Optimal Investment in Equity and Credit Default Swaps in the Presence of Default

ArXiv ID: 2504.08085 “View on arXiv”

Authors: Unknown

Abstract

We consider an equity market subject to risk from both unhedgeable shocks and default. The novelty of our work is that to partially offset default risk, investors may dynamically trade in a credit default swap (CDS) market. Assuming investment opportunities are driven by functions of an underlying diffusive factor process, we identify the certainty equivalent for a constant absolute risk aversion investor with a semi-linear partial differential equation (PDE) which has quadratic growth in both the function and gradient coefficients. For general model specifications, we prove existence of a solution to the PDE which is also the certainty equivalent. We show the optimal policy in the CDS market covers not only equity losses upon default (as one would expect), but also losses due to restricted future trading opportunities. We use our results to price default dependent claims though the principal of utility indifference, and we show that provided the underlying equity market is complete absent the possibility of default, the equity-CDS market is complete accounting for default. Lastly, through a numerical application, we show the optimal CDS policies are essentially static (and hence easily implementable) and that investing in CDS dramatically increases investor indirect utility.

Keywords: Credit Default Swaps (CDS), Utility Indifference Pricing, Partial Differential Equation (PDE), Default Risk, Portfolio Selection

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematics including multi-dimensional diffusions, non-linear PDEs with quadratic growth, and rigorous existence proofs, but provides only a single numerical example without any backtesting, dataset, or statistical metrics, making it heavily theoretical rather than implementation-focused.

  flowchart TD
    A["Research Goal: <br>Optimal investment in equity & CDS<br>in presence of default risk"] --> B["Methodology: <br>Dynamic Stochastic Optimization"]
    B --> C["Inputs: <br>Diffusive factor process,<br>Equity & CDS market data"]
    C --> D["Computational Process: <br>Solve Semi-linear PDE<br>with Quadratic Coefficients"]
    D --> E["Key Findings: <br>1. CDS covers equity losses<br>& restricted trading losses<br>2. Equity-CDS market is complete<br>3. CDS policies are static/effective"]