On the relative performance of some parametric and nonparametric estimators of option prices

Paper: arXiv 2412.00135

Abstract

We examine the empirical performance of some parametric and nonparametric estimators of prices of options with a fixed time to maturity, focusing on variance-gamma and Heston models on one side, and on expansions in Hermite functions on the other side. The latter class of estimators can be seen as perturbations of the classical Black-Scholes model. The comparison between parametric and Hermite-based models having the same “degrees of freedom” is emphasized. The main criterion is the out-of-sample relative pricing error on a dataset of historical option prices on the S&P500 index. Prior to the main empirical study, the approximation of variance-gamma and Heston densities by series of Hermite functions is studied, providing explicit expressions for the coefficients of the expansion in the former case, and integral expressions involving the explicit characteristic function in the latter case. Moreover, these approximations are investigated numerically on a few test cases, indicating that expansions in Hermite functions with few terms achieve competitive accuracy in the estimation of Heston densities and the pricing of (European) options, but they perform less effectively with variance-gamma densities. On the other hand, the main large-scale empirical study show that parsimonious Hermite estimators can even outperform the Heston model in terms of pricing errors. These results underscore the trade-offs inherent in model selection and calibration, and their empirical fit in practical applications.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.2/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced functional analysis (Hermite function expansions in L2, stochastic calculus) and explicit derivations, yet validates these with a large-scale empirical study on historical S&P500 options, emphasizing out-of-sample pricing error metrics.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Compare Parametric vs.<br>Nonparametric Option Pricing<br>Estimators"] --> B{"Key Methodology Steps"}
  B --> B1["Analytic Density Approximation<br>Variance-Gamma & Heston<br>by Hermite Functions"]
  B --> B2["Main Empirical Study<br>Out-of-sample Relative<br>Pricing Errors"]
  B --> C["Dataset<br>Historical Option Prices<br>S&P 500 Index"]
  B --> D["Computational Processes<br>Calibration & Pricing"]
  B1 --> D
  B2 --> D
  C --> D
  D --> E{"Key Findings/Outcomes"}
  E --> E1["Hermite Expansions<br>Competitive w/ Heston<br>for Fewer Terms"]
  E --> E2["Variance-Gamma<br>Less Effective with<br>Hermite Approximation"]
  E --> E3["Empirical Results<br>Parsimonious Hermite Estimators<br>Can Outperform Heston"]

Cited References (14)

1. Nonparametric estimates of option prices via Hermite basis functions (2022) — link
2. Polynomial approximation of discounted moments (2021) — link
3. On certain representations of pricing functionals (2021) — link
4. On the properties of hermite series based distribution function estimators (2020) — link
5. Option pricing with orthogonal polynomial expansions (2017) — link
6. Optimal Fourier Inversion in Semi-Analytical Option Pricing (2007) — link
7. NIST Digital Library of Mathematical Functions (2003) — link
8. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993) — link

0 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.