SVIJ Model

In this subpackage (ajdmom.mdl_svij), we consider the following SV model, which adds independent jump components in the price and variance of the Heston model:

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

where

• \(z^v(t)\) is a CPP with constant arrival rate \(\lambda_v\) and jumps distributed according to an exponential distribution with scale parameter \(\mu_v\) (= 1/rate), i.e., \(J_i^v \sim \text{exp}(\mu_v)\) (which will help to make sure the variance always be non-negative, i.e., \(v(t) \ge 0, \forall t\ge 0\)).

• \(z^s(t)\) is another CPP (independent of \(z^v(t)\)) with constant arrival rate \(\lambda_s\) and jumps distributed according to a normal distribution with mean \(\mu_s\) and variance \(\sigma_s^2\), i.e., \(J_i^s \sim \mathcal{N}(\mu_s, \sigma_s^2)\).

Define \(y_t \triangleq \log s(t) - \log s(0)\), and \(I\!Z_t^s\triangleq \int_0^t dz^s(u)\). Then, we have

\[y_t = y_{svvj,t} + I\!Z_t^s,\]

where \(y_{svvj,t}\) denotes the yield \(y_t\) in Equation (1) from the SVVJ model.

For models including jumps in the variance, it seems that only conditional moments and conditional central moments (given \(v_0, z^v(u), 0\le u \le t\)) can be derived in closed-form for any order. Therefore, for those models, the package will focus on the derivation of conditional moments and conditional central moments.

Conditional Moments - II

Given the initial variance \(v_0\) and the CPP in the variance over interval \([0,t]\), \(z^v(u), 0\le u \le t\), we are going to derive the conditional moments and conditional central moments of return over this interval \([0,t]\).

We define two centralized variables

\[\begin{split}\begin{align*} \overline{y}_{svvj,t} &\triangleq y_{svvj,t} - \mathbb{E}[y_{svvj,t}|v_0,z^v_u, 0\le u \le t],\\ \overline{I\!Z^s_t} &\triangleq I\!Z^s_t - \mathbb{E}[I\!Z^s_t], \end{align*}\end{split}\]

to introduce the (conditionally) centralized return

\[\overline{y}_t \triangleq \overline{y}_{svvj, t} + \overline{I\!Z^s_t}.\]

Thus, the conditional moments and central moments can be derived through the following equations,

\[\begin{split}\begin{align*} &\mathbb{E}[y_t^m|v_0, z^v_u, 0\le u\le t] \\ &= \sum_{i=0}^{m}C_m^i \mathbb{E}[y_{svvj, t}^i|v_0, z^v_u, 0\le u\le t] \mathbb{E}[(I\!Z^s_t)^{m-i}],\\ &\mathbb{E}[\overline{y}_t^m|v_0, z^v_u, 0\le u\le t] \\ &= \sum_{i=0}^{m}C_m^i \mathbb{E}[\overline{y}_{svvj, t}^i |v_0, z^v_u, 0\le u\le t] \mathbb{E}[(\overline{I\!Z^s_t})^{m-i}]. \end{align*}\end{split}\]

They are implementd in functions moments_y_to() and cmoments_y_to() in this subpackage (ajdmom.mdl_svij), respectively.

API

ajdmom.mdl_svij.cond2_cmom	Conditional Central Moments for the SVIJ model, given \(v_0\) and the realized jumps in the variance.
ajdmom.mdl_svij.cond2_mom	Conditional Moments for the SVIJ model, given \(v_0\) and the realized jumps in the variance.