Joint calibration of the volatility surface and variance term structure

Paper: arXiv 2509.08096

Authors: Jiwook Yoo

Abstract

This article proposes a calibration framework for complex option pricing models that jointly fits market option prices and the term structure of variance. Calibrated models under the conventional objective function, the sum of squared errors in Black-Scholes implied volatilities, can produce model-implied variance term structures with large errors relative to those observed in the market and implied by option prices. I show that this can occur even when the model-implied volatility surface closely matches the volatility surface observed in the market. The proposed joint calibration addresses this issue by augmenting the conventional objective function with a penalty term for large deviations from the observed variance term structure. This augmented objective function features a hyperparameter that governs the relative weight placed on the volatility surface and the variance term structure. I test this framework on a jump-diffusion model with stochastic volatility in two calibration exercises: the first using volatility surfaces generated under a Bates model, and the second using a panel of S&P 500 equity index options covering the 1996-2023 period. I demonstrate that the proposed method is able to fit observed option prices well while delivering realistic term structures of variance. Finally, I provide guidance on the choice of hyperparameters based on the results of these numerical exercises.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a well-motivated calibration framework that bridges the gap between matching option prices and preserving realistic term structures of variance. It combines solid continuous-time stochastic calculus foundations with extensive empirical validation using a 1996–2023 S&P 500 option dataset.

Research Flowchart

  flowchart TD
    A[Research Goal: Jointly calibrate volatility surface & variance term structure] --> B[Input Data: S&P 500 Options 1996-2023 & Synthetic Bates Surfaces]
    B --> C[Augment Objective Function: BS Implied Volatility SSE + Variance Term Structure Penalty]
    C --> D[Calibrate Stochastic Volatility Jump-Diffusion Model via Hyperparameter Optimization]
    D --> E[Outcome: Accurate Option Price Fits with Realistic Variance Term Structures]
    D --> F[Outcome: Hyperparameter Selection Guidance for Complex Pricing Models]

Cited References (273)

1. Common factors affecting bond returns (1991)
2. Anticipating jumps: Decomposition of straddle price (2023)
3. Volatility surface calibration to illiquid options (2019)
4. Pricing and hedging with smiles (1997)
5. A closed-form GARCH option valuation model (2000)
6. Just what you need to know about variance swaps (2005)
7. Sequential calibration of options (2008)
8. Which moments to match? (1996)

0 of 273 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.