PyDerivatives 5.0

A Modern Toolkit for Option Pricing, Densities, and Econometric Analysis

PyDerivatives is an easy-to-use Python toolbox for option pricing and financial econometrics, with a particular emphasis on implementing state-of-the-art methodologies from the academic literature and making them readily accessible to researchers, practitioners, and data scientists.

The package is designed to provide a unified and extensible framework for estimating, analyzing, and visualizing option-implied objects across multiple asset classes.

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Core Capabilities

PyDerivatives supports full estimation and calibration of:

• Call price surfaces
• Implied volatility (IV) surfaces
• Risk-neutral density (RND) surfaces
• Pricing kernel surfaces
• Physical density (PD) surfaces

These objects can be constructed using a broad class of advanced option pricing models and nonparametric techniques, with a strong focus on robustness, flexibility, and empirical relevance.

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Installation

PyDerivatives can be installed directly from PyPI:

pip install pyderivatives
from pyderivatives import*

Pricing Kernel and Physical Density Estimation

PyDerivatives includes tools for estimating physical densities and pricing kernels, allowing researchers to study risk preferences, risk premia, and state dependence:

• Conditional pricing kernel estimation via exponential polynomials
  Schreindorfer, D., & Sichert, T. (2025).
  Conditional risk and the pricing kernel.
  Journal of Financial Economics, 171, 104106.

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Arbitrage Detection and Removal

The package provides functionality for enforcing static no-arbitrage conditions on option price surfaces:

• Static arbitrage detection and repair
  Cohen, S. N., Reisinger, C., & Wang, S. (2020).
  Detecting and repairing arbitrage in traded option prices.
  Applied Mathematical Finance, 27(5), 345–373.

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Econometric Toolbox

Wavelet Analysis

• Crowley, P. M. (2007).
  A guide to wavelets for economists.
  Journal of Economic Surveys, 21(2), 207–267.

  Quantile Time-Varying VAR (QTVP-VAR)

• Raza, S. A., Ahmed, M., & Ali, S. (2026).
  Untangling market links: A QVAR–TVP VAR analysis of precious metals and oil amid the pandemic.
  Journal of Futures Markets, 46(1), 101–120.

Quantile Regression Analysis

• Badshah, I., et al. (2016).
  Asymmetries of the intraday return–volatility relation.
  International Review of Financial Analysis, 48, 182–192.

Call Surface, Implied Volatility Surface, and Risk-Neutral Density Estimation

• Two-factor stochastic volatility with double-exponential jumps (Double Heston–Kou)
  Guohe, D. (2020). Option pricing under two-factor stochastic volatility jump-diffusion model.
  Complexity, Hindawi.

• Two-factor stochastic volatility model (Double Heston)
  Christoffersen, P., Heston, S., & Jacobs, K. (2009).
  The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well.
  Management Science, 55(12), 1914–1932.

• Stochastic volatility with double-exponential jumps (Heston–Kou)
  Ahlip, R., & Rutkowski, M. (2015).
  Semi-analytical pricing of currency options in the Heston/CIR jump-diffusion hybrid model.
  Applied Mathematical Finance, 22(1), 1–27.

• Jump-diffusion with double-exponential jumps (Kou model)
  Kou, S. G. (2002). A jump-diffusion model for option pricing.
  Management Science, 48(8), 1086–1101.

• Stochastic volatility with lognormal jumps (Bates model)
  Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options.
  Review of Financial Studies, 9(1), 69–107.

• Stochastic volatility model (Heston)
  Heston, S. L. (1993). A closed-form solution for options with stochastic volatility, with applications to bond and currency options.
  Review of Financial Studies, 6(2), 327–343.

• Black–Scholes model

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