# Unlocking Explicit Moments for Affine Jump Diffusions
[](http://yyschools.com/ajdmom/)
[](https://pypi.org/project/ajdmom/)
[](https://github.com/xmlongan/ajdmom/blob/main/LICENSE)
## 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
| Model | Unconditional Moments | Conditional Moments - I | Conditional Moments - II |
|:-----:|:---------------------:|:-----------------------:|:------------------------:|
| Heston| ✅ | ✔️ | N/A |
| 1FSVJ | ✅ | ✔️ | N/A |
| 2FSV | ✅ | ✔️ | N/A |
| 2FSVJ | ✅ | ✔️ | N/A |
| SRJD | ✅ | ✅ | ✅ |
| SVVJ | ✅ | ✅ | ✅ |
| SVCJ | ✅ | ✅ | ✅ |
| SVIJ | ✔️ | ✔️ | ✅ |
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:
```python
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:
```python
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](https://github.com/xmlongan/ajdmom).
## Support
For support in using this software, submit an [issue](https://github.com/xmlongan/ajdmom/issues/new).
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"description": "# Unlocking Explicit Moments for Affine Jump Diffusions\n\n[](http://yyschools.com/ajdmom/)\n[](https://pypi.org/project/ajdmom/)\n[](https://github.com/xmlongan/ajdmom/blob/main/LICENSE)\n\n\n## Description\n\n`ajdmom` is a Python library for **automatically deriving explicit, closed-form\nmoment formulas** for well-established Affine Jump Diffusion (AJD) processes. \nIt significantly enhances the usability of AJD models by providing both \n**unconditional moments** and **conditional moments**, up to any positive integer\norder.\n\nIt also serves as a valuable tool for sensitivity analysis, **providing partial\nderivatives** of these moments with respect to model parameters. The package \nfeatures a **modular architecture**, facilitating easy adaptation and extension \nby researchers. `ajdmom` is open-source and readily available for installation \nfrom the Python Package Index (PyPI):\n```\npip install ajdmom\n```\nor GitHub:\n```\npip install git+https://github.com/xmlongan/ajdmom\n```\n\nThe moments derived by `ajdmom` have **broad applications** in quantitative finance \nand stochastic modeling, including:\n\n- **Density Approximation**: Accurately approximating unknown probability densities \n (e.g., through Pearson distributions) by matching derived moments. This enables \n efficient European option pricing under the concerned models.\n\n- **Exact Simulation**: Facilitating the exact simulation of AJD models in an \n efficient way when compared to characteristic function inversion methods.\n\n- **Parameter Estimation**: Formulating explicit moment estimators for AJD models \n whose likelihood functions are not analytically solvable.\n\nConsequently, `ajdmom` has the potential to become an essential instrument for \nresearchers and practitioners demanding comprehensive AJD model analysis.\n\n### Supported Models & Moment Types\n\n| Model | Unconditional Moments | Conditional Moments - I | Conditional Moments - II |\n|:-----:|:---------------------:|:-----------------------:|:------------------------:|\n| Heston| \u2705 | \u2714\ufe0f | N/A |\n| 1FSVJ | \u2705 | \u2714\ufe0f | N/A |\n| 2FSV | \u2705 | \u2714\ufe0f | N/A |\n| 2FSVJ | \u2705 | \u2714\ufe0f | N/A |\n| SRJD | \u2705 | \u2705 | \u2705 |\n| SVVJ | \u2705 | \u2705 | \u2705 |\n| SVCJ | \u2705 | \u2705 | \u2705 |\n| SVIJ | \u2714\ufe0f | \u2714\ufe0f | \u2705 |\n\nNotes: \n\n- \u2705 **Implemented:** The feature is fully implemented.\n- \u2714\ufe0f **Applicable:** The feature is applicable to this model but not yet implemented. \n- **N/A Not Applicable:** The feature is not relevant or applicable for this model. \n- **Unconditional Moments:** Include raw moments ($\\mathbb{E}[y_n^l]$), \n central moments (), \n and autocovariances\n ()). \n - *Note: Autocovariances are not yet available for SRJD and SVCJ.*\n- **Conditional Moments - I:** Derivation where the initial state of the variance \n process ($v_0$) is given.\n- **Conditional moments - II:** Derivation where both the initial state ($v_0$) and \n the realized jump times and jump sizes in the variance process over the concerned\n interval are given beforehand.\n\n## Simple Usage\n\nTo get the formula for the first moment $\\mathbb{E}[y_n]$ for the Heston Stochastic\nVolatility (SV) model ( $y_n$ denotes the return over the nth interval of length $h$ ), \nrun the following code snippet:\n\n```python\nfrom ajdmom import mdl_1fsv # mdl_1fsv -> mdl_1fsvj, mdl_2fsv, mdl_2fsvj\nfrom pprint import pprint\n\nm1 = mdl_1fsv.moment_y(1) # 1 in moment_y(1) -> 2,3,4...\n\n# moment_y() -> cmoment_y() : central moment\n# dpoly(m1, wrt), wrt = 'k','theta',... : partial derivative\n\nmsg = \"which is a Poly with attribute keyfor = \\n{}\"\nprint(\"moment_y(1) = \"); pprint(m1); print(msg.format(m1.keyfor))\n```\n\nwhich produces:\n\n```python\nmoment_y(1) = \n{(0, 1, 0, 0, 1, 0, 0, 0): Fraction(-1, 2),\n (0, 1, 0, 1, 0, 0, 0, 0): Fraction(1, 1)}\nwhich is a Poly with attribute keyfor = \n('e^{-kh}', 'h', 'k^{-}', 'mu', 'theta', 'sigma_v', 'rho', 'sqrt(1-rho^2)')\n```\n\nWithin the produced results, the two key-value pairs, namely (0,1,0,0,1,0,0,0): Fraction(-1,2) and \n(0,1,0,1,0,0,0,0): Fraction(1,1), correspond to the following expressions:\n\n$$\n-\\frac{1}{2}\\times e^{-0kh}h^1k^{-0}\\mu^0\\theta^1\\sigma_v^0\\rho^0\\left(\\sqrt{1-\\rho^2}\\right)^0,\n$$\n\n$$\n1\\times e^{-0kh}h^1k^{-0}\\mu^1\\theta^0\\sigma_v^0\\rho^0\\left(\\sqrt{1-\\rho^2}\\right)^0,\n$$\n\nrespectively. 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