The characteristic function of the VGSA model is defined as a specific parameterization of the characteristic function of the CIR (Cox-Ingersol-Ross mean reverting process) time-change:
$ \mathbb{E}e^{iuY(t)} = \varphi_{VGSA}(u,t,y(0),\kappa, \eta, \lambda) = A(u,t,\kappa,\eta,\lambda)e^{B(u,t,\kappa,\lambda)y(0)} $
Where \begin{align*} A(u,t,\kappa,\eta,\lambda) &= \frac{\exp\Big( \frac{\kappa^2\eta t}{\lambda^2} \Big) }{\Big( \cosh(\gamma t/2) + \frac{\kappa}{\gamma}\sinh(\gamma t/2) \Big)^{\frac{2\kappa\eta}{\lambda^2}}}\\ B(u,t,\kappa,\lambda) &= \frac{2iu}{\kappa + \gamma\coth(\gamma t/2)}\\ \gamma &= \sqrt{\kappa^2-2\lambda^2iu} \end{align*}
The VGSA characteristic function is given by: $ \mathbb{E}e^{iuZ_{VGSA}(t)} = \varphi_{VGSA}(-i\Psi_{VG}(u), t, \nu^{-1}, \kappa, \eta, \lambda) $
Where $\Psi_{VG}(u)$ is the log characteristic function of Variance Gamma at unit time:
$ \Psi_{VG}(u) = -\frac{1}{\nu}\log(1-iu\nu\theta + \sigma^2\nu u^2/2) $
The cumulant generating function satisfies:
$ G(w) = \ln \mathbb{E}e^{wX} = \ln(\varphi_{VGSA}(-iw)) $
Its $n$-th cumulant is the $n$-th derivative of the cumulant generating function evaluated at zero: $ c_n = G^{(n)}(0) = \frac{-i\varphi_{VGSA}^{(n)}(0)}{\varphi_{VGSA}(0)} $
I need the first, second, and fourth cumulants in order to price VGSA using the Fourier-cosine method.
Does anyone have these cumulants or know if they even exist in closed form?