SVCJ Model

In this subpackage (ajdmom.mdl_svcj), we consider the following SV model, which adds contemporaneous jumps 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 (\(J_i^v\)) 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 sharing the same arrival process with \(z^v(t)\), and with jumps distributed according to a normal distribution with mean \(\mu_s + \rho_J J_i^v\) (dependent on the realized jump \(J_i^v\) in the varance process) and variance \(\sigma_s^2\), i.e., \(J_i^s|J_i^v \sim \mathcal{N}(\mu_s+\rho_J J_i^v, \sigma_s^2)\).

For models including jumps in the variance, both conditional and unconditional moments can be derived.

Conditional Moments - I

We now introduce the following notation:

\[\begin{split}\begin{align*} I\!E_t &\mathrel{:=} \int_0^t e^{ks}\sqrt{v(s)}\mathrm{d} w^v(s), &I_t &\mathrel{:=} \int_0^t \sqrt{v(s)} \mathrm{d} w^v(s),\\ I\!E\!Z_t &\mathrel{:=} \int_0^t e^{ks}\mathrm{d} z^v(s), ~ &I\!Z_t &\mathrel{:=} \int_0^t \mathrm{d} z^v(s). \end{align*}\end{split}\]

Additionally, we define

\[\quad I\!Z_t^s \mathrel{:=} \int_0^t \mathrm{d} z^s(t), \quad I\!Z_t^* \mathrel{:=} \int_0^t\mathrm{d} z^*(t),\]

where \(z^*(t)\) is another compound Poisson process sharing the same arrival process as \(z^v(t)\) and \(z^s(t)\), but with independent jumps \(J_i^*\sim \mathcal{N}(\mu_s, \sigma_s^2)\). Since the jumps of \(z^s(t)\) are distributed as \(J_i^s|J_i^v \sim \mathcal{N}(\mu_s + \rho_J J_i^v, \sigma_s^2)\), the compound Poisson process in the return can be decomposed into another two compound Poisson processes, i.e., \(I\!Z_{t}^s = \rho_J I\!Z_t + I\!Z_t^*\). The solution to the variance process now is given by:

\[\begin{equation*} e^{kt}v_t = (v_0-\theta) + e^{kt}\theta + \sigma_v I\!E_t + I\!E\!Z_t. \end{equation*}\]

For the return \(y_t\) (\(y_t \mathrel{:=} p(t) - p(0)\)), we have the following decomposition:

\[\begin{split}\begin{align*} y_t &= \frac{\sigma_v}{2k} e^{-kt}I\!E_t + \left(\rho -\frac{\sigma_v}{2k} \right)I_t + \sqrt{1-\rho^2}I_t^{*} + \frac{1}{2k}e^{-kt}I\!E\!Z_t + \left(\rho_J - \frac{1}{2k}\right)I\!Z_t + I\!Z_t^{*}\\ &\quad + \left(\mu-\frac{\theta}{2}\right)t - (v_0 - \theta)\beta_t, \end{align*}\end{split}\]

where \(\beta_t = (1-e^{-kt})/(2k)\) by definition. The \(m\)-th conditional moment of \(y_t\), given \(v_0\), can be derived through

\[\begin{align*} \mathbb{E}[y_t^m|v_0] &= \sum_{m_1+\cdots+m_8=m}c_{\boldsymbol{m}} b_{\boldsymbol{m}} \mathbb{E}[I\!E_t^{m_1}I_t^{m_2}I_t^{*m_3}I\!E\!Z_t^{m_4}I\!Z_t^{m_5}I\!Z_t^{*m_6}|v_0], \end{align*}\]

where \(\boldsymbol{m} \mathrel{:=} (m_1, \dots, m_8)\), \(c(\boldsymbol{m}) \mathrel{:=} \binom{m}{m_1,\dots,m_8}\) and

\[\begin{split}\begin{align*} &b(\boldsymbol{m}) \\ &\mathrel{:=} \frac{e^{-(m_1+m_4)kt}}{(2k)^{m_1+m_4}}\sigma_v^{m_1}\left(\rho - \frac{\sigma_v}{2k}\right)^{m_2}\left(\sqrt{1-\rho^2}\right)^{m_3}\left(\rho_J - \frac{1}{2k} \right)^{m_5}\left[\left(\mu-\frac{\theta}{2}\right)t\right]^{m_7} \left[-(v_0-\theta)\beta_t\right]^{m_8}\\ &=\sum_{i_1=0}^{m_2}\sum_{i_2=0}^{m_5}\sum_{i_3=0}^{m_7}\sum_{i_4=0}^{m_8} c_{\boldsymbol{i}} b_{\boldsymbol{m},\boldsymbol{i}}, \end{align*}\end{split}\]

where \(\boldsymbol{i} \mathrel{:=} (i_1,i_2, i_3,i_4)\),

\[\begin{split}\begin{align*} c_{\boldsymbol{i}} &\mathrel{:=} \binom{m_2}{i_1}\binom{m_5}{i_2}\binom{m_7}{i_3}\binom{m_8}{i_4} \frac{(-1)^{i_1+\cdots+i_4}}{2^{m_1+m_4+i_1+i_2+i_3+m_8}},\\ b_{\boldsymbol{m},\boldsymbol{i}} &\mathrel{:=} \frac{e^{-(m_1+m_4+m_8-i_4)kt} t^{m_7}}{k^{m_1+m_4+i_1+i_2+m_8}} \mu^{m_7-i_3}(v_0-\theta)^{m_8}\theta^{i_3}\sigma_v^{m_1+i_1} \rho^{m_2-i_1} \left(\sqrt{1-\rho^2}\right)^{m_3} \rho_J^{m_5-i_2}. \end{align*}\end{split}\]

The first conditional moment is straightforward to compute and is given by:

\[\begin{equation*} \mathbb{E}[y_t|v_0] %= \frac{1- e^{-kt}}{2k^2}\lambda \mu_v + \left(\rho_J - \frac{1}{2k} \right) \lambda t \mu_v + \lambda t \mu_s + (\mu -\theta/2)t - (v_0-\theta)\beta_t. = (\mu - \mathbb{E}[v]/2)t - (v_0 - \mathbb{E}[v])\beta_{t} + \lambda t (\mu_s + \rho_J\mu_v), \end{equation*}\]

where \(\mathbb{E}[v] = \theta + \lambda \mu_v /k\). To compute higher-order conditional moments of \(y_t\), it suffices to evaluate the conditional joint moment:

\[\begin{equation}%\label{eqn:joint-ieii-ieziziz} \mathbb{E}[I\!E_t^{m_1}I_t^{m_2}I_t^{*m_3}I\!E\!Z_t^{m_4}I\!Z_t^{m_5}I\!Z_t^{*m_6}|v_0]. \end{equation}\]

Before addressing the computation of this joint moment, we outline the derivation of conditional central moment of \(y_t\). Define the centralized return as

\[\begin{equation*} \bar{y}_t \mathrel{:=} y_t - \mathbb{E}[y_t|v_0]. \end{equation*}\]

The \(m\)-th conditional central moment of \(\bar{y}_t\) can then be expressed as:

\[\begin{align*} \mathbb{E}[\bar{y}_t^m|v_0] = \sum_{i=0}^m\binom{m}{i}(-1)^i \mathbb{E}^i[y_t|v_0]\mathbb{E}[y_t^{m-i}|v_0]. \end{align*}\]

This decomposition demonstrates that the computation of conditional central moments relies on the computation of conditional moments.

Unconditional Moments

Consequently, the conditional moments of the return, \(\mathbb{E}[y_t^m|v_0]\), are also polynomials in \(v_0\). This property allows us to leverage the polynomial structure to compute the unconditional moments of the return, \(\mathbb{E}[y_t^m]\), as demonstrated in the SRJD Model.

Conditional Moments - II

For some circumstances, the conditional is that both the initial variance and the jumps in the variance are given. This section is devoted to deriving the conditional moments and central moments under these situations.

Note that \(y_t \triangleq \log s(t) - \log s(0)\). Define \(I\!Z_t^s\triangleq \int_0^t dz^s(u)\). 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.

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

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

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,

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

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

Note that

\[\begin{split}\begin{align*} &\mathbb{E}[(I\!Z_t^{s})^m|z^v(u), 0\le u \le t]\\ &= \mathbb{E}\left[\left(\sum_{i=1}^{N(t)} J_i^{s}|J_i^v \right)^m \bigg|z^v(u), 0\le u \le t\right]\\ &= \sum_{m_1,\cdots, m_{N(t)}} \binom{m}{m_1,\dots,m_{N(t)}} \mathbb{E}[(J_1^s)^{m_1}|J_1^v] \cdots \mathbb{E}[(J_{N(t)}^s)^{m_{N(t)}} |J_{N(t)}^v]. \end{align*}\end{split}\]

And \(\mathbb{E}[(\overline{I\!Z_t^{s}})^m|z^v(u), 0\le u \le t]\) is derived similarly. They are implemented in functions moment_IZs() and cmoment_IZs() in this subpackage (ajdmom.mdl_svcj), respectively.

API

ajdmom.mdl_svcj.cmom	Central Moments of the SVCJ model
ajdmom.mdl_svcj.mom	Moments of the SVCJ model
ajdmom.mdl_svcj.cond_cmom	Conditional Central Moments of the SVCJ model, given initial variance
ajdmom.mdl_svcj.cond_mom	Conditional Moments of the SVCJ model, given initial variance
ajdmom.mdl_svcj.cond2_cmom	Conditional Central Moments for the SVCJ model, given \(v_0\) and the realized jumps in the variance.
ajdmom.mdl_svcj.cond2_mom	Conditional Moments for the SVCJ model, given \(v_0\) and the realized jumps in the variance.
ajdmom.mdl_svcj.cond_ieii_ieziziz_mom	Joint Conditional Moment of \(I\!E_t,I_t,I_t^{*},I\!E\!Z_t,I\!Z_t,I\!Z_t^{*}|v_0\)
ajdmom.mdl_svcj.ieziziz_mom	Joint Moments of CPPs \(I\!E\!Z_{s,t}, I\!Z_{s,t}, I\!Z_{s,t}^{*}\)