Unlocking Explicit Moments for Affine Jump Diffusions

Description

ajdmom is a Python library for automatically deriving explicit, closed-form moment formulas for well-established Affine Jump Diffusion (AJD) processes. It significantly enhances the usability of AJD models by providing both unconditional moments and conditional moments, up to any positive integer order.

It also serves as a valuable tool for sensitivity analysis, providing partial derivatives of these moments with respect to model parameters. The package features a modular architecture, facilitating easy adaptation and extension by researchers. ajdmom is open-source and readily available for installation from the Python Package Index (PyPI):

pip install ajdmom

or GitHub:

pip install git+https://github.com/xmlongan/ajdmom

The moments derived by ajdmom have broad applications in quantitative finance and stochastic modeling, including:

• Density Approximation: Accurately approximating unknown probability densities (e.g., through Pearson distributions) by matching derived moments. This enables efficient European option pricing under the concerned models.

• Exact Simulation: Facilitating the exact simulation of AJD models in an efficient way when compared to characteristic function inversion methods.

• Parameter Estimation: Formulating explicit moment estimators for AJD models whose likelihood functions are not analytically solvable.

Consequently, ajdmom has the potential to become an essential instrument for researchers and practitioners demanding comprehensive AJD model analysis.

Supported Models & Moment Types

Notes:

• ✅ Implemented: The feature is fully implemented.
• ✔️ Applicable: The feature is applicable to this model but not yet implemented.
• N/A Not Applicable: The feature is not relevant or applicable for this model.
• Unconditional Moments: Include raw moments ($\mathbb{E}[y_n^l]$), central moments (), and autocovariances ().
  • Note: Autocovariances are not yet available for SRJD and SVCJ.
• Conditional Moments - I: Derivation where the initial state of the variance process ($v_0$) is given.
• Conditional moments - II: Derivation where both the initial state ($v_0$) and the realized jump times and jump sizes in the variance process over the concerned interval are given beforehand.

Simple Usage

To get the formula for the first moment $\mathbb{E}[y_n]$ for the Heston Stochastic Volatility (SV) model ( $y_n$ denotes the return over the nth interval of length $h$ ), run the following code snippet:

from ajdmom import mdl_1fsv # mdl_1fsv -> mdl_1fsvj, mdl_2fsv, mdl_2fsvj
from pprint import pprint

m1 = mdl_1fsv.moment_y(1)   # 1 in moment_y(1) -> 2,3,4...

# moment_y() -> cmoment_y()             : central moment
# dpoly(m1, wrt), wrt = 'k','theta',... : partial derivative

msg = "which is a Poly with attribute keyfor = \n{}"
print("moment_y(1) = "); pprint(m1); print(msg.format(m1.keyfor))

which produces:

moment_y(1) = 
{(0, 1, 0, 0, 1, 0, 0, 0): Fraction(-1, 2),
 (0, 1, 0, 1, 0, 0, 0, 0): Fraction(1, 1)}
which is a Poly with attribute keyfor = 
('e^{-kh}', 'h', 'k^{-}', 'mu', 'theta', 'sigma_v', 'rho', 'sqrt(1-rho^2)')

Within the produced results, the two key-value pairs, namely (0,1,0,0,1,0,0,0): Fraction(-1,2) and (0,1,0,1,0,0,0,0): Fraction(1,1), correspond to the following expressions:

$$ -\frac{1}{2}\times e^{-0kh}h^1k^{-0}\mu^0\theta^1\sigma_v^0\rho^0\left(\sqrt{1-\rho^2}\right)^0, $$

$$ 1\times e^{-0kh}h^1k^{-0}\mu^1\theta^0\sigma_v^0\rho^0\left(\sqrt{1-\rho^2}\right)^0, $$

respectively. The summation of these terms reproduces the first moment of the Heston SV model: $\mathbb{E}[y_n] = (\mu-\theta/2)h$. This demonstrates that the ajdmom package successfully encapsulates the model's dynamics into a computationally manipulable form, specifically leveraging a custom dictionary data structure, referred to as Poly, to encode the moment's expression. This structure allows ajdmom to perform symbolic differentiation and other advanced operations directly on the moment formulas.

Documentation

The documentation is hosted on http://www.yyschools.com/ajdmom/

Ongoing Development

This code is being developed on an on-going basis at the author's Github site.

Support

For support in using this software, submit an issue.