Time-Consistent Asset Allocation for Risk Measures in a Lévy Market

Paper: arXiv 2305.09471

Authors: Felix Fießinger, Mitja Stadje

Abstract

Focusing on gains & losses relative to a risk-free benchmark instead of terminal wealth, we consider an asset allocation problem to maximize time-consistently a mean-risk reward function with a general risk measure which is i) law-invariant, ii) cash- or shift-invariant, and iii) positively homogeneous, and possibly plugged into a general function. Examples include (relative) Value at Risk, coherent risk measures, variance, and generalized deviation risk measures. We model the market via a generalized version of the multi-dimensional Black-Scholes model using $α$-stable Lévy processes and give supplementary results for the classical Black-Scholes model. The optimal solution to this problem is a Nash subgame equilibrium given by the solution of an extended Hamilton-Jacobi-Bellman equation. Moreover, we show that the optimal solution is deterministic under appropriate assumptions.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for time-consistent asset allocation under Lévy processes using an extended Hamilton-Jacobi-Bellman equation. However, it is purely mathematical with no empirical backtesting, datasets, or code provided to validate the theoretical findings in real market conditions.

Research Flowchart

  flowchart TD
    A[Research Goal: Time-Consistent Mean-Risk Asset Allocation] --> B[Inputs: Multi-Asset Lévy Market & Risk-Free Benchmark]
    B --> C[Formulate Reward Function: Law-Invariant, Shift-Invariant & Positively Homogeneous Risk Measures]
    C --> D[Derive Extended Hamilton-Jacobi-Bellman Equation]
    D --> E[Compute Nash Subgame Equilibrium Strategy]
    E --> F[Outcomes: Time-Consistent & Deterministic Optimal Allocation Strategy]