Paper: arXiv 2411.06484

Authors: Yan-Feng Wu, Jian-Qiang Hu

Abstract

We introduce ajdmom, a Python package designed for automatically deriving moment formulae for the well-established affine jump diffusion processes with state-independent jump intensities. ajdmom can produce explicit closed-form expressions for conditional and unconditional moments of any order, significantly enhancing the usability of these models. Additionally, ajdmom can compute partial derivatives of these moments with respect to the model parameters, offering a valuable tool for sensitivity analysis. The package’s modular architecture makes it easy for adaptation and extension by researchers. ajdmom is open-source and readily available for installation from GitHub or the Python package index (PyPI).

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly useful, mathematically sophisticated symbolic computation package for automated moment derivation in affine jump-diffusion models. While theoretical completeness and software reproducibility are outstanding, empirical application is limited to software validation rather than real-world quantitative trading backtests or econometric estimations.

Research Flowchart

  flowchart TD
    A[Goal: Automate Moment Formulas & Derivatives for Affine Jump Diffusion Processes] --> B[Inputs: Model Parameters & State-Independent Jump Intensities]
    B --> C[Modular Python Engine: Symbolic/Algorithmic Moment Derivation]
    C --> D[Computational Engine: Differentiation & Conditioning Modules]
    D --> E[Outcome 1: Explicit Closed-Form Conditional & Unconditional Moments]
    D --> F[Outcome 2: Exact Partial Derivatives for Sensitivity Analysis]
    E & F --> G[Deliverable: Open-Source 'ajdmom' Python Package on PyPI/GitHub]

Cited References (78)

  1. Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes (2006)
  2. Estimating Stochastic Volatility Diffusion Using Conditional Moments of Integrated Volatility (2002)
  3. Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach (2002)
  4. Exact and Computationally Efficient Likelihood-Based Estimation for Discretely Observed Diffusion Processes (with Discussion) (2006)
  5. Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics (2004)
  6. Monte Carlo Maximum Likelihood Estimation for Discretely Observed Diffusion Processes (2009)
  7. MCMC Analysis of Diffusion Models with Application to Finance (2001)
  8. Non-Gaussian Ornstein–Uhlenbeck-Based Models and Some of Their Uses in Financial Economics (2001)
and 70 more
  1. Specification Analysis of Affine Term Structure Models (2000)
  2. Simulated Moments Estimation of Markov Models of Asset Prices (1993)
  3. Transform Analysis and Asset Pricing for Affine Jump-Diffusions (2000)
  4. Which Moments to Match? (1996)
  5. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993)
  6. Realized Jumps on Financial Markets and Predicting Credit Spreads (2011)
  7. Moment-Based Estimation of Stochastic Volatility (2013)
  8. The Impact of Jumps in Volatility and Returns (2003)
  9. MCMC Methods for Continuous-Time Financial Econometrics (2010)
  10. Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study (1999) — link
  11. Estimation of Affine Asset Pricing Models Using the Empirical Characteristic Function (2001)
  12. Estimation of Continuous-Time Processes via the Empirical Characteristic Function (2002)
  13. Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes (1995)
  14. A Method for Calculating Bounds on the Asymptotic Covariance Matrices of Generalized Method of Moments Estimators (1985)
  15. Estimation of Objective and Risk-Neutral Distributions Based on Moments of Integrated Volatility (2011)
  16. Exact Simulation of Diffusions (2005)
  17. Retrospective Exact Simulation of Diffusion Sample Paths with Applications (2006)
  18. Econometric Analysis of Jump-Driven Stochastic Volatility Models (2011)
  19. Moment-Based Estimation of Stochastic Volatility Models (2009)
  20. Simulation-Based Method of Moments and Efficiency (2002)
  21. Stochastic Calculus for Finance II: Continuous-Time Models (2004)
  22. A Theory of the Term Structure of Interest Rates (1985)
  23. Estimation in the Cox-Ingersoll-Ross Model (1997)
  24. Asymptotic Statistics (2000)
  25. Risk-Neutral Moment-Aased Estimation of Affine Option Pricing Models (2018) — link
  26. Spectral GMM Estimation of Continuous-Time Processes (2003)
  27. Gradient-Based Simulated Maximum Likelihood Estimation for Stochastic Volatility Models Using Characteristic Functions (2016) — link
  28. Gradient-Based Simulated Maximum Likelihood Estimation for Lévy-Driven Ornstein–Uhlenbeck Stochastic Volatility Models (2014) — link
  29. Moment Estimators for the Parameters of Ornstein-Uhlenbeck Processes Driven by Compound Poisson Processes (2019)
  30. Moment Estimators for Parameters of Lévy-driven Ornstein–Uhlenbeck Processes (2022) — link
  31. Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions (2005)
  32. Some Limit Theorems for Stationary Processes (1962)
  33. Time Series: Theory and Methods (1991)
  34. MCMC for Estimating Heston Stochastic Volatility Models (2023)
  35. Method of Moments Estimation for Affine Stochastic Volatility Models (2024) — link
  36. Method of moments estimation for the superposition of square-root diffusions (2025)
  37. \pkg SymPy : Symbolic Computing in \proglang Python (2017)
  38. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options (1996)
  39. Model Specification and Risk Premia: Evidence from Futures Options (2007)
  40. \pkg ajdmom : a \proglang Python Package for Deriving Moments of Affine Jump Diffusion Processes (2023)
  41. Unified Moment-Based Modeling of Integrated Stochastic Processes (2024)
  42. Counterexamples in Probability (2013)
  43. Method of Moments Estimation for L 'e vy-Driven Ornstein-Uhlenbeck Stochastic Volatility Models (2021)
  44. Exact Simulation of Point Processes with Stochastic Intensities (2011)
  45. Efficient Simulation of Clustering Jumps with CIR Intensity (2017)
  46. An Inventory Model Where Customer Demand is Dependent on a Stochastic Price Process (2019)
  47. Bayesian Forecasting of an Inhomogeneous Poisson Process with Applications to Call Center Data (2007)
  48. Forecasting Time Series of Inhomogeneous Poisson Processes with Application to Call Center Workforce Management (2008)
  49. Simulation Schemes for the Heston Model with Poisson Conditioning — → 2301.02800
  50. Affine Point Processes: Approximation and Efficient Simulation (2015)
  51. Simulating the Cox-Ingersoll-Ross and Heston Processes: Matching the First Four Moments (2022)
  52. Sphinx 4.0.0+ Documentation (2020) — link
  53. MATLAB version: 9.13.0 (R2022b) (2022) — link
  54. Python 3 Reference Manual (2009)
  55. Mathematica, V ersion 14.1 (2024) — link
  56. R: A Language and Environment for Statistical Computing (2024) — link
  57. The C++ programming language (2013)
  58. Julia: A fresh approach to numerical computing (2017) — link
  59. \pkg PROJ_Option_Pricing_Matlab : Option Pricing PROJ Method (Exotic/Vanilla Options) (2023) — link
  60. \pkg DerivativesPricing : Derivatives Pricing and Optimisation (2020) — link
  61. \pkg Jdmbs : An \proglang R Package for Monte Carlo Option Pricing Algorithms for Jump Diffusion Models with Correlational Companies (2020) — link
  62. \pkg jumpdiff : A \proglang Python Library for Statistical Inference of Jump-Diffusion Processes in Observational or Experimental Data Sets (2023)
  63. Arbitrary-Order Finite-Time Corrections for the Kramers–Moyal Operator (2021)
  64. \pkg kramersmoyal : Kramers–Moyal Coefficients for Stochastic Processes (2019) — link
  65. The Langevin Approach: An \proglang R Package for Modeling Markov Processes (2016)
  66. Extending \pkg JumpProcesses.jl for fast point process simulation with time-varying intensities (2024) — link
  67. \pkg DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in \proglang Julia (2017) — link
  68. The Pricing of Options and Corporate Liabilities (1973)
  69. Density Approximation of Affine Jump Diffusions via Closed-Form Moment Matching — → 2504.06942
  70. Polynomial processes and their applications to mathematical finance (2012)

2 of 78 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.