Paper: arXiv 2311.10021

Authors: Sascha Desmettre, Sebastian Merkel, Annalena Mickel, Alexander Steinicke

Abstract

We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott (2002) of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in 2010. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston’s stochastic volatility model, Bates’ stochastic volatility model including jumps, and Kim-Omberg’s model for a stochastic excess return.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous theoretical analysis utilizing advanced mathematical tools such as backward stochastic differential equations (BSDEs) and martingale methods to solve worst-case portfolio problems under market crashes. However, it is fundamentally a theoretical math-finance paper lacking empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A["Goal: Solve Worst-Case Optimal Portfolio Problem<br/>(Log Preferences & Market Crashes)"] --> B["Inputs: Market Models<br/>(Heston, Bates, Kim-Omberg)"]
    B --> C["Method: Enhanced Martingale Approach<br/>& Crash-Exposure Transformation"]
    C --> D["Formulate BSDEs & Associated PDEs<br/>for Indifference Strategies"]
    D --> E["Solve Systems for Unbounded<br/>Market Price of Risk"]
    E --> F["Outcomes: Characterized Worst-Case Strategies<br/>& Proven BSDE Existence"]

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