ajdmom.mdl_svcj.cond_ieii_ieziziz_mom

Joint Conditional Moment of \(I\!E_t,I_t,I_t^{*},I\!E\!Z_t,I\!Z_t,I\!Z_t^{*}|v_0\)

In this module, we focus on computing the following conditional joint moment:

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

While the recursive computation of \(\mathbb{E}[I\!E_t^{m_1} I_t^{m_2} I_t^{*m_3}|v_0]\) is well-established for models such as the Heston model, we encounter a new challenge: the last three quantities \(I\!E\!Z_t^{m_4} I\!Z_t^{m_5} I\!Z_t^{*m_6}\) are not independent of the first three quantities \(I\!E_t^{m_1} I_t^{m_2} I_t^{*m_3}\) in above equation. For \(m_4 + m_5 + m_6 \ge 1\), we must evaluate integrals of the form

\[\begin{equation} \int_0^t e^{lks} \mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3} I\!E\!Z_t^{m_4} I\!Z_t^{m_5} I\!Z_t^{*m_6}|v_0]\mathrm{d} s. \end{equation}\]

The dependence between \(I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3}\) and \(I\!E\!Z_t^{m_4} I\!Z_t^{m_5} I\!Z_t^{*m_6}\) motivates us to decompose the latter as follows:

\[\begin{split}\begin{align*} &I\!E\!Z_t^{m_4}I\!Z_t^{m_5}I\!Z_t^{*m_6}\\ &= \sum_{i_1=0}^{m_4}\sum_{i_2=0}^{m_5}\sum_{i_3=0}^{m_6} \binom{m_4}{i_1}\binom{m_5}{i_2} \binom{m_6}{i_3} I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3} I\!E\!Z_{s,t}^{m_4-i_1} I\!Z_{s,t}^{m_5-i_2} I\!Z_{s,t}^{*m_6-i_3}, \quad \forall s \le t, \end{align*}\end{split}\]

where \(I\!E\!Z_t\) is split into two independent parts \(I\!E\!Z_s, I\!E\!Z_{s,t}\), i.e., \(I\!E\!Z_t = I\!E\!Z_s + I\!E\!Z_{s,t}\). Similarly, \(I\!Z_t\) and \(I\!Z_t^*\) are decomposed as \(I\!Z_t = I\!Z_s + I\!Z_{s,t}\) and \(I\!Z_t^* = I\!Z_s^* + I\!Z_{s,t}^*\), respectively. Here, the new terms \(I\!E\!Z_{s,t}\), \(I\!Z_{s,t}\) and \(I\!Z_{s,t}^{*}\) are defined as

\[\begin{equation*} I\!E\!Z_{s,t} \mathrel{:=} \int_s^te^{ku}\mathrm{d} z^v(u), \quad I\!Z_{s,t} \mathrel{:=} \int_s^t\mathrm{d} z^v(u), \quad I\!Z^{*}_{s,t} \mathrel{:=} \int_s^t\mathrm{d} z^{s}(u). \end{equation*}\]

Consequently, we have

\[\begin{split}\begin{align*} &\int_0^t e^{lks} \mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3} I\!E\!Z_t^{m_4} I\!Z_t^{m_5} I\!Z_t^{*m_6}|v_0]\mathrm{d} s\\ &= \sum_{i_1=0}^{m_4}\sum_{i_2=0}^{m_5}\sum_{i_3=0}^{m_6}\binom{m_4}{i_1}\binom{m_5}{i_2}\binom{m_6}{i_3} \int_0^t e^{lks} \mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3} I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0] \cdot M(i_1,i_2,i_3)\mathrm{d} s. \end{align*}\end{split}\]

where \(M(i_1,i_2,i_3)\mathrel{:=} \mathbb{E}[I\!E\!Z_{s,t}^{m_4-i_1} I\!Z_{s,t}^{m_5-i_2} I\!Z_{s,t}^{*m_6-i_3}|v_0]\). We have established that the conditional joint moment \(\mathbb{E}[I\!E\!Z_{s,t}^{m_4}I\!Z_{s,t}^{m_5}I\!Z_{s,t}^{*m_6}|v_0]\) can be computed as the following “polynomial”:

\[\begin{equation} \mathbb{E}[I\!E\!Z_{s,t}^{m_4}I\!Z_{s,t}^{m_5}I\!Z_{s,t}^{*m_6}|v_0] = \sum_{\boldsymbol{j}}c_{\boldsymbol{j}}e^{j_1kt}t^{j_2}e^{j_3js}s^{j_4}k^{-j_5}\lambda^{j_6}\mu_v^{j_7}\mu_s^{j_8}\sigma_s^{j_9}, \end{equation}\]

where \(\boldsymbol{j}\mathrel{:=} (j_1,\dots, j_9), j_1,\dots,j_9\) are integers, \(c_{\boldsymbol{j}}\) represents the corresponding monomial coefficient. For detailed derivations, please refer to ajdmom.mdl_svcj.ieziziz_mom.

With above equation, the conditional joint moment can be computed recursively as follows:

\[\begin{split}\begin{align} &\mathbb{E}[I\!E_t^{m_1} I_t^{m_2} I_t^{*m_3} I\!E\!Z_t^{m_4} I\!Z_t^{m_5} I\!Z_t^{*m_6}|v_0]\nonumber\\ &=\sum_{i_1=0}^{m_4}\sum_{i_2=0}^{m_5}\sum_{i_3=0}^{m_6}\binom{m_4}{i_1}\binom{m_5}{i_2}\binom{m_6}{i_3} \sum_{\boldsymbol{j}}c_{\boldsymbol{j}}e^{j_1kt}t^{j_2}k^{-j_5}\lambda^{j_6}\mu_v^{j_7}\mu_s^{j_8} \sigma_s^{j_9}F(m_1, m_2, m_3),%\label{eqn:recursive-ieii-ieziziz} \end{align}\end{split}\]

where

\[\begin{align*} F(m_1,m_2,m_3) &\mathrel{:=} \sum_{i=1}^4\left[\frac{m_1(m_1-1)}{2}f_{6i} + \frac{m_2(m_2-1)}{2}g_{6i} + m_1m_2h_{6i} + \frac{m_3(m_3-1)}{2}q_{6i}\right], \end{align*}\]

and the terms \(f_{6i}, g_{6i}, h_{6i}, q_{6i}, i=1,2,3,4\) are defined as:

\[\begin{split}\begin{align*} f_{61} &\mathrel{:=} \int_0^te^{(j_3+1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-2}I_s^{m_2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times (v_0-\theta),\\ f_{62} &\mathrel{:=} \int_0^te^{(j_3+2)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-2}I_s^{m_2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \theta,\\ f_{63} &\mathrel{:=} \int_0^te^{(j_3+1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-1}I_s^{m_2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \sigma_v,\\ f_{64} &\mathrel{:=} \int_0^te^{(j_3+1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-2}I_s^{m_2}I_s^{*m_3}I\!E\!Z_s^{i_1+1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s, \end{align*}\end{split}\]

\[\begin{split}\begin{align*} g_{61} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2-2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times (v_0-\theta),\\ g_{62} &\mathrel{:=} \int_0^te^{(j_3-0)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2-2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \theta,\\ g_{63} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1+1}I_s^{m_2-2}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \sigma_v,\\ g_{64} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2-2}I_s^{*m_3}I\!E\!Z_s^{i_1+1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s, \end{align*}\end{split}\]

\[\begin{split}\begin{align*} h_{61} &\mathrel{:=} \int_0^te^{(j_3+0)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-1}I_s^{m_2-1}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times (v_0-\theta),\\ h_{62} &\mathrel{:=} \int_0^te^{(j_3+1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-1}I_s^{m_2-1}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \theta,\\ h_{63} &\mathrel{:=} \int_0^te^{(j_3+0)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-0}I_s^{m_2-1}I_s^{*m_3}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \sigma_v,\\ h_{64} &\mathrel{:=} \int_0^te^{(j_3+0)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1-1}I_s^{m_2-1}I_s^{*m_3}I\!E\!Z_s^{i_1+1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s, \end{align*}\end{split}\]

and

\[\begin{split}\begin{align*} q_{61} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3-2}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times (v_0-\theta),\\ q_{62} &\mathrel{:=} \int_0^te^{(j_3-0)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3-2}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \theta,\\ q_{63} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1+1}I_s^{m_2}I_s^{*m_3-2}I\!E\!Z_s^{i_1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s \times \sigma_v,\\ q_{64} &\mathrel{:=} \int_0^te^{(j_3-1)ks}s^{j_4}\mathbb{E}[I\!E_s^{m_1}I_s^{m_2}I_s^{*m_3-2}I\!E\!Z_s^{i_1+1}I\!Z_s^{i_2}I\!Z_s^{*i_3}|v_0]\mathrm{d} s. \end{align*}\end{split}\]

Before closing this subsection, we highlight that the final expression for the conditional joint moment takes the form of a polynomial in \(v_0-\theta\). Specifically, it can be expressed as:

\[\begin{equation}%\label{eqn:ieii-ieziziz-polynomial} \mathbb{E}[I\!E_t^{m_1}I_t^{m_2}I_t^{*m_3}I\!E\!Z_t^{m_4}I\!Z_t^{m_5}I\!Z_t^{*m_6}|v_0] = \sum_{i=0}^{\lfloor (m_1+m_2)/2 \rfloor + \lfloor m_3/2 \rfloor} c_i (v_0-\theta)^i, \end{equation}\]

where \(\lfloor x \rfloor\) denotes the floor function, i.e., the greatest integer less than or equal to \(x\). Here, with a slight abuse of notation, \(c_i\) represents the coefficient, which can be computed via the recursive equation.

Functions

expand_ieziziz(poly)
ieziziz_to_ieii_ieziziz(poly)
int_e_s_poly(c, tp, m1, m2, poly)
key_times_poly(k, poly)
m_ieii_ieziziz(order, par)
moment_ieii_ieziziz(n1, n2, n3, n4, n5, n6)	joint conditional moment of \(IE_t,I_t,I_t^{*},IEZ_t,IZ_t,IZ_t^{*}|v_0\)
poly2num(poly, par)
recursive_ieii_ieziziz(n1, n2, n3, n4, n5, ...)