A Portfolio’s Common Causal Conditional Risk-neutral PDE

ArXiv ID: 2401.00949 “View on arXiv”

Authors: Unknown

Abstract

Portfolio’s optimal drivers for diversification are common causes of the constituents’ correlations. A closed-form formula for the conditional probability of the portfolio given its optimal common drivers is presented, with each pair constituent-common driver joint distribution modelled by Gaussian copulas. A conditional risk-neutral PDE is obtained for this conditional probability as a system of copulas’ PDEs, allowing for dynamical risk management of a portfolio as shown in the experiments. Implied conditional portfolio volatilities and implied weights are new risk metrics that can be dynamically monitored from the PDEs or obtained from their solution.

Keywords: Gaussian copulas, Conditional probability, Stochastic PDE, Portfolio diversification, Implied volatilities, Portfolio (Equities/General)

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats
  • Why: The paper is dense with advanced mathematics, including PDEs derived from copulas and Ito calculus, but the empirical validation relies on visual plots without detailed backtesting metrics, datasets, or executable code.
  flowchart TD
    A["Research Goal<br>Portfolio's optimal diversification drivers are common causes<br>of constituents' correlations"] --> B["Methodology: Gaussian Copulas<br>Model joint distributions of constituent<br>and common driver pairs"]
    B --> C["Data/Inputs<br>Portfolio constituents (Equities/General)<br>Historical correlation data<br>Optimal common drivers"]
    C --> D["Computational Process<br>Derive closed-form formula for<br>conditional probability given optimal drivers"]
    D --> E["Computational Process<br>Obtain conditional risk-neutral PDE<br>as system of copulas' PDEs"]
    E --> F["Key Findings: New Risk Metrics<br>Implied conditional portfolio volatilities<br>Implied weights dynamically monitored"]
    F --> G["Outcome<br>Enables dynamic risk management<br>of portfolio via PDE solutions"]