SVVJ Model

In this subpackage (ajdmom.mdl_svvj), we consider the following SV model, which adds a jump component in the variance of the Heston model:

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

where \(z(t)\) is a CPP with constant arrival rate \(\lambda\) and jumps distributed according to distribution \(F_j(\cdot,\boldsymbol{\theta}_j)\) with parameter \(\boldsymbol{\theta}_j\). Usually, the jump distribution is set as an exponential distribution with scale parameter \(\mu_v\) (= 1/rate), which will help to assure the variance always be non-negative, i.e., \(v(t) \ge 0, \forall t\ge 0\).

Define \(y_t \equiv \log s(t) - \log s(0)\). Then, we have

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

where \(\beta_t \equiv (1-e^{-kt})/(2k)\). Please refer to ajdmom.ito_cond2_mom for the definitions of \(I\!E\!Z_t\) and \(I\!Z_t\). Please also note that \(I\!E_t\equiv I\!E_{0,t}, I_t\equiv I_{0,t}, I_t^{*} \equiv I_{0,t}^{*}\), and refer to Theory for the definitions of \(I\!E_{s,t}, I_{s,t}, I_{s,t}^{*}\).

Conditional Moments - I

Given the initial variance \(v_0\), the conditional mean of \(y_t\) is given as

\[\begin{split}\begin{align*} \mathbb{E}[y_t|v_0] &= (\mu-\theta/2)t - (v_0 - \theta)\beta_t + \frac{1}{2k}e^{-kt} \mathbb{E}[I\!E\!Z_t] - \frac{1}{2k}\mathbb{E}[I\!Z_t]\\ &= (\mu-\theta/2)t - (v_0 - \theta)\beta_t + \frac{\lambda \mu_v}{k}\beta_t - \frac{1}{2k}\lambda t \mu_v. \end{align*}\end{split}\]

Let us define \(\overline{y}_t \triangleq y_t - \mathbb{E}[y_t|v_0]\), then we have

\[\begin{split}\begin{align*} \overline{y}_t &= \frac{\sigma_v}{2k} e^{-kt}I\!E_t + \left(\rho -\frac{\sigma_v}{2k} \right)I_t + \sqrt{1-\rho^2}I_t^{*} + \frac{1}{2k}e^{-kt}I\!E\!Z_t - \frac{1}{2k}I\!Z_t\\ &\quad - \frac{\lambda \mu_v}{2k^2}(1-e^{-kt}) + \frac{1}{2k}\lambda t\mu_v. \end{align*}\end{split}\]

Central Moments

Then, the \(l\)-th conditional central moment can be computed via

\[\mathbb{E}[\overline{y}_t^l|v_0] = \sum_{\mathbf{n}}c(\mathbf{n}) b(\mathbf{n})\mathbb{E}[(e^{-kt}I\!E_t)^{n_1}I_t^{n_2}I_t^{*n_3} (e^{-kt}I\!E\!Z_t)^{n_4} I\!Z_t^{n_5}|v_0],\]

where \(\mathbf{n} = (n_1,n_2,n_3,n_4,n_5,n_6,n_7)\), \(n_1+\cdots+n_7=l\),

\[\begin{split}\begin{eqnarray*} c(\mathbf{n}) &=& \binom{l}{n_1,\cdots, n_7},\\ b(\mathbf{n}) &=& \left(\frac{\sigma_v}{2k}\right)^{n_1} \left(\rho -\frac{\sigma_v}{2k} \right)^{n_2} \left(\sqrt{1-\rho^2}\right)^{n_3} \frac{(-1)^{n_5+n_6}}{(2k)^{n_4+n_5+n_7}} \left[\frac{\lambda \mu_v}{2k^2}(1-e^{-kt})\right]^{n_6} (\lambda t\mu_v)^{n_7}. \end{eqnarray*}\end{split}\]

Moments

Given conditional central moments, it is easy to compute conditional moments as the following

\[\mathbb{E}[y_t^l|v_0] = \sum_{i=0}^l\binom{n}{i} \mathbb{E}^i[y_t|v_0] \mathbb{E}[\overline{y}_t^{l-i}|v_0].\]

Unconditional Moments

By substituting the terms \(\mathbb{E}[(v_0 - \theta)^l]\) within the conditional central moments and conditional moments, we get the unconditional central moments and unconditional moments. Please refer to the SRJD model page for the computation of the terms \(\mathbb{E}[(v_0 - \theta)^l]\).

However, it is a little complicated to derive the unconditional covariances. It is necessary to introduce more notations as \(y_{n+1} \equiv y((n+1)h) - y(nh)\),

\[\begin{split}\begin{align*} y_{n+1} &= - (v_n-\theta)\beta + \frac{\sigma_v}{2k} e^{-k(n+1)h}I\!E_{n+1} + \left(\rho -\frac{\sigma_v}{2k} \right)I_{n+1} + \sqrt{1-\rho^2}I_{n+1}^{*}\\ &\quad ~ + \frac{1}{2k}e^{-k(n+1)h}I\!E\!Z_{n+1} - \frac{1}{2k}I\!Z_{n+1} + (\mu - \theta/2) h , \end{align*}\end{split}\]

where \(\beta \equiv (1-e^{-kh})/(2k)\), and \(v_n - \theta = e^{-kh}(v_{n-1} - \theta) + \sigma_ve^{-knh}I\!E_n + e^{-knh}I\!E\!Z_n\).

When expanding \(y_{n+1}^{l_2}\), the indexing (\(n_0+\cdots+n_6=l_2\)) is organized as

\((v_n-\theta)\)	\(I\!E_{n+1}\)	\(I_{n+1}\)	\(I_{n+1}^{*}\)	\(I\!E\!Z_{n+1}\)	\(I\!Z_{n+1}\)	\((\mu - \theta/2) h\)
\(n_0\)	\(n_1\)	\(n_2\)	\(n_3\)	\(n_4\)	\(n_5\)	\(n_6\)

Covariances

Covariances can be computed via

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

Therefore, we only need to compute \(\mathbb{E}[y_n^{l_1}y_{n+1}^{l_2}]\).

\[\begin{split}\begin{align*} &\mathbb{E}[y_n^{l_1}y_{n+1}^{l_2}]\\ &=\sum_{\mathbf{n}} c(\mathbf{n}) b(\mathbf{n}) \sum_{\mathbf{m}} c(\mathbf{m}) b(\mathbf{m})\\ &\quad\mathbb{E}[(v_{n-1} - \theta)^{m_0}(e^{-knh}I\!E_n)^{m_1}I_n^{m_2}I_n^{*m_3} (e^{-knh}I\!E\!Z_n)^{m_4} I\!Z_n^{m_5} \cdot \\ &\qquad (v_n - \theta)^{n_0}(e^{-k(n+1)h}I\!E_{n+1})^{n_1}I_{n+1}^{n_2}I_{n+1}^{*n_3} (e^{-k(n+1)h}I\!E\!Z_{n+1})^{n_4} I\!Z_{n+1}^{n_5}]. \end{align*}\end{split}\]

Note that

\[\begin{split}\begin{align*} &\mathbb{E}[I\!E_{n+1}^{n_1} I_{n+1}^{n_2} I_{n+1}^{*n_3} I\!E\!Z_{n+1}^{n_4} I\!Z_{n+1}^{n_5}]\\ &= \sum_{n_1,n_4,i,j,l,o,p,q,r,s}b_{n_1,n_4,i,j,l,o,p,q,r,s} e^{(n_1+n_4)knh} e^{ikh} h^j k^{-l} (v_n-\theta)^o \theta^p \sigma_v^q \lambda^r \mu_v^s,\\ &(v_n-\theta)^{n_0} e^{-(n_1+n_4)k(n+1)h} \mathbb{E}[I\!E_{n+1}^{n_1} I_{n+1}^{n_2} I_{n+1}^{*n_3} I\!E\!Z_{n+1}^{n_4} I\!Z_{n+1}^{n_5}]\\ &= \sum_{n_1,n_4,i,j,l,o,p,q,r,s}b_{n_1,n_4,i,j,l,o,p,q,r,s} e^{-(n_1+n_4)kh} e^{ikh} h^j k^{-l} (v_n-\theta)^{o+n_0} \theta^p \sigma_v^q \lambda^r \mu_v^s. \end{align*}\end{split}\]

A function is defined to implement the corresponding derivation and the expansion of \((v_n-\theta)\), resulting in

\[\begin{split}\begin{align*} &ve\_I\!EII\_I\!E\!ZI\!Z\_vn(n_0,n_1,n_2,n_3,n_4,n_5)\\ &= \sum_{m_1,m_2,i,j,l,o,p,q,r,s} b_{m_1,m_2,i,j,l,o,p,q,r,s} e^{-(m_1+m_2)knh} I\!E_n^{m_1} I\!E\!Z_n^{m_2} e^{-ikh} h^j k^{-l} (v_n-\theta)^{o} \theta^p \sigma_v^q \lambda^r \mu_v^s. \end{align*}\end{split}\]

The expansion of \((v_n-\theta)\) is done via,

\[(v_n-\theta)^m = \sum_{\mathbf{m}} \binom{m}{m_1,m_2,m_3} [e^{-kh}(v_{n-1} - \theta)]^{m_1} (\sigma_ve^{-knh}I\!E_n)^{m_2} (e^{-knh}I\!E\!Z_n)^{m_3}.\]

Conditional Moments - II

Given the initial variance \(v_0\) and the CPP over interval \([0,t]\), \(z_s, 0\le s \le t\), the conditional mean of return over this interval is given by

\[\begin{split}\begin{align*} &\mathbb{E}[y_t|v_0,z_s, 0\le s\le t] \\ &= (\mu-\theta/2)t - (v_0 - \theta)\beta_t + \frac{1}{2k} e^{-kt}I\!E\!Z_t - \frac{1}{2k}I\!Z_t. \end{align*}\end{split}\]

Let us define \(\overline{y}_t \triangleq y_t - \mathbb{E}[y_t|v_0,z_s, 0\le s\le t]\), then we have

\[\overline{y}_t = \frac{\sigma_v}{2k} e^{-kt}I\!E_t + \left( \rho -\frac{\sigma_v}{2k} \right)I_t + \sqrt{1-\rho^2}I_t^{*}.\]

Conditional Central Moments

For conditional central moments, we have

(2)\[\begin{split} \begin{align*} &\mathbb{E}[{\overline{y}_t}^l|v_0,z_s, 0\le s\le t]\\ &= \sum_{\boldsymbol{n}} c(\boldsymbol{n}) b(\boldsymbol{n}) \mathbb{E}[(e^{-kt}I\!E_t)^{n_1} I_t^{n_2} I_t^{*n_3} |v_0,z_s, 0\le s\le t], \end{align*}\end{split}\]

where \(\boldsymbol{n} = (n_1, n_2, n_3)\), \(n_1+n_2+n_3=l\), \(c(\boldsymbol{n}) = C_l^{n_1} C_{l-n_1}^{n_2}\), and

\[b(\boldsymbol{n}) = \left( \frac{\sigma_v}{2k} \right)^{n_1} \left( \rho -\frac{\sigma_v}{2k} \right)^{n_2} \left( \sqrt{1-\rho^2} \right)^{n_3}\]

The derivation for \(\mathbb{E}[E_t^{n_1} I_t^{n_2} I_t^{*n_3}|v_0,z_s, 0\le s\le t]\) has been implemented in moment_IEII() from module ajdmom.ito_cond2_mom. The conditional central moments in (2) is implemented in cmoments_y_to() in the subpackage ajdmom.mdl_svvj, noting that the function now derives simultaneously the conditional central moments with orders from 1 to \(l\).

Conditional Moments

Now we rewrite \(y_t\) as the following

\[y_t = (\mu - \theta/2)t - (v_0 - \theta)\beta_t + \frac{1}{2k} e^{-kt}I\!E\!Z_t - \frac{1}{2k}I\!Z_t + \overline{y}_t,\]

to enable us to exploit cmoment_y() for the derivation of those involved conditional central moments. With this expression, the conditional moments can be derived through the following equation

(3)\[\begin{split} \begin{align*} &\mathbb{E}[y_t^l|v_0,z_s, 0\le s\le t] \\ &= \sum_{\boldsymbol{n}} c_2(\boldsymbol{n}) b_2(\boldsymbol{n}) (e^{-kt}I\!E\!Z_t)^{n_3} I\!Z_t^{n_4} \mathbb{E}[\overline{y}_t^{n_5}|v_0, z_s, 0\le s \le t], \end{align*}\end{split}\]

where \(\boldsymbol{n} = (n_1, n_2, n_3, n_4, n_5)\), \(\sum_{i=1}^5 n_i = l\),

\[\begin{split}\begin{align*} c_2(\boldsymbol{n}) &= C_l^{n_1} C_{l-n_1}^{n_2} C_{l-n_1-n_2}^{n_3} C_{l-n_1-n_2-n_3}^{n_4} C_{l-n_1-n_2-n_3-n_4}^{n_5},\\ b_2(\boldsymbol{n}) &= (-1)^{n_2} (2k)^{-(n_3+n_4)} [(\mu-\theta/2)t]^{n_1} (v_0-\theta)^{n_2}. \end{align*}\end{split}\]

The conditional moments in (3) is implemented in moments_y_to() in the subpackage ajdmom.mdl_svvj.

API

ajdmom.mdl_svvj.cmom	Central Moments of the SVVJ model
ajdmom.mdl_svvj.mom	Moments of the SVVJ model
ajdmom.mdl_svvj.cov	Covariance for the SVVJ model
ajdmom.mdl_svvj.cond_cmom	Conditional Central Moments of the SVVJ model, given initial variance
ajdmom.mdl_svvj.cond_mom	Conditional Moment of the SVVJ model, given initial variance
ajdmom.mdl_svvj.cond2_cmom	Conditional Central Moments for the SVVJ model, given the initial state of the variance and the realized jumps in the variance.
ajdmom.mdl_svvj.cond2_mom	Conditional Moments for the SVVJ model, given the initial state of the variance and the realized jumps in the variance.