1FSVJ Model

In this subpackage (ajdmom.mdl_1fsvj), we consider the following SV model, which adds a jump component in the log price process of the Heston model:

\[\begin{split}d\log s(t) &= (\mu- v(t)/2) dt + \sqrt{v(t)}dw^s(t) + dz(t),\\ dv(t) &= k(\theta - v(t))dt + \sigma_v \sqrt{v(t)}dw^v(t),\end{split}\]

where \(z(t)\) is a CPP with a constant arrival rate \(\lambda\) and jump distribution \(F_j(\cdot,\boldsymbol{\theta}_j)\) with parameter \(\boldsymbol{\theta}_j\). We take normal distribution with mean \(\mu_j\) and variance \(\sigma_j^2\) as an example of \(F_j(\cdot,\boldsymbol{\theta}_j)\). For this model,

\[y_n = y_{o,n} + J_n,\]

where

\[\begin{split}y_{o,n} &\triangleq \mu h - \frac{1}{2}IV_{n} + \rho I_n + \sqrt{1-\rho^2}I_n^{*}, \\ J_n &\triangleq z(nh) - z((n-1)h) = \sum_{i=N((n-1)h)+1}^{N(nh)}j_i,\end{split}\]

where \(N(t)\) is a Poisson process with rate \(\lambda\), \(j_i\sim\mathcal{N}(\mu_j,\sigma_j^2)\).

Moments

Moments and Central Moments

\[\begin{split}E[\overline{y}_{n}^l] &= E[(\overline{y}_{o,n} + \overline{J}_n)^l] = \sum_{i=0}^{l} C_l^i E[\overline{y}_{o,n}^i]E[\overline{J}_n^{l-i}],\\ E[y_n^l] &= E[(y_{o,n} + J_n)^l] = \sum_{i=0}^{l} C_l^i E[y_{o,n}^i] E[J_n^{l-i}].\end{split}\]

Functions moment_y() and cmoment_y() can be used to compute \(E[y_{o,n}^i]\) and \(E[\overline{y}_{o,n}^i]\), respectively. Meanwhile, functions mcpp() and cmcpp() can be used to compute \(E[J_n^{l-i}]\) and \(E[\overline{J}_n^{l-i}]\), respectively.

Covariances

\[cov(y_n^{l_1}, y_{n+1}^{l_2}) = E[y_n^{l_1}y_{n+1}^{l_2}] - E[y_n^{l_1}]E[y_{n+1}^{l_2}]\]

which reduces to

\[\begin{split}&E[y_n^{l_1}y_{n+1}^{l_2}]\\ &= \sum_{i=0}^{l_2}C_{l_2}^i E[y_n^{l_1}y_{o,n+1}^i]E[J_{n+1}^{l_2-i}]\\ &= \sum_{i=0}^{l_2}C_{l_2}^i \sum_{j=0}^{l_1}C_{l_1}^j E[y_{o,n}^jy_{o,n+1}^i] E[J_n^{l_1-j}]E[J_{n+1}^{l_2-i}].\end{split}\]

Function moment_yy() in module ajdmom.mdl_1fsv.cov can be used to compute \(E[y_{o,n}^jy_{o,n+1}^i]\).

In summary, I defined

1. moment_y() for moment \(E[y_n^l]\).

2. cmoment_y() for central moment \(E[\overline{y}_{n}^l]\).

3. cov_yy() for covariance \(cov(y_n^{l_1}, y_{n+1}^{l_2})\).

API

ajdmom.mdl_1fsvj.cmom	Central Moments for One-Factor SV with jumps
ajdmom.mdl_1fsvj.mom	Moments for One-Factor SV with jumps
ajdmom.mdl_1fsvj.cov	Covariance for One-Factor SV with jumps
ajdmom.mdl_1fsvj.euler	Module for generating a trajectory of samples from mdl_1fsvj by Euler approximation