ajdmom.mdl_1fsv.cond_cmom

Conditional Central Moment

Conditional central moments of the Heston SV model, given the initial variance.

Note that the keyfor attribute is different from that in the module ajdmom.mdl_1fsv.cmom, which now is (‘e^{kt}’, ‘t’, ‘k^{-}’, ‘v_0-theta’, ‘theta’, ‘sigma’, ‘sigma/2k’, ‘rho-sigma/2k’, ‘sqrt(1-rho^2)’)

\[y_{n-1,t} = (\mu - \theta/2)[t-(n-1)h] - \beta_{n-1,t}(v_{n-1}-\theta) + \bar{y}_{n-1,t}\]

where \(\beta_{n-1,t} = (1-e^{-k[t-(n-1)h]})/(2k)\) and the centralized term

\[\bar{y}_{n-1,t} = \frac{\sigma_v}{2k}e^{-kt}I\!E_{n-1,t} + \left( \rho - \frac{\sigma_v}{2k} \right)I_{n-1,t} + \sqrt{1-\rho^2}I_{n-1,t}^{*}.\]

The module implements the derivation of conditional moments of the centralized return

\[\mathbb{E}[\bar{y}_{n-1,t}^m|v_{n-1}] = \sum_{m_1+m_2+m_3=m} c(\boldsymbol{m}) b(\boldsymbol{m}) (e^{-kt}I\!E_{n-1,t})^{m_1}(I_{n-1,t})^{m_2}(I_{n-1,t}^{*})^{m_3}\]

with \(c(\boldsymbol{m}) = C_m^{m_1}C_{m-m_1}^{m_2}\), \(b(\boldsymbol{m}) = \left(\frac{\sigma_v}{2k}\right)^{m_1} \left(\rho - \frac{\sigma_v}{2k}\right)^{m_2} \left(\sqrt{1-\rho^2}\right)^{m_3}\) and \(\boldsymbol{m} = (m_1,m_2,m_3)\).

Functions

cmoments_y_to(l[, show])	Conditional central moments from order 1 to \(l\).
cond_cm(l, par)	Conditional central moment in scalar
int_mIEII(c, tp, m, poly)	Integral of \(c \times tp \times \int_{(n-1)h}^t e^{mks} poly ds\)
poly2num(poly, par)	Decode poly back to scalar
recursive_IEII(n3, n4, n5, IEII)	Recursive step in equation (3)
simplify(poly[, tp])	Simplify polynomials differently