_{t \in [0, T]}$ be a family of stochastic processes indexed by the strategy $\pi \in A$ such that Then, $\hat{\pi}$ is a solution to the maximization problem pbAgent.

Figures (4)

  • Figure 1: Wealth and strategies for our models and the case without any default.
  • Figure 2: Compensation and optimal incentives evolution in different models.
  • Figure 3: Wealth, strategy, and average incentive with respect to the default under uniform default between the more general contract and the linear one.
  • Figure 4: Comparison of no default, uniform default, and exponential default for wealth, trading strategy, average incentive with respect to the default, and portfolio evolution.

Theorems & Definitions (22)

  • Definition 1
  • Remark 1
  • Definition 2: Admissible strategy with constraints
  • Definition 3: Admissible contract with contractible variables
  • Remark 2
  • Remark 3
  • Lemma 1: Martingale Optimality Principle
  • proof
  • Theorem 1
  • proof
  • ...and 12 more