Paper: arXiv 2501.05975

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: 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.

Research Flowchart

  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"]

Cited References (42)

  1. State Spaces of Multifactor Approximations of Nonnegative Volterra Processes — → 2412.17526
  2. A Heath–Jarrow–Morton framework for energy markets: review and applications for practitioners (2024) — link
  3. Gaussian Volterra Processes as Models of Electricity Markets (2023) — link
  4. An infinite‐dimensional affine stochastic volatility model (2021) — link
  5. The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach (2021) — link
  6. A survey of electricity spot and futures price models for risk management applications (2021) — link
  7. Infinite dimensional pathwise Volterra processes driven by Gaussian noise – Probabilistic properties and applications – (2020) — link
  8. Capturing the power options smile by an additive two-factor model for overlapping futures prices (2019) — link
and 34 more
  1. Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework (2019) — link
  2. Lifting the Heston model (2018) — link
  3. A Multifactor Stochastic Volatility Model of Commodity Prices (2017) — link
  4. The Heston stochastic volatility model in Hilbert space (2017) — link
  5. A space-time random field model for electricity forward prices (2017) — link
  6. Modelling electricity futures prices using seasonal path-dependent volatility (2016) — link
  7. Calibration of a multifactor model for the forward markets of several commodities (2013) — link
  8. Arbitrage-free SVI volatility surfaces (2012) — link
  9. Markov models for commodity futures: theory and practice (2010) — link
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  31. Appendix 1. Estimation of historical covariances Given two rolling futures contracts’
  32. iii) a stochastic volatility component coming from the lifted Heston model with three time-scales to match the implied volatility
  33. An exact calibration correction of the term structure of the implied VS volatility not captured in step 1)
  34. Estimation of the number of factors in a multi-factorial Heath–Jarrow–Morton model in power markets

1 of 42 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.