Semi-Static Variance-Optimal Hedging of Covariance Risk in Multi-Asset Derivatives

Paper: arXiv 2603.25320

Authors: Konstantinos Chatziandreou, Sven Karbach

Abstract

We develop a semi-static framework for the variance-optimal hedging of multi-asset derivatives exposed to correlation and covariance risk. The approach combines continuous-time dynamic trading in the underlying assets with a static portfolio of auxiliary contingent claims. Using a multivariate Galtchouk–Kunita–Watanabe decomposition, we show that the resulting global mean-variance problem decouples naturally into an inner continuous-time projection onto the space spanned by the underlying assets and an outer finite-dimensional quadratic optimization over the static hedging instruments. To systematically select suitable auxiliary claims, we leverage multidimensional functional spanning theory, establishing that otherwise unhedgeable cross-gamma exposures can be structurally mitigated through static strips of vanilla, product, and spread options. As a central application, we derive explicit semi-static replication formulas for covariance swaps and geometric dispersion trades. Our framework accommodates a broad class of asset dynamics, including quadratic and stochastic Volterra covariance models, as well as affine stochastic covariance models with jumps, yielding tractable semi-closed-form solutions via Fourier transform techniques. Extensive numerical experiments demonstrate that incorporating optimally weighted static strips of cross-asset instruments substantially reduces the mean-squared hedging error relative to purely dynamic benchmark strategies across various model classes.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous mathematical framework utilizing multivariate Galtchouk-Kunita-Watanabe decompositions and Volterra processes to solve semi-static hedging problems for multi-asset derivatives. While the theoretical novelties and functional spanning results are strong, the empirical evaluation focuses primarily on simulated numerical experiments rather than backtests on historical market data.

Research Flowchart

  flowchart TD
    A[Goal: Variance-Optimal Semi-Static Hedging of Multi-Asset Covariance Risk] --> B[Input: Derivative Target, Underlying Assets & Static Aux Claims]
    B --> C[Method: Multivariate Galtchouk-Kunita-Watanabe Decomposition]
    C --> D[Decoupled Optimization: Inner Dynamic Projection + Outer Static QP]
    D --> E[Selection: Multidimensional Functional Spanning via Option Strips]
    E --> F[Computation: Fourier Transforms for Volterra/Affine Jump Models]
    F --> G[Outcome: Substantial Reduction in Hedging Error vs Purely Dynamic Strategies]

Cited References (179)

1. Approximating Random Variables by Stochastic Integrals (1994) — link
2. Markowitz Portfolio Selection for Multivariate Affine and Quadratic Volterra Models (2021) — link
3. Deep Quadratic Hedging — → 2212.12725
4. Option Pricing in Multivariate Stochastic Volatility Models of OU Type (2012) — link
5. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options (1996) — link
6. A characterization of the martingale property of exponentially affine processes (2011) — link
7. UTILITY MAXIMIZATION IN AFFINE STOCHASTIC VOLATILITY MODELS (2010) — link
8. Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models (2014) — link

5 of 179 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.