Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration

ArXiv ID: 2501.05975 “View on arXiv”

Authors: Unknown

Abstract

In energy markets, joint historical and implied calibration is of paramount importance for practitioners yet notoriously challenging due to the need to align historical correlations of futures contracts with implied volatility smiles from the option market. We address this crucial problem with a parsimonious multiplicative multi-factor Heath-Jarrow-Morton (HJM) model for forward curves, combined with a stochastic volatility factor coming from the Lifted Heston model. We develop a sequential fast calibration procedure leveraging the Kemna-Vorst approximation of futures contracts: (i) historical correlations and the Variance Swap (VS) volatility term structure are captured through Level, Slope, and Curvature factors, (ii) the VS volatility term structure can then be corrected for a perfect match via a fixed-point algorithm, (iii) implied volatility smiles are calibrated using Fourier-based techniques. Our model displays remarkable joint historical and implied calibration fits - to both German power and TTF gas markets - and enables realistic interpolation within the implied volatility hypercube.

Keywords: Heath-Jarrow-Morton (HJM) Model, Stochastic Volatility, Forward Curves, Implied Volatility Smiles, Energy Markets, Energy

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail
  • Why: The paper deploys highly advanced mathematical constructs including multiplicative HJM, Lifted Heston, fixed-point algorithms, and Fourier techniques, establishing high math complexity. It demonstrates strong empirical rigor through joint calibration to real market data (German power and TTF gas), explicit historical covariance estimation, and validation of model approximations, though it stops short of presenting live trading results.
  flowchart TD
    A["Research Goal<br>Joint Historical & Implied Calibration<br>in Energy Markets"] --> B["Methodology<br>Multiplicative HJM + Lifted Heston"]
    B --> C["Key Data Inputs<br>German Power & TTF Gas Futures/Options"]
    C --> D["Step 1: Historical Calibration<br>Level, Slope, Curvature Factors<br>via Kemna-Vorst Approx."]
    D --> E["Step 2: VS Volatility Correction<br>Fixed-point algorithm for perfect match"]
    E --> F["Step 3: Implied Smile Calibration<br>Fourier-based techniques"]
    F --> G{"Key Findings & Outcomes"}
    G --> G1["Remarkable joint fits"]
    G --> G2["Realistic implied volatility<br>hypercube interpolation"]