quant research // summarizing the paperLOADING
Riccardo Alberti, Sven Karbach · 2026-06-27
The paper develops a mathematical method for hedging European options written on forward curves (like commodity or energy forwards) when both the whole term structure and its volatility evolve randomly. It derives the variance-optimal hedge and shows how the leftover hedging error can be split exactly into three parts: risk from specific maturity buckets, risk from the number of volatility factors used, and a residual that can't be removed. It proves that hedges using finite maturities and a finite number of volatility factors converge to the ideal hedge.
Why it matters: For anyone hedging derivatives on forward curves — especially in commodities, energy, or rates — this offers a framework for understanding which risks can be hedged with traded instruments and which cannot. The residual 'stochastic-volatility floor' formalizes that some volatility-linked risk is inherently untradeable, which matters for setting realistic hedging expectations and reserves.
⚠ This is a theoretical/mathematical result relying on specific HJMM stochastic-covariance model assumptions, illustrated only with model examples rather than tested on real market data or live hedging.
We study the variance-optimal hedging of European contingent claims written on forwards. We assume that the dynamics of the underlying forward curves follow a Heath--Jarrow--Morton--Musiela stochastic partial differential equation modulated by an infinite-rank stochastic covariance component. The variance-optimal hedge is then given by the Galtchouk--Kunita--Watanabe projection with respect to some covariance-norm quotient generated by the forward curve martingale. We show density of finite-maturity and delivery-window strategies, convergence of spectral finite-rank hedge projections and an exact decomposition of the quadratic hedging error into bucket, rank and residual risk components. In enlarged filtrations, the residual risk is a stochastic-volatility floor for claims loading on non-traded covariance noise. We illustrate the hedging framework in affine stochastic covariance and multiplicative HJMM models, and give a concrete example of the decomposition in a CIR stochastic covariance model.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.