User Guide

Installing ajdmom

Install ajdmom from the Python Package Index through Command Prompt (cmd.exe) or Anaconda Prompt, depending on how Python is installed in your Windows OS:

pip install ajdmom

The package can be installed similarly through Terminal if you are working on a Linux or macOS.

Alternatively, you can install directly from the GitHub repository:

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

Heston SV model

The most well-known example of AJD model is probably the Heston SV model which serves here as a demonstration example. In the ajdmom package, it is treated as the baseline model and described by the following SDEs [1] ,

\[\begin{split}ds(t) &= \mu s(t)dt + \sqrt{v(t)}s(t)dw^s(t),\\ dv(t) &= k(\theta - v(t))dt + \sigma_v\sqrt{v(t)}dw^v(t),\end{split}\]

where \(s(t)\) is the asset price at time \(t\), and refer to the Theory page for details. The return \(y_i\) over the i-th interval of length \(h\) is defined as,

\[y_i \triangleq \log s(ih) - \log s((i-1)h).\]

The moment derivation for the Heston SV model is implemented in the mdl_1fsv subpackage of the ajdmom package.

Formula Derivation

The moment and covariance formulas are encoded in objects of class Poly which is a customized dictionary data structure derived from the UserDict class in the Python Standard Library collections.

Moment formulas

To get the formula for the first moment \(E[y_n]\):

>>> 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))
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)')

where the two key-value pairs within the returned poly of moment_y(1) stand for

\[\begin{split}-\frac{1}{2}\times & e^{-0kh}h^1k^{-0}\mu^0\theta^1\sigma_v^0\rho^0 \left(\sqrt{1-\rho^2}\right)^0,\\ 1\times & e^{-0kh}h^1k^{-0}\mu^1\theta^0\sigma_v^0\rho^0 \left(\sqrt{1-\rho^2}\right)^0,\end{split}\]

respectively. Adding together these two terms gives the first moment of the Heston SV model, i.e., \(E[y_n] = (\mu-\theta/2)h\).

Covariance formulas

The covariances considered in this package are that between \(y_n\) and its lag-1 counterpart \(y_{n+1}\) with orders \((l_1,l_2)\), i.e., \(cov(y_n^{l_1}, y_{n+1}^{l_2})\).

To derive the formula for the covariance \(cov(y_n^2,y_{n+1})\):

>>> from ajdmom import mdl_1fsv # mdl_1fsv -> mdl_1fsvj, mdl_2fsv, mdl_2fsvj
>>> from pprint import pprint
>>>
>>> cov21 = mdl_1fsv.cov_yy(2,1) # (2,1) -> (1,1), (1,2), (2,2), (3,2), ...
>>>
>>> # dpoly(cov21, wrt), wrt = 'k','theta',... : partial derivative
>>>
>>> msg = "which is a Poly with attribute keyfor =\n{}"
>>> print("cov_yy(2,1) = "); pprint(cov21); print(msg.format(cov21.keyfor))
cov_yy(2,1) =
{(0, 0, 3, 0, 1, 2, 0, 2): Fraction(-1, 4),
 (0, 0, 3, 0, 1, 2, 2, 0): Fraction(-5, 4),
 (0, 0, 4, 0, 1, 3, 1, 0): Fraction(3, 4),
 (0, 0, 5, 0, 1, 4, 0, 0): Fraction(-1, 8),
 (0, 1, 2, 0, 2, 1, 1, 0): Fraction(1, 2),
 (0, 1, 2, 1, 1, 1, 1, 0): Fraction(-1, 1),
 (0, 1, 3, 0, 2, 2, 0, 0): Fraction(-1, 8),
 (0, 1, 3, 1, 1, 2, 0, 0): Fraction(1, 4),
 (1, 0, 3, 0, 1, 2, 0, 2): Fraction(1, 2),
 (1, 0, 3, 0, 1, 2, 2, 0): Fraction(5, 2),
 (1, 0, 4, 0, 1, 3, 1, 0): Fraction(-3, 2),
 (1, 0, 5, 0, 1, 4, 0, 0): Fraction(1, 4),
 (1, 1, 2, 0, 1, 2, 2, 0): Fraction(1, 1),
 (1, 1, 2, 0, 2, 1, 1, 0): Fraction(-1, 1),
 (1, 1, 2, 1, 1, 1, 1, 0): Fraction(2, 1),
 (1, 1, 3, 0, 1, 3, 1, 0): Fraction(-3, 4),
 (1, 1, 3, 0, 2, 2, 0, 0): Fraction(1, 4),
 (1, 1, 3, 1, 1, 2, 0, 0): Fraction(-1, 2),
 (1, 1, 4, 0, 1, 4, 0, 0): Fraction(1, 8),
 (2, 0, 3, 0, 1, 2, 0, 2): Fraction(-1, 4),
 (2, 0, 3, 0, 1, 2, 2, 0): Fraction(-5, 4),
 (2, 0, 4, 0, 1, 3, 1, 0): Fraction(3, 4),
 (2, 0, 5, 0, 1, 4, 0, 0): Fraction(-1, 8),
 (2, 1, 2, 0, 1, 2, 2, 0): Fraction(-1, 1),
 (2, 1, 2, 0, 2, 1, 1, 0): Fraction(1, 2),
 (2, 1, 2, 1, 1, 1, 1, 0): Fraction(-1, 1),
 (2, 1, 3, 0, 1, 3, 1, 0): Fraction(3, 4),
 (2, 1, 3, 0, 2, 2, 0, 0): Fraction(-1, 8),
 (2, 1, 3, 1, 1, 2, 0, 0): Fraction(1, 4),
 (2, 1, 4, 0, 1, 4, 0, 0): Fraction(-1, 8)}
which is a Poly with attribute keyfor =
('e^{-kh}', 'h', 'k^{-}', 'mu', 'theta', 'sigma_v', 'rho', 'sqrt(1-rho^2)')

Moment Values

Given an exact set of parameter values, values of the central moments, moments and covariances, and their partial derivatives w.r.t. a parameter can also be computed.

Moments and Central Moments

To compute the exact value of the third moment \(E[y_n^3]\), given \((\mu=0.125, k=0.1, \theta=0.25, \sigma_v=0.1, \rho=-0.7, h=1)\):

>>> ## Moments and Central Moments
>>> from ajdmom.mdl_1fsv.mom import m, dm       # for moments
>>> from ajdmom.mdl_1fsv.cmom import cm, dcm    # for central moments
>>>
>>> parameters = {'mu':0.125, 'k':0.1, 'theta':0.25,
...   'sigma_v':0.1, 'rho':-0.7, 'h': 1}
>>>
>>> # 3rd moment as an example
>>> moment = m(l=3, par=parameters)             #  cm: central moment
>>> # partial derivative w.r.t. parameter 'k'
>>> dmoment = dm(l=3, par=parameters, wrt='k')  # dcm: central moment
>>> moment
-0.04489260315929133
>>> dmoment
0.20556366585696395

Covariances

To compute the exact value of covariance \(cov(y_n^2, y_{n+1}^2)\), given \((\mu=0.125, k=0.1, \theta=0.25, \sigma_v=0.1, \rho=-0.7, h=1)\):

>>> ## Covariance
>>> from ajdmom.mdl_1fsv.cov import cov, dcov
>>>
>>> parameters = {'mu':0.125, 'k':0.1, 'theta':0.25,
...   'sigma_v':0.1, 'rho':-0.7, 'h': 1}
>>>
>>> # covariance cov(y_n^2, y_{n+1}^2) as an example
>>> covariance = cov(l1=2, l2=2, par=parameters)
>>> # partial derivative w.r.t. parameter 'k'
>>> dcovariance = dcov(l1=2, l2=2, par=parameters, wrt='k')
>>> covariance
0.0149529894520537
>>> dcovariance
-0.15904979864793667

AJD Extensions

In addition to the Heston SV model, there are several extensions, which are summarized in the following table:

Model	Description
mdl_1fsv	baseline model, i.e., the Heston SV model refers to Theory or 1FSV Model
mdl_1fsvj	with jumps in the return process of the model mdl_1fsv refers to 1FSVJ Model
mdl_2fsv	with volatility consisting of superposition of two SRDs refers to 2FSV Model
mdl_2fsvj	with jumps in the return process of the model mdl_2fsv refers to 2FSVJ Model
mdl_svvj	with jumps in the variance of the Heston model refers to SVVJ Model
mdl_svij	with independent jumps in the price and variance of Heston refers to SVIJ Model
mdl_svcj	with contemporaneous jumps in the price and variance of Heston refers to SVCJ Model
mdl_srjd	Square-Root Jump Diffusion refers to SRJD Model

Notes: SRD is short for Square-Root Diffusion.

Besides unconditional moment derivation, the ajdmom package also supports

• conditional moment derivation with the initial state (\(v_0\)) of the variance process given beforehand,

• conditional moment derivation with both the initial state (\(v_0\)) and the realized jump times and jump sizes of the variance process given beforehand, for those models having the corresponding jump component.

Most of the derivations have already been implemented within the ajdmom package. Please refer to the individual subpackages for details. Under some cases, you probably need to implement by yourself with additional efforts for untypical models.

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[1]

Whose exact equations vary according to different authors. One simplified setting is \(dp(t) = \mu dt + \sqrt{v(t)}dw^s(t)\) where \(p(t) = \log s(t)\) while all other settings keep as the same. \(v(t)\) is the instantaneous return variance at time \(t\), and \(w^s(t)\) and \(w^v(t)\) are two Wiener processes with correlation \(\rho\). In order to make sure \(v(t) >0\) for \(t>0\), it is required that the parameters \(k>0,\theta>0,\sigma_v>0\) and satisfy \(\sigma_v^2 \leq 2k\theta\), along with an initial \(v(0)>0\).