2FSV Model¶
In this subpackage (ajdmom.mdl_2fsv), we consider the following
SV model in which volatility is
a superposition of two SRD processes,
\[\begin{split}d\log s(t) &= (\mu- v(t)/2) dt + \sqrt{v(t)}dw(t),\\
v(t) &= v_1(t) + v_2(t),\\
dv_1(t) &= k_1(\theta_1 - v_1(t))dt + \sigma_{1v} \sqrt{v_1(t)}dw_1(t),\\
dv_2(t) &= k_2(\theta_2 - v_2(t))dt + \sigma_{2v} \sqrt{v_2(t)}dw_2(t),\end{split}\]
where \(w_1(t)\), \(w_2(t)\) are two independent Wiener processes, which are also independent of \(w(t)\).
Notations¶
We define
\[\begin{split}I_{1,n} &\triangleq \int_{(n-1)h}^{nh}\sqrt{v_1(t)}dw_1(t),
&I_{2,n} &\triangleq \int_{(n-1)h}^{nh}\sqrt{v_2(t)}dw_2(t),\\
I\!E_{1,n} &\triangleq \int_{(n-1)h}^{nh}e^{k_1t}\sqrt{v_1(t)}dw_1(t),\quad
&I\!E_{2,n} &\triangleq \int_{(n-1)h}^{nh}e^{k_2t}\sqrt{v_2(t)}dw_2(t),\\
IV_{1,n}&\triangleq \int_{(n-1)h}^{nh}v_1(t)dt,
&IV_{2,n}&\triangleq \int_{(n-1)h}^{nh}v_2(t)dt,\\
IV_{n} &\triangleq IV_{1,n} + IV_{2,n},
&I_n^* &\triangleq \int_{(n-1)h}^{nh}\sqrt{v(t)}dw(t).\end{split}\]
Return of the nth interval \(y_n\) can be expressed as \(y_n = \mu h - \frac{1}{2}IV_n + I_n^*\). We have
\[IV_{i,n}
= (h-\tilde{h}_i)\theta_i + \tilde{h}v_{i,n-1} -
\frac{\sigma_{vi}}{k_i}e^{-k_inh}I\!E_{i,n} + \frac{\sigma_{vi}}{k_i}I_{i,n},
i=1,2,\]
where \(\tilde{h}_i \triangleq (1-e^{-k_ih})/k_i, i=1,2\).
Central Moments¶
Similarly, we define \(\overline{y}_n \triangleq y_n - E[y_n]\) and we have
\[\begin{split}\overline{y}_n
&=\frac{1}{2}(\tilde{h}_1\theta_1 + \tilde{h}_2\theta_2)
- \frac{1}{2}\tilde{h}_1v_{1,n-1} - \frac{1}{2}\tilde{h}_2v_{2,n-1}\\
&\quad + \frac{\sigma_{v1}}{2k_1}e^{-k_1 nh}I\!E_{1,n}
- \frac{\sigma_{v1}}{2k_1}I_{1,n}
+ \frac{\sigma_{v2}}{2k_2}e^{-k_2 nh}I\!E_{2,n}
- \frac{\sigma_{v2}}{2k_2}I_{2,n}
+ I_{n}^{*}.\end{split}\]
According to above expansion, central moment \(\overline{y}_n\) with order \(l\) reduces to
\[\begin{split}&E[\overline{y}_n^l]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3} (e^{-k_1 nh}I\!E_{1,n})^{m_4} I_{1,n}^{m_5}
(e^{-k_2 nh}I\!E_{2,n})^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1 nh}e^{-m_6k_2 nh}
I\!E_{1,n}^{m_4} I_{1,n}^{m_5} I\!E_{2,n}^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1 nh}e^{-m_6k_2 nh}\\
&\qquad \times \text{moment_IEI_IEII}(m_4,m_5,m_6,m_7,m_8)]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b(\boldsymbol{m})
\times \text{vvee_IEI_IEII}(m_2, m_3, m_4, m_5, m_6, m_7, m_8)\\
&= \sum_{\boldsymbol{m}} \text{moment_comb}(l,m_1,m_2,m_3,m_4,m_5,m_6,m_7,m_8)\end{split}\]
where \(\boldsymbol{m} = (m_1,\cdots,m_8), \sum_{i=1}^8m_i = l\),
\[\begin{split}c(\boldsymbol{m})
&= C_{l}^{m_1}C_{l-m_1}^{m_2}C_{l-m_1-m_2}^{m_3}
C_{l-m_1-m_2-m_3}^{m_4} C_{l-m_1-m_2-m_3-m_4}^{m_5}
C_{l-m_1-m_2-m_3-m_4-m_5}^{m_6} \\
&\quad C_{l-m_1-m_2-m_3-m_4-m_5-m_6}^{m_7},\end{split}\]
\[\begin{split}b(\boldsymbol{m})
&= (-1)^{m_2+m_3+m_5+m_7}2^{-(l-m_8)}
(\tilde{h}_1\theta_1+\tilde{h}_2\theta_2)^{m_1}
\tilde{h}_1^{m_2} \tilde{h}_2^{m_3}
(\sigma_{v1}/k_1)^{m_4+m_5} (\sigma_{v2}/k_2)^{m_6+m_7}\\
&=\sum_{i_1+i_2+i_3+i_4=m_1}\sum_{j_1=0}^{m_2}\sum_{j_2=0}^{m_3}
c_1(\boldsymbol{i},\boldsymbol{j})
(-1)^{i_2+i_4+m_2+m_3+j_1+j_2+m_5+m_7}2^{-(l-m_8)}\\
&\quad e^{-[(i_2+j_1)k_1 + (i_4+j_2)k_2]h}
k_1^{-(i_1+i_2+m_2+m_4+m_5)} k_2^{-(i_3+i_4+m_3+m_6+m_7)}
\theta_1^{i_1+i_2}\theta_2^{i_3+i_4}
\sigma_{v1}^{m_4+m_5} \sigma_{v2}^{m_6+m_7}\end{split}\]
where
\[c_1(\boldsymbol{i},\boldsymbol{j})
= C_{m_1}^{i_1}C_{m_1-i_1}^{i_2}C_{m_1-i_1-i_2}^{i_3}
C_{m_2}^{j_1} C_{m_3}^{j_2}.\]
Function
moment_IEI_IEII(m_4,m_5,m_6,m_7,m_8) returns a poly with attribute
keyfor = ('(n_1m*k1+n_2m*k2)^{-i_m},...,(n_11*k1+n_21*k2)^{-i_1}',
'e^{(m_4*k1+m_6*k2)(n-1)h}','e^{(j_1*k1+j_2*k2)[t-(n-1)h]}','[t-(n-1)h]',
'v_{1,n-1}','theta1','sigma_v1', 'v_{2,n-1}','theta2','sigma_v2').
In summary, I defined
Moments¶
We have \(y_n = \overline{y}_n + E[y_n]\) and \(E[y_n] = \mu h - \frac{1}{2}(\theta_1 + \theta_2)h\), thus
\[\begin{split}y_n
&= \frac{1}{2}(\tilde{h}_1\theta_1 + \tilde{h}_2\theta_2)
- \frac{1}{2}(\theta_1 + \theta_2)h + \mu h
- \frac{1}{2}\tilde{h}_1v_{1,n-1} - \frac{1}{2}\tilde{h}_2v_{2,n-1}\\
&\quad + \frac{\sigma_{v1}}{2k_1}e^{-k_1 nh}I\!E_{1,n}
- \frac{\sigma_{v1}}{2k_1}I_{1,n}
+ \frac{\sigma_{v2}}{2k_2}e^{-k_2 nh}I\!E_{2,n}
- \frac{\sigma_{v2}}{2k_2}I_{2,n}
+ I_{n}^{*}.\end{split}\]
Similarly,
\[\begin{split}E[y_n^l]
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3} (e^{-k_1 nh}I\!E_{1,n})^{m_4} I_{1,n}^{m_5}
(e^{-k_2 nh}I\!E_{2,n})^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1 nh}e^{-m_6k_2 nh}
I\!E_{1,n}^{m_4} I_{1,n}^{m_5} I\!E_{2,n}^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
E[v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1 nh}e^{-m_6k_2 nh}\\
&\qquad \times \text{moment_IEI_IEII}(m_4,m_5,m_6,m_7,m_8)]\\
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
\times \text{vvee_IEI_IEII}(m_2, m_3, m_4, m_5, m_6, m_7, m_8)\\
&= \sum_{\boldsymbol{m}} \text{moment_comb}(l,m_1,m_2,m_3,m_4,m_5,m_6,m_7,m_8)\end{split}\]
where
\[\begin{split}b_2(\boldsymbol{m})
&= (-1)^{m_2+m_3+m_5+m_7}2^{-(l-m_8)}
(\tilde{h}_1\theta_1+\tilde{h}_2\theta_2 - h\theta_1 -h\theta_2
+2\mu h)^{m_1}
\tilde{h}_1^{m_2} \tilde{h}_2^{m_3}\\
&\quad (\sigma_{v1}/k_1)^{m_4+m_5} (\sigma_{v2}/k_2)^{m_6+m_7}\\
&=\sum_{i_1+i_2+i_3+i_4+i_5+i_6+i_7=m_1}\sum_{j_1=0}^{m_2}\sum_{j_2=0}^{m_3}
c_2(\boldsymbol{i},\boldsymbol{j})
(-1)^{i_2+i_4+i_5+i_6+m_2+m_3+j_1+j_2+m_5+m_7} 2^{-(l-m_8)+i_7}\\
&\quad e^{-[(i_2+j_1)k_1 + (i_4+j_2)k_2]h}
k_1^{-(i_1+i_2+m_2+m_4+m_5)} k_2^{-(i_3+i_4+m_3+m_6+m_7)}
\theta_1^{i_1+i_2+i_5}\theta_2^{i_3+i_4+i_6} \\
&\quad \sigma_{v1}^{m_4+m_5} \sigma_{v2}^{m_6+m_7}
h^{i_5+i_6+i_7}\mu^{i_7}\end{split}\]
where
\[c_2(\boldsymbol{i},\boldsymbol{j})
= C_{m_1}^{i_1}C_{m_1-i_1}^{i_2}C_{m_1-i_1-i_2}^{i_3}
C_{m_1-i_1-i_2-i_3}^{i_4}C_{m_1-i_1-i_2-i_3-i_4}^{i_5}
C_{m_1-i_1-i_2-i_3-i_4-i_5}^{i_6}
C_{m_2}^{j_1} C_{m_3}^{j_2}.\]
In summary, I defined
One alternative way,
\[E[y_n^l]
= \sum_{i=0}^l C_l^i E[\overline{y}_n^i] E^{l-i}[y_n],
\quad
E^l[y_n]
= \sum_{i,j} C_l^i C_{l-i}^j (-1)^{l-i} \frac{1}{2^{l-i}} h^l \mu^i \theta_1^j \theta_2^{l-i-j}.\]
Covariances¶
\[cov(y_n^{l_1}, y_{n+1}^{l_2})
= E[y_n^{l_1}y_{n+1}^{l_2}] - E[y_n^{l_1}]E[y_{n+1}^{l_2}]\]
Co-Moments¶
\[\begin{split}&E[y_n^{l_1}y_{n+1}^{l_2}]\\
&= \sum_{\boldsymbol{n}}c(\boldsymbol{n})b_2(\boldsymbol{n})
E[y_n^{l_1}v_{1,n}^{n_2}v_{2,n}^{n_3} (e^{-k_1 (n+1)h}I\!E_{1,n+1})^{n_4}
I_{1,n+1}^{n_5} (e^{-k_2 (n+1)h}I\!E_{2,n+1})^{n_6} I_{2,n+1}^{n_7}
I_{n+1}^{*n_8}]\\
&= \sum_{\boldsymbol{n}}c(\boldsymbol{n})b_2(\boldsymbol{n})
E[y_n^{l_1}\color{teal}v_{1,n}^{n_2}v_{2,n}^{n_3}e^{-n_4k_1(n+1)h}
e^{-n_6k_2(n+1)h} \\
&\quad \color{teal} E[I\!E_{1,n+1}^{n_4} I_{1,n+1}^{n_5} I\!E_{2,n+1}^{n_6}
I_{2,n+1}^{n_7}I_{n+1}^{*n_8}|v_{1,n},v_{2,n}]]\\
&= \sum_{\boldsymbol{n}}c(\boldsymbol{n})b_2(\boldsymbol{n})
E[y_n^{l_1}\color{teal}
\text{vvee_IEI_IEII_vnvn}(n_2,n_3,n_4,n_5,n_6,n_7,n_8)]\\
&= \sum_{\boldsymbol{n}}c(\boldsymbol{n})b_2(\boldsymbol{n})
\color{magenta}
\sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
E[ v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1nh}e^{-m_6k_2nh}I\!E_{1,n}^{m_4}
I_{1,n}^{m_5} I\!E_{2,n}^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8}\\
&\quad \color{teal}
\text{vvee_IEI_IEII_vnvn}(n_2,n_3,n_4,n_5,n_6,n_7,n_8)]\end{split}\]
where I used
\[\begin{split}y_n^{l_1}
&= \sum_{\boldsymbol{m}}c(\boldsymbol{m})b_2(\boldsymbol{m})
v_{1,n-1}^{m_2}v_{2,n-1}^{m_3}e^{-m_4k_1nh}e^{-m_6k_2nh}I\!E_{1,n}^{m_4}
I_{1,n}^{m_5} I\!E_{2,n}^{m_6} I_{2,n}^{m_7} I_{n}^{*m_8},\\
y_{n+1}^{l_2}
&= \sum_{\boldsymbol{n}}c(\boldsymbol{n})b_2(\boldsymbol{n})
v_{1,n}^{n_2}v_{2,n}^{n_3}e^{-n_4k_1(n+1)h}e^{-n_6k_2(n+1)h}
I\!E_{1,n+1}^{n_4} I_{1,n+1}^{n_5} I\!E_{2,n+1}^{n_6} I_{2,n+1}^{n_7}
I_{n+1}^{*n_8}.\end{split}\]
Note that
\[\begin{split}&E[I\!E_{1,n+1}^{n_4} I_{1,n+1}^{n_5} I\!E_{2,n+1}^{n_6} I_{2,n+1}^{n_7}
I_{n+1}^{*n_8}|v_{1,n},v_{2,n}]\\
&= \sum_{t0,(n_4,n_6),(i,i'),j,l,p,q,l',p',q'}
b_{t0(n_4,n_6)(i,i')jlpql'p'q'} \cdot \\
&\quad (n_{1m}k_1+n_{2m}k_2)^{-i_m}
\cdots (n_{11}k_1+n_{21}k_2)^{-i_1}\cdot
e^{(n_4k_1+n_6k_2)nh}\cdot\\
&\quad e^{(ik_1+i'k_2)(t-nh)} (t-nh)^{j}
v_{1,n}^{l}\theta_1^{p}\sigma_{v1}^{q}
v_{2,n}^{l'}\theta_2^{p'}\sigma_{v2}^{q'},\\
%
&E[v_{1,n}^{n_2}v_{2,n}^{n_3}e^{-n_4k_1(n+1)h}e^{-n_6k_2(n+1)h}
I\!E_{1,n+1}^{n_4} I_{1,n+1}^{n_5} I\!E_{2,n+1}^{n_6} I_{2,n+1}^{n_7}
I_{n+1}^{*n_8}|v_{1,n},v_{2,n}]\\
&= \sum_{t0,(n_4,n_6),(i,i'),j,l,p,q,l',p',q'}
b_{t0(n_4,n_6)(i,i')jlpql'p'q'} \cdot \\
&\quad (n_{1m}k_1+n_{2m}k_2)^{-i_m}
\cdots (n_{11}k_1+n_{21}k_2)^{-i_1}\cdot
e^{-(n_4k_1+n_6k_2)h}\cdot\\
&\quad e^{(ik_1+i'k_2)(t-nh)} (t-nh)^{j}
v_{1,n}^{l+n_2}\theta_1^{p}\sigma_{v1}^{q}
v_{2,n}^{l'+n_3}\theta_2^{p'}\sigma_{v2}^{q'},\end{split}\]
where \(t0 = ((n_{1m},n_{2m},i_{m}),...,(n_{11},n_{21},i_{1}))\) and \(t=(n+1)h\).
Function vvee_IEI_IEII_vnvn()
is defined to
accomplish above computation and expand \(v_{1,n}\) and
\(v_{2,n}\)
which returns a poly with attribute
keyfor = ('e^{-k1*nh}IE_{1,n}','e^{-k2*nh}IE_{2,n}',
'(n_1m*k1+n_2m*k2)^{-i_m},...,(n_11*k1+n_21*k2)^{-i_1}',
'e^{-(n1*k1+n2*k2)h}','h',
'v_{1,n-1}','theta1','sigma_v1', 'v_{2,n-1}','theta2','sigma_v2'), i.e.,
\[\begin{split}&\text{vvee_IEI_IEII_vnvn}(n_2,n_3,n_4,n_5,n_6,n_7,n_8)\\
&=\sum_{o,o',t0,i,i',j,l,p,q,l',p',q'}b_{oo't0(i,i')jlpql'p'q'}
e^{-ok_1nh}I\!E_{1,n}^o e^{-o'k_2nh}I\!E_{2,n}^{o'}\\
&\quad (n_{1m}k_1+n_{2m}k_2)^{-i_m} \cdots (n_{11}k_1+n_{21}k_2)^{-i_1}
e^{-(ik_1+i'k_2)h} h^j v_{1,n-1}^{l}\theta_1^{p}\sigma_{v1}^{q}
v_{2,n-1}^{l'}\theta_2^{p'}\sigma_{v2}^{q'}.\end{split}\]
Expansion of \(v_{1,n}\) is done through
\[\begin{split}v_{1,n}
&= e^{-k_1h}v_{1,n-1} + (1 - e^{-k_1h})\theta_1 + \sigma_{v1}
e^{-k_1nh}I\!E_{1,n},\\
v_{1,n}^m
&= \sum_{\boldsymbol{m}} c_v(\boldsymbol{m}) b_v(\boldsymbol{m}) \cdot
v_{1,n-1}^{m_1}(e^{-k_1nh}I\!E_{1,n})^{m_3},\end{split}\]
(taking \(v_{1,n}^m\) as an example), where \(\boldsymbol{m} = (m_1,m_2,m_3)\), \(m_1+m_2+m_3 = m\), and
\[c_v(\boldsymbol{m})
\triangleq C_m^{m_1}C_{m-m_1}^{m_2},
\quad
b_v(\boldsymbol{m})
\triangleq e^{-m_1 k_1h} \cdot [(1-e^{-k_1h})\theta_1]^{m_2} \cdot
\sigma_{v1}^{m_3}.\]
Expansion of \(v_{2,n}\) is done similarly.
In summary, I defined
API¶
Central Moments for Two-Factor SV model |
|
Moments for Two-Factor SV |
|
Covariances for Two-Factor SV |
|
Module for generating a trajectory of samples from mdl_2fsv by Euler approximation |