A Proof of Theorem 1
To obtain the joint characteristic function for the log-asset price \(\ln S(T)\) and the log-assets value \(\ln V(T)\), we first take the following transformation
$$\begin{aligned} d \ln S(t)= & {} (r(t)-\frac{1}{2}Z(t))dt+\sqrt{Z(t)}d W_{s}^{T}(t), \end{aligned}$$
(52)
$$\begin{aligned} d \ln V(t)= & {} \left( r(t)-\frac{1}{2}\sigma _{v}^{2}\right) dt +\sigma _{v}\left( \rho _{2}d W_{s}^{T}(t)+\sqrt{(1-\rho _{2}^{2})}dW_{v}^{T}(t)\right) , \end{aligned}$$
(53)
Then, Feynman–Kac theorem gives the following partial differential equation (PDE) for the characteristic function
$$\begin{aligned}&-\frac{\partial f}{\partial \tau }+\left( r-\frac{1}{2}z\right) \frac{\partial f}{\partial x}+\frac{1}{2}z\frac{\partial ^{2}f}{\partial x^{2}}+ \kappa _{z}(\theta -z)\frac{\partial f}{\partial z}+\frac{1}{2} \sigma _{z}^{2}z^{2}\frac{\partial ^{2}f}{\partial z^{2}}+ \left( r-\frac{1}{2}\sigma _{v}^{2}\right) \frac{\partial f}{\partial y}\nonumber \\&\quad +\frac{1}{2}\sigma _{v}^{2}\frac{\partial ^{2}f}{\partial y^{2}}+(b-\sigma _{r}^{2}B_{r}(\tau )-ar)\frac{\partial f}{\partial r}+\frac{1}{2}\sigma _{r}^{2}\frac{\partial ^{2}f}{\partial r^{2}}+\frac{\partial ^{2}f}{\partial x\partial z} z^{\frac{3}{2}}\sigma _{z}\rho _{1}+\frac{\partial ^{2}f}{\partial x\partial y} z^{\frac{1}{2}}\sigma _{v}\rho _{2}\nonumber \\&\quad +\frac{\partial ^{2}f}{\partial y\partial z} z\sigma _{z}\sigma _{v}\rho _{1}\rho _{2}=0, \end{aligned}$$
(54)
The boundary condition for Eq. (54) is given by
$$\begin{aligned} f(x,y,r,z,0;\varphi _{1},\varphi _{2})=e^{i\varphi _{1}\ln S(T)+i\varphi _{2}\ln V(T)}. \end{aligned}$$
(55)
Equation (54) is a non-linear PDE. The general results for affine processes no longer apply here, and there is no known exact analytical solution to this equation. However, an approximate solution is provided by using perturbation method (Kevorkian et al. 1982; Chacko and Viceira 2003). The idea is to approximate \(z^{2}\), \(z^{\frac{3}{2}}\) and \(z^{\frac{1}{2}}\) in the PDE using Taylor expansions around the long-run mean of variance as follows:
$$\begin{aligned} z^{2}= & {} 2\theta z-\theta ^{2}, \end{aligned}$$
(56)
$$\begin{aligned} z^{\frac{3}{2}}= & {} \frac{3}{2}\theta ^{\frac{1}{2}} z-\frac{1}{2}\theta ^{\frac{3}{2}}, \end{aligned}$$
(57)
$$\begin{aligned} z^{\frac{1}{2}}= & {} \frac{1}{2}\theta ^{-\frac{1}{2}} z+\frac{1}{2}\theta ^{\frac{1}{2}}, \end{aligned}$$
(58)
Substituting Eqs. (56), (57) and (58) into the PDE in Eq. (54), we have
$$\begin{aligned}&-\frac{\partial f}{\partial \tau }+ \left( r-\frac{1}{2}z\right) \frac{\partial f}{\partial x}+\frac{1}{2}z\frac{\partial ^{2}f}{\partial x^{2}}+ \kappa _{z}(\theta -z)\frac{\partial f}{\partial z}+\frac{1}{2}\sigma _{z}^{2}(2\theta z-\theta ^{2})\frac{\partial ^{2}f}{\partial z^{2}}+ \left( r-\frac{1}{2}\sigma _{v}^{2}\right) \frac{\partial f}{\partial y}\nonumber \\&\quad +\frac{1}{2}\sigma _{v}^{2}\frac{\partial ^{2}f}{\partial y^{2}}+(b-\sigma _{r}^{2}B_{r}(\tau )-ar)\frac{\partial f}{\partial r}+\frac{1}{2}\sigma _{r}^{2}\frac{\partial ^{2}f}{\partial r^{2}}+\frac{\partial ^{2}f}{\partial x\partial z} \left( \frac{3}{2}\theta ^{\frac{1}{2}}z -\frac{1}{2}\theta ^{\frac{3}{2}}\right) \sigma _{z}\rho _{1}\nonumber \\&\quad +\frac{\partial ^{2}f}{\partial x\partial y} \left( \frac{1}{2}\theta ^{-\frac{1}{2}}z +\frac{1}{2}\theta ^{\frac{1}{2}}\right) \sigma _{v}\rho _{2}+\frac{\partial ^{2}f}{\partial y\partial z} z\sigma _{z}\sigma _{v}\rho _{1}\rho _{2}=0, \end{aligned}$$
(59)
This PDE has an exponential-affine solution of the form
$$\begin{aligned} f(x,y,r,z,\tau ;\varphi _{1},\varphi _{2})=e^{i\varphi _{1} x+i\varphi _{2} y+D(\tau ,\varphi _{1},\varphi _{2})r+E(\tau ,\varphi _{1},\varphi _{2})z+F(\tau ,\varphi _{1},\varphi _{2})}, \end{aligned}$$
(60)
with the boundary conditions
$$\begin{aligned} D(0,\varphi _{1},\varphi _{2})=E(0,\varphi _{1},\varphi _{2}) =F(0,\varphi _{1},\varphi _{2})=0. \end{aligned}$$
(61)
Substituting Eq. (60) into PDE in Eq. (59) yields
$$\begin{aligned}&-\,(D_{\tau }(\tau ,\varphi _{1},\varphi _{2})r+E_{\tau }(\tau ,\varphi _{1}, \varphi _{2})z+F_{\tau }(\tau ,\varphi _{1},\varphi _{2}))+\left( r-\frac{1}{2}z\right) i \varphi _{1} \nonumber \\&\quad -\,\frac{1}{2}\varphi _{1}^{2}z+\kappa _{z}(\theta -z)E(\tau ,\varphi _{1}, \varphi _{2}) \nonumber \\&\quad +\,\frac{1}{2}\sigma _{z}^{2}(2\theta z-\theta ^{2})E^{2}(\tau ,\varphi _{1},\varphi _{2}) +\left( r-\frac{1}{2}\sigma _{v}^{2}\right) i\varphi _{2} \nonumber \\&\quad -\,\frac{1}{2}\varphi _{2}^{2}\sigma _{v}^{2}+ (b-\sigma _{r}^{2}B_{r}(\tau )-ar)D(\tau ,\varphi _{1},\varphi _{2}) +\frac{1}{2}\sigma _{r}^{2}D^{2}(\tau ,\varphi _{1},\varphi _{2})\nonumber \\&\quad +\,i\varphi _{1}E(\tau ,\varphi _{1},\varphi _{2})(\frac{3}{2}\theta ^{\frac{1}{2}}z -\frac{1}{2}\theta ^{\frac{3}{2}}) \sigma _{z}\rho _{1}-\varphi _{1}\varphi _{2}\left( \frac{1}{2}\theta ^{-\frac{1}{2}}z +\frac{1}{2}\theta ^{\frac{1}{2}}\right) \sigma _{v}\rho _{2} \nonumber \\&\qquad i\varphi _{2}E(\tau ,\varphi _{1},\varphi _{2})\sigma _{z}\sigma _{v}\rho _{1} \rho _{2}z=0. \end{aligned}$$
(62)
By matching cofficients, we can derive the following three ordinary differential equations (ODEs)
$$\begin{aligned} D_{\tau }(\tau ,\varphi _{1},\varphi _{2})= & {} -a D(\tau ,\varphi _{1},\varphi _{2})+i(\varphi _{1}+\varphi _{2}), \end{aligned}$$
(63)
$$\begin{aligned} E_{\tau }(\tau ,\varphi _{1},\varphi _{2})= & {} \sigma _{z}^{2}\theta E^{2}(\tau ,\varphi _{1},\varphi _{2}) +\left( \frac{3}{2}i\varphi _{1}\theta ^{\frac{1}{2}} \sigma _{z}\rho _{1}+i\varphi _{2}\sigma _{z}\sigma _{v}\rho _{1} \rho _{2}-\kappa _{z}\right) E(\tau ,\varphi _{1},\varphi _{2}) \nonumber \\&-\frac{1}{2}\left( i\varphi _{1}+\varphi _{1}^{2} +\varphi _{1}\varphi _{2}\theta ^{-\frac{1}{2}}\sigma _{v}\rho _{2}\right) , \end{aligned}$$
(64)
$$\begin{aligned} F_{\tau }(\tau ,\varphi _{1},\varphi _{2})= & {} -\frac{1}{2}\sigma _{z}^{2}\theta ^{2}E(\tau ,\varphi _{1},\varphi _{2})^{2} +\left( \kappa _{z}\theta -\frac{1}{2}i\varphi _{1}\sigma _{z} \theta ^{\frac{3}{2}}\rho _{1}\right) E(\tau ,\varphi _{1},\varphi _{2})\nonumber \\&+(b-\sigma _{r}^{2}B_{r}(\tau ))D(\tau ,\varphi _{1},\varphi _{2})\nonumber \\&+\frac{1}{2}\sigma _{r}^{2}D^{2} (\tau ,\varphi _{1},\varphi _{2})-\frac{1}{2} \left( i\varphi _{2}\sigma _{v}^{2}+\varphi _{2}^{2}\sigma _{v}^{2}+\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2}\right) , \end{aligned}$$
(65)
Accord to the boundary condition (61), we obtain
$$\begin{aligned} D(\tau ,\varphi _{1},\varphi _{2})=\frac{i(\varphi _{1}+\varphi _{2})}{a}(1-\exp \{-a\tau \}), \end{aligned}$$
(66)
We consider the ODE (64), which the general solution can be derived for Riccati equation.
$$\begin{aligned} E_{\tau }(\tau ,\varphi _{1},\varphi _{2})=k_{2} E^{2}(\tau ,\varphi _{1},\varphi _{2})+k_{1}E(\tau ,\varphi _{1},\varphi _{2})+k_{0}, \end{aligned}$$
(67)
where \(k_{0}=-\frac{1}{2}(i\varphi _{1}+\varphi _{1}^{2}+\varphi _{1}\varphi _{2}\theta ^{-\frac{1}{2}}\sigma _{v}\rho _{2})\), \(k_{1}=(\frac{3}{2}i\varphi _{1}\theta ^{\frac{1}{2}}\sigma _{z}\rho _{1}+i\varphi _{2}\sigma _{z}\sigma _{v}\rho _{1}\rho _{2}-\kappa _{z})\), \(k_{2}=\sigma _{z}^{2}\theta \).
Making the subsitution
$$\begin{aligned} E(\tau ,\varphi _{1},\varphi _{2})=-\frac{O{'}(\tau )}{k_{2}O(\tau )} \end{aligned}$$
(68)
Hence,
$$\begin{aligned} O^{''}(\tau )-k_{1}O{'}(\tau )+k_{0}k_{2}O(\tau )=0. \end{aligned}$$
(69)
The ODE (69) has a general solution of the form
$$\begin{aligned} O(\tau )=d_{1}e^{\omega _{+}\tau }+d_{2}e^{\omega _{-}\tau }, \end{aligned}$$
(70)
where
$$\begin{aligned} \omega _{\pm }=\frac{k_{1}\pm \zeta }{2},\nonumber \\ \zeta =\sqrt{k_{1}^{2}-4k_{0}k_{2}}. \end{aligned}$$
(71)
According to (70) and the initial condition \(E(0,\varphi _{1},\varphi _{2})=0\), \(d_{1}\) and \(d_{2}\) are two constants to be determind from boundary conditions,
$$\begin{aligned} \left\{ \begin{array}{l} O(0)=d_{1}+d_{2},\\ O{'}(0)=d_{+}\omega _{+}+d_{2}\omega _{-}=0. \end{array} \right. \end{aligned}$$
(72)
We obtain the solution to (72) is \(d_{1}=\frac{-O(0)\omega _{-}}{\zeta }\) and \(d_{2}=\frac{O(0)\omega _{+}}{\zeta }\). Therefore, the general solution for (68) is
$$\begin{aligned} E(\tau ,\varphi _{1},\varphi _{2})= & {} -\frac{1}{k_{2}}\frac{-\frac{O(0)\omega _{-}\omega _{+}}{\zeta }e^{\omega _{+}\tau } +\frac{O(0)\omega _{+}\omega _{-}}{\zeta }e^{\omega _{-}\tau }}{-\frac{O(0)\omega _{-}}{\zeta }e^{\omega _{+}\tau }+\frac{O(0)\omega _{+}}{\zeta }e^{\omega _{-}\tau }}\nonumber \\= & {} -k_{0}\frac{-e^{\omega _{+}\tau }+e^{\omega _{-}\tau }}{-\omega _{-}e^{\omega _{+}\tau }+\omega _{+}e^{\omega _{-}\tau }}\nonumber \\= & {} k_{0}\frac{1-e^{-\zeta \tau }}{-\omega _{-}+\omega _{+}e^{-\zeta \tau }}. \end{aligned}$$
(73)
multiplying \(-\frac{1}{2}\theta \) on both sides of the Eq. (64), we can obtain
$$\begin{aligned} -\frac{1}{2}\sigma _{z}^{2}\theta ^{2}E(\tau ,\varphi _{1},\varphi _{2})^{2}= & {} -\frac{1}{2}\theta E_{\tau }(\tau ,\varphi _{1},\varphi _{2})+ \left( \frac{3}{4}i\varphi _{1}\sigma _{z}\theta ^{\frac{3}{2}}\rho _{1} +\frac{1}{2}i\varphi _{2}\theta \sigma _{z}\sigma _{v}\rho _{1}\rho _{2} -\frac{1}{2}\kappa _{z}\theta \right) \nonumber \\&\quad \times E(\tau ,\varphi _{1},\varphi _{2})-\frac{1}{4} \left( i\varphi _{1}\theta +\varphi _{1}^{2}\theta +\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2}\right) , \end{aligned}$$
(74)
Substituting (74) into (65), we have
$$\begin{aligned} F_{\tau }(\tau ,\varphi _{1},\varphi _{2})= & {} -\frac{1}{2}\theta E_{\tau }(\tau ,\varphi _{1},\varphi _{2}) +\left( \frac{1}{4}i\varphi _{1}\theta ^{\frac{3}{2}}\sigma _{z}\rho _{1}+\frac{1}{2}i\varphi _{2}\theta \sigma _{z}\sigma _{v}\rho _{1}\rho _{2} +\frac{1}{2}\kappa _{z}\theta \right) \nonumber \\&\times E(\tau ,\varphi _{1},\varphi _{2})-\frac{1}{2}\left( i\varphi _{2}\sigma _{v}^{2} +\varphi _{2}^{2}\sigma _{v}^{2}\right) \nonumber \\&-\,\frac{1}{4}(i\varphi _{1}\theta \nonumber +\varphi _{1}^{2}\theta ) -\frac{3}{4}\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2} +(b-\sigma _{r}B_{r}(\tau ))D(\tau ,\varphi _{1},\varphi _{2})\nonumber \\&+\frac{1}{2}\sigma _{r}^{2}D^{2}(\tau ,\varphi _{1},\varphi _{2}). \end{aligned}$$
(75)
By solving Eq. (75), we find the following expression for \(F(\tau ,\varphi _{1},\varphi _{2})\)
$$\begin{aligned} F(\tau ,\varphi _{1},\varphi _{2})= & {} -\frac{1}{2}\theta E(\tau ,\varphi _{1},\varphi _{2})+\int _{0}^{\tau } \left( \frac{1}{4}i\varphi _{1}\theta ^{\frac{3}{2}}\sigma _{z}\rho _{1}+\frac{1}{2}i\varphi _{2} \theta \sigma _{z}\sigma _{v}\rho _{1}\rho _{2} +\frac{1}{2}\kappa _{z}\theta \right) \nonumber \\&\quad \times E(u,\varphi _{1},\varphi _{2})du - \left[ \frac{1}{2}\left( i\varphi _{2}\sigma _{v}^{2}+\varphi _{2}^{2}\sigma _{v}^{2}\right) +\frac{1}{4}(i\varphi _{1}\theta +\varphi _{1}^{2}\theta ) +\frac{3}{4}\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2}\right] \tau \nonumber \\&\quad +\int _{0}^{\tau }(b-\sigma _{r}B_{r}(u))D(u,\varphi _{1},\varphi _{2})du +\int _{0}^{\tau }\frac{1}{2}\sigma _{r}^{2}D^{2} (u,\varphi _{1},\varphi _{2})d(u). \end{aligned}$$
(76)
we denote
$$\begin{aligned} \varLambda (\tau )=\int _{0}^{\tau } (b-\sigma _{r}^{2}B_{r}(u))D(u,\varphi _{1},\varphi _{2})du +\int _{0}^{\tau }\frac{1}{2}\sigma _{r}^{2}D^{2} (u,\varphi _{1},\varphi _{2})d(u), \end{aligned}$$
(77)
then
$$\begin{aligned}&\varLambda (\tau )=\int _{0}^{\tau }(b-\sigma _{r}^{2}B_{r}(u)) D(u,\varphi _{1},\varphi _{2})du+\int _{0}^{\tau }\frac{1}{2} \sigma _{r}^{2}D^{2}(u,\varphi _{1},\varphi _{2})d(u) \nonumber \\&\quad =\frac{i(\varphi _{1}+\varphi _{2})b}{a}\tau +\frac{i(\varphi _{1}+\varphi _{2})b}{a^{2}}(e^{-a\tau }-1) \nonumber \\&\quad -\frac{1}{a^{2}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}+\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \tau \nonumber \\&\quad +\frac{1}{a^{3}}\left( i(\varphi _{1}+\varphi _{2}) \sigma _{r}^{2}+\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \left( \frac{1}{2}e^{-2a\tau }-2e^{-a\tau }\right) \nonumber \\&\quad +\frac{3}{2a^{3}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}+\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \nonumber \\&\quad =\left( \frac{i(\varphi _{1}+\varphi _{2})b}{a} -\frac{1}{a^{2}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}+\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \right) \tau \nonumber \\&\quad +\frac{i(\varphi _{1}+\varphi _{2})b}{a^{2}}e^{-a\tau } +\frac{1}{a^{3}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}\right. \nonumber \\&\qquad \left. +\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \left( \frac{1}{2}e^{-2a\tau }-2e^{-a\tau }\right) +\frac{3}{2a^{3}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}\right. \nonumber \\&\qquad \left. +\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) -\frac{i(\varphi _{1}+\varphi _{2})b}{a^{2}}. \end{aligned}$$
(78)
Let \(k_{3}=(\frac{1}{4}i\varphi _{1}\theta ^{\frac{3}{2}}\sigma _{z}\rho _{1}+\frac{1}{2}i\varphi _{2}\theta \sigma _{z}\sigma _{v}\rho _{1}\rho _{2} +\frac{1}{2}\kappa _{z}\theta )\), we can obtain
$$\begin{aligned}&\int _{0}^{\tau }\left( \frac{1}{4}i\varphi _{1}\theta ^{\frac{3}{2}} \sigma _{z}\rho _{1}+\frac{1}{2}i\varphi _{2}\sigma _{z}\sigma _{v}\rho _{1} \rho _{2} +\frac{1}{2}\kappa _{z}\theta \right) E(u,\varphi _{1},\varphi _{2}) du=k_{3}\int _{0}^{\tau }E(u,\varphi _{1},\varphi _{2})du\nonumber \\&\quad =-\frac{k_{3}}{k_{2}}\int _{0}^{\tau }\frac{O{'}(u)}{O(u)}du =-\frac{k_{3}}{k_{2}}(\ln O(u)|_{u=0}^{u=\tau }) \nonumber \\&\quad =-\frac{k_{3}}{k_{2}}\ln [\frac{O(\tau )}{O(0)}] =-\frac{k_{3}}{k_{2}}\ln \left( \frac{-\frac{O(0)\omega _{-}}{\zeta }e^{\omega _{+}\tau }+\frac{O(0)\omega _{+}}{\zeta }e^{\omega _{-}\tau }}{-\frac{O(0)\omega _{-}}{\zeta }+\frac{O(0)\omega _{+}}{\zeta }}\right) \nonumber \\&\quad =-\frac{k_{3}}{k_{2}} \left[ \omega _{+}\tau +\ln \left( \frac{-\omega _{-}+\omega _{+}e^{-\zeta \tau }}{\zeta }\right) \right] \end{aligned}$$
(79)
Finally, we get the following expression for \(F(\tau ,\varphi _{1},\varphi _{2})\)
$$\begin{aligned} F(\tau ,\varphi _{1},\varphi _{2})= & {} -\frac{1}{2}\theta E(\tau ,\varphi _{1},\varphi _{2})-\frac{k_{3}}{k_{2}} \left[ \omega _{+}\tau +\ln \left( \frac{-\omega _{-}+\omega _{+}e^{-\zeta \tau }}{\zeta }\right) \right] -\left[ \frac{1}{2}\left( i\varphi _{2}\sigma _{v}^{2}+\varphi _{2}^{2}\sigma _{v}^{2}\right) \right. \nonumber \\&\left. +\frac{1}{4}\left( i\varphi _{1}\theta +\varphi _{1}^{2}\theta \right) +\frac{3}{4}\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2}\right] \tau +\varLambda (\tau ). \end{aligned}$$
(80)
B Proof of Theorem 2
Proof
For \({\widehat{C}}_{1}(k,\xi )\), by the Fubini’s theorem we have
$$\begin{aligned}&H_{1c}(\phi _{1},\phi _{2}):=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{i\phi _{1} k}e^{i \phi _{2} \xi }{\widehat{C}}_{1} (k,\xi )d k d \xi \nonumber \\&\quad =\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{(i\phi _{1}+\alpha _{1}) k}e^{(i \phi _{2} +\beta _{1})\xi }E^{Q^{T}}[(e^{X_{T}}-e^{k})^{+}{\mathbf {1}}_{\{V(T)\ge e^{\xi }\}}|{\mathcal {F}}_{t}]d k d \xi \nonumber \\&\quad =E^{Q^{T}}\left[ \int _{-\infty }^{+\infty }e^{\beta _{1}\xi }e^{i\phi _{2}\xi }{\mathbf {1}}_{\{e^{Y_{T}}\ge e^{\xi }\}}\times \int _{-\infty }^{X_{T}}(e^{\alpha _{1}k}e^{i\phi _{1}k}e^{X_{T}} -e^{\alpha _{1}k}e^{i\phi _{1}k}e^{k})dkd\xi |{\mathcal {F}}_{t}\right] \nonumber \\&\quad =\frac{1}{\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{1}+1)\phi _{1}}E^{Q^{T}} \left[ \int _{-\infty }^{+\infty }e^{\beta _{1}\xi }e^{i\phi _{2}\xi }{\mathbf {1}}_{\{e^{Y_{T}}\ge e^{\xi }\}}e^{(\alpha _{1}+1+i\phi _{1})X_{T}}d\xi |{\mathcal {F}}_{t}\right] \nonumber \\&\quad =\frac{1}{\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i (2\alpha _{1}+1)\phi _{1}}E^{Q^{T}} \left[ \int _{-\infty }^{Y_{T}}e^{\beta _{1}\xi }e^{i\phi _{2}\xi } e^{(\alpha _{1}+1+i\phi _{1})X_{T}}d\xi |{\mathcal {F}}_{t} \right] \nonumber \\&\quad =\frac{1}{(\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{1}+1)\phi _{1})(\beta _{1}+i\phi _{2})}E^{Q^{T}} \left[ e^{(\alpha _{1}+1+i\phi _{1})X_{T}} e^{(\beta _{1}\xi +i\phi _{2}\xi )Y_{T}}|{\mathcal {F}}_{t}\right] \nonumber \\&\quad =\frac{f(x,y,r,z,T-t;\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=\phi _{2}-\beta _{1}i)}{(\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2} +i(2\alpha _{1}+1)\phi _{1})(\beta _{1}+i\phi _{2})}, \end{aligned}$$
(81)
Meanwhile, for \({\widehat{C}}_{2}(k,{\widetilde{\xi }})\), with similar calculation process, we can obtain
$$\begin{aligned}&H_{2c}(\phi _{1},{\widetilde{\phi }}_{2}):=\int _{-\infty }^{+\infty } \int _{-\infty }^{+\infty }e^{i\phi _{1} k}e^{i {\widetilde{\phi }}_{2}{\widetilde{\xi }}}{\widehat{C}}_{2}(k,{\widetilde{\xi }})d k d {\widetilde{\xi }}\nonumber \\&\quad =\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{i\phi _{1} k}e^{i {\widetilde{\phi }}_{2}{\widetilde{\xi }}} E^{Q^{T}}[e^{Y_{T}}(e^{X_{T}}-e^{k})^{+}{\mathbf {1}}_{\{e^{Y_{T}}<e^{-{\widetilde{\xi }}}\}}|{\mathcal {F}}_{t}] d k d {\widetilde{\xi }}\nonumber \\&\quad =E^{Q^{T}} \left[ \int _{-\infty }^{+\infty }e^{Y_{T}}e^{\beta _{2}{\widetilde{\xi }}}e^{i\widetilde{\phi _{2}}{\widetilde{\xi }}}{\mathbf {1}}_{\{e^{Y_{T}}< e^{-{\widetilde{\xi }}}\}}\times \int _{-\infty }^{X_{T}}(e^{\alpha _{1}k}e^{i\phi _{1}k}e^{X_{T}} -e^{\alpha _{1}k}e^{i\phi _{1}k}e^{k})dkd{\widetilde{\xi }}|{\mathcal {F}}_{t} \right] \nonumber \\&\quad =\frac{1}{(\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{1}+1) \phi _{1})}E^{Q^{T}}\left[ \int _{-\infty }^{+\infty }e^{Y_{T}}e^{\beta _{2} {\widetilde{\xi }}}e^{i{\widetilde{\phi }}_{2}{\widetilde{\xi }}} {\mathbf {1}}_{\{e^{Y_{T}}< e^{-{\widetilde{\xi }}}\}} e^{(\alpha _{1}+1+i\phi _{1})X_{T}}d{\widetilde{\xi }}|{\mathcal {F}}_{t} \right] \nonumber \\&\quad =\frac{1}{(\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{2}+1) \phi _{1})}E^{Q^{T}}\left[ \int _{-\infty }^{-Y_{T}}e^{Y_{T}}e^{\beta _{2} {\widetilde{\xi }}}e^{i{\widetilde{\phi }}_{2}{\widetilde{\xi }}} e^{(\alpha _{1}+1+i\phi _{1})X_{T}}d{\widetilde{\xi }} |{\mathcal {F}}_{t}\right] \nonumber \\&\quad =\frac{1}{(\alpha _{1}^{2}+\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{1}+1)\phi _{1}) (\beta _{2}+i{\widetilde{\phi }}_{2})}E^{Q^{T}} \left[ e^{(\alpha _{1}+1+i\phi _{1})X_{T}}e^{(1-\beta _{2}-i{\widetilde{\phi }}_{2}) Y_{T}}|{\mathcal {F}}_{t}\right] \nonumber \\&\quad =\frac{f(x,y,r,z,T-t;\varphi _{1} =\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=-{\widetilde{\phi }}_{2}-(1-\beta _{2})i)}{(\alpha _{1}^{2} +\alpha _{1}-\phi _{1}^{2}+i(2\alpha _{1}+1)\phi _{1}) (\beta _{2}+i{\widetilde{\phi }}_{2})}. \end{aligned}$$
(82)
C Proof of Theorem 4
According to the Carr and Madan (1999), the hedge ratio Delta \(\varDelta _{call}\) is given as follows: Delta:
$$\begin{aligned} \varDelta _{call}= & {} \frac{\partial C(t,S(t),V(t),r(t),Z(t),T)}{\partial S(t)}\nonumber \\= & {} P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{\partial H_{1c}(\phi _{1}, \phi _{2})}{\partial S(t)}d\phi _{1}d\phi _{2} \nonumber \\&+P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{\partial H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})}{\partial S(t)}d\phi _{1}d{\widetilde{\phi }}_{2} \nonumber \\= & {} P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{S(t)(2\pi )^{2}}\varPi _{1}(t,T) +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{S(t)(2\pi )^{2}} \varPi _{2}(t,T), \end{aligned}$$
(83)
where
$$\begin{aligned} \varPi _{1}(t,T)= & {} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{f(x,y,r,z,T-t;\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=\phi _{2}-\beta _{1}i)}{(\alpha _{1}+i\phi _{1}) (\beta _{1}+i\phi _{2})}d\phi _{1}d\phi _{2}, \end{aligned}$$
and
$$\begin{aligned} \varPi _{2}(t,T)= & {} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{f(x,y,r,z,T-t;\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=-{\widetilde{\phi }}_{2}-(1-\beta _{2})i)}{(\alpha _{1}+i\phi _{1}) (\beta _{2}+i{\widetilde{\phi }}_{2})}d\phi _{1}d{\widetilde{\phi }}_{2}. \end{aligned}$$
The Gamma \(\varGamma _{call}\) of the vulnerable option price is given by the second derivative of the option price with the respect to the underlying asset price:
Gamma:
$$\begin{aligned} \varGamma _{call}= & {} \frac{\partial \varDelta _{call}}{\partial S(t)} \nonumber \\= & {} -P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{S(t)^{2}(2\pi )^{2}}\varPi _{1}(t, T)+ P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{S(t)(2\pi )^{2}}\frac{\partial \varPi _{1}(t,T)}{\partial S(t)} \nonumber \\&-P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{S(t)^{2}(2\pi )^{2}} \varPi _{2}(t,T)+P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{S(t)(2\pi )^{2}} \frac{\partial \varPi _{2}(t,T)}{\partial S(t)} \nonumber \\= & {} P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{S(t)^{2}(2\pi )^{2}}\varPi _{3}(t,T) +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{S(t)^{2}(2\pi )^{2}} \varPi _{4}(t,T), \end{aligned}$$
(84)
where
$$\begin{aligned} \varPi _{3}(t,T)= \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{f(x,y,r,z,T-t;\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=\phi _{2}-\beta _{1}i)}{ \beta _{1}+i\phi _{2}}d\phi _{1}d\phi _{2}, \end{aligned}$$
and
$$\begin{aligned} \varPi _{4}(t,T)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{f(x,y,r,z,T-t;\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=-{\widetilde{\phi }}_{2}-(1-\beta _{2})i)}{\beta _{2}+i{\widetilde{\phi }}_{2}}d\phi _{1}d{\widetilde{\phi }}_{2}. \end{aligned}$$
The general form for Vega \(\nu _{call}\) is given by
$$\begin{aligned} \nu _{call}= & {} \frac{\partial C(t,S(t),V(t),r(t),Z(t),T)}{\partial Z(t)}\nonumber \\= & {} P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{\partial H_{1c}(\phi _{1}, \phi _{2})}{\partial Z(t)}d\phi _{1}d\phi _{2} \nonumber \\&+P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{\partial H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})}{\partial Z(t)}d\phi _{1}d{\widetilde{\phi }}_{2} \nonumber \\= & {} P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\varPi _{5}(t,T) +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \varPi _{6}(t,T), \end{aligned}$$
(85)
where
$$\begin{aligned} \varPi _{5}(t,T)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}E(\tau ,\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i,\varphi _{2}=\phi _{2}-\beta _{1}i)H_{1c} (\phi _{1},\phi _{2})d\phi _{1}d\phi _{2}, \end{aligned}$$
and
$$\begin{aligned} \varPi _{6}(t,T)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}E(\tau ,\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i, \varphi _{2}=-{\widetilde{\phi }}_{2}-(1-\beta _{2})i)H_{2c}(\phi _{1},{\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}. \end{aligned}$$
For Rho \(\rho _{call}\) yields
$$\begin{aligned} \rho _{call}= & {} \frac{\partial C(t,S(t),V(t),r(t),Z(t),T)}{\partial r(t)}\nonumber \\= & {} P(t,T)(-B_{r}(\tau ))\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}H_{1c}(\phi _{1}, \phi _{2})d\phi _{1}\phi _{2}\nonumber \\&+ P(t,T)(-B_{r}(\tau ))\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}\nonumber \\&+P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{\partial H_{1c}(\phi _{1}, \phi _{2})}{\partial r(t)}d\phi _{1}d\phi _{2} \nonumber \\&+P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{\partial H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})}{\partial r(t)}d\phi _{1}d{\widetilde{\phi }}_{2} \nonumber \\= & {} P(t,T)(-B_{r}(\tau ))\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}H_{1c}(\phi _{1}, \phi _{2})d\phi _{1}\phi _{2}\nonumber \\&+ P(t,T)(-B_{r}(\tau ))\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}\nonumber \\&+ P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\varPi _{7}(t,T) +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \varPi _{8}(t,T), \end{aligned}$$
(86)
where
$$\begin{aligned} \varPi _{7}(t,T)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }&e^{-i(\phi _{1}k +\phi _{2}\xi )}D(\tau ,\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i,\\&\varphi _{2} =\phi _{2}-\beta _{1}i) H_{1c}(\phi _{1},\phi _{2})d\phi _{1}d\phi _{2}, \end{aligned}$$
and
$$\begin{aligned} \varPi _{8}(t,T)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }&e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}D(\tau ,\varphi _{1}= \phi _{1}-(\alpha _{1}+1)i,\\&\varphi _{2} =-{\widetilde{\phi }}_{2}-(1-\beta _{2})i)H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}. \end{aligned}$$
The Theta \(\varTheta _{call}\) of the vulnerable option price is given by
$$\begin{aligned} \varTheta _{call}= & {} \frac{\partial C(t,S(t),V(t),r(t),Z(t),T)}{\partial t}=-\frac{\partial C(t,S(t),V(t),r(t),Z(t),T)}{\partial \tau }\nonumber \\= & {} -\left[ \frac{\partial P(t,T)}{\partial \tau }\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k +\phi _{2}\xi )}H_{1c}(\phi _{1}, \phi _{2})d\phi _{1}\phi _{2}\right. \nonumber \\&+ P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-i(\phi _{1}k+\phi _{2}\xi )}\frac{\partial H_{1c}(\phi _{1}, \phi _{2})}{\partial \tau }d\phi _{1}\phi _{2}\nonumber \\&+ \frac{\partial P(t,T)}{\partial \tau }\frac{(1-\gamma )}{D} \frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2} \nonumber \\&\left. +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2} {\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k +{\widetilde{\phi }}_{2}{\widetilde{\xi }})}\frac{\partial H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})}{\partial \tau }d\phi _{1}d{\widetilde{\phi }}_{2}\right] \nonumber \\= & {} -\left[ P(t,T)(M(\tau ))\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+\phi _{2}\xi )}H_{1c}(\phi _{1}, \phi _{2})d\phi _{1}\phi _{2}\right. \nonumber \\&+ P(t,T)(M(\tau ))\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}H_{2c}(\phi _{1}, {\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}\nonumber \\&\left. + P(t,T)\frac{e^{-\alpha _{1}k-\beta _{1}\xi }}{(2\pi )^{2}}\varPi _{9}(t,T) +P(t,T)\frac{(1-\gamma )}{D}\frac{e^{-\alpha _{1}k-\beta _{2}{\widetilde{\xi }}}}{(2\pi )^{2}} \varPi _{10}(t,T)\right] , \end{aligned}$$
(87)
where
$$\begin{aligned} \varPi _{9}(t,T)= & {} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k +\phi _{2}\xi )}G(\tau ,\varphi _{1}=\phi _{1}-(\alpha _{1}+1)i,\varphi _{2}\\= & {} \phi _{2}-\beta _{1}i) H_{1c}(\phi _{1},\phi _{2})d\phi _{1}d\phi _{2},\\ \varPi _{10}(t,T)= & {} \int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-i(\phi _{1}k+{\widetilde{\phi }}_{2}{\widetilde{\xi }})}G(\tau ,\varphi _{1}= \phi _{1}-(\alpha _{1}+1)i,\varphi _{2}\\= & {} -{\widetilde{\phi }}_{2}-(1-\beta _{2})i)H_{2c}(\phi _{1},{\widetilde{\phi }}_{2})d\phi _{1}d{\widetilde{\phi }}_{2}.\\ M(\tau )= & {} \frac{\sigma _{r}^{2}-2ab}{2a^{2}}+\frac{ab-\sigma _{r}^{2}}{a^{2}}e^{-a\tau }+\frac{\sigma _{r}^{2}}{4a^{2}}e^{-2a\tau }-r(t)e^{-a\tau },\\ G(\tau ,\varphi _{1},\varphi _{2})= & {} i(\varphi _{1}+\varphi _{2})e^{-a\tau }r(t)+ k_{0}\frac{\zeta ^{2} e^{-\zeta \tau }}{(-\omega _{-}+\omega _{+}e^{-\zeta \tau })^{2}}\left( Z(t)-\frac{1}{2}\theta \right) \\&\quad -\frac{k_{3}}{k_{2}}\left[ \omega _{+}-\frac{\omega _{+}\zeta ^{2}e^{-\zeta \tau }}{(-\omega _{-}+\omega _{+}e^{-\zeta \tau })}\right] \\&\quad -\left[ \frac{1}{2}(i\varphi _{2}\sigma _{v}^{2}+\varphi _{2}^{2}\sigma _{v}^{2}) +\frac{1}{4}(i\varphi _{1}\theta +\varphi _{1}^{2}\theta ) +\frac{3}{4}\varphi _{1}\varphi _{2}\theta ^{\frac{1}{2}}\sigma _{v}\rho _{2}\right] \\&\quad +\left( \frac{i(\varphi _{1}+\varphi _{2})b}{a}-\frac{1}{a^{2}} \left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2} +\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) \right) \\&\quad -\frac{i(\varphi _{1}+\varphi _{2})b}{a}e^{-a\tau } +\frac{1}{a^{2}}\left( i(\varphi _{1}+\varphi _{2})\sigma _{r}^{2}+\frac{1}{2}(\varphi _{1} +\varphi _{2})^{2}\sigma _{r}^{2}\right) (2e^{-a\tau }-e^{-2a\tau }), \end{aligned}$$
\(k_{0}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), \(\omega _{\pm }\) and \(\zeta \) are defined in Theorem 1.
D The Characteristic Function \(g(x,r,z,\tau ,\varphi _{3})\)
The characteristic function \(g(x,r,z,\tau ;\varphi _{3})\) for \(\ln S(T)\) is defined as
$$\begin{aligned} g(x,r,z,\tau ;\varphi _{3})=E^{Q^{T}}[e^{i\varphi _{3}\ln S(T)}|\ln S(t)=x, r(t)=r,Z(t)=z], \end{aligned}$$
(88)
where \(T\ge t\), \(\tau =T-t\), \(i=\sqrt{-1}\). Then, the following lemma holds.
Then, Feynman–Kac theory gives the following PDE for the characteristic function
$$\begin{aligned}&-\frac{\partial g}{\partial \tau }+\left( r-\frac{1}{2}z\right) \frac{\partial g}{\partial x}+\frac{1}{2}z\frac{\partial ^{2}g}{\partial x^{2}}+ \kappa _{z}(\theta -z)\frac{\partial g}{\partial z}+\frac{1}{2}\sigma _{z}^{2}z^{2}\frac{\partial ^{2}g}{\partial z^{2}}\nonumber \\&\quad +\,\left( b-\sigma _{r}^{2}B_{r}(\tau )-ar\right) \frac{\partial g}{\partial r} +\frac{1}{2}\sigma _{r}^{2}\frac{\partial ^{2}g}{\partial r^{2}}+\frac{\partial ^{2}g}{\partial x\partial z} z^{\frac{3}{2}}\sigma _{z}\rho _{1}=0, \end{aligned}$$
(89)
The boundary condition for Eq. (89) is given by
$$\begin{aligned} g(x,r,z,0;\varphi _{3})=e^{i\varphi _{3}\ln S(T)}. \end{aligned}$$
(90)
Similar to the proof of Theorem 1 in “Appendix A”. We use the Taylor expansions around the long-run mean of variance \(\theta \) to approximate \(z^{2}\) and \(z^{\frac{3}{2}}\) in above the PDE as follows:
$$\begin{aligned} z^{2}= & {} 2\theta z-\theta ^{2}, \end{aligned}$$
(91)
$$\begin{aligned} z^{\frac{3}{2}}= & {} \frac{3}{2}\theta ^{\frac{1}{2}} z-\frac{1}{2}\theta ^{\frac{3}{2}}, \end{aligned}$$
(92)
Substituting Eqs. (91) and (92) into the PDE in Eq. (89), we have
$$\begin{aligned}&-\frac{\partial g}{\partial \tau }+\left( r-\frac{1}{2}z\right) \frac{\partial g}{\partial x}+\frac{1}{2}z\frac{\partial ^{2}g}{\partial x^{2}}+ \kappa _{z}(\theta -z)\frac{\partial g}{\partial z}\nonumber \\&\quad +\frac{1}{2}\sigma _{z}^{2}(2\theta z-\theta ^{2})\frac{\partial ^{2}g}{\partial z^{2}}+(b-\sigma _{r}^{2}B_{r}(\tau )-ar)\frac{\partial g}{\partial r}\nonumber \\&\quad +\frac{1}{2}\sigma _{r}^{2}\frac{\partial ^{2}g}{\partial r^{2}}+\frac{\partial ^{2}f}{\partial x\partial z} \left( \frac{3}{2}\theta ^{\frac{1}{2}}z-\frac{1}{2}\theta ^{\frac{3}{2}}\right) \sigma _{z}\rho _{1}=0, \end{aligned}$$
(93)
This PDE has an exponential-affine solution of the form
$$\begin{aligned} g(x,r,z,\tau ;\varphi _{3})=e^{i\varphi _{3} x+{\widetilde{D}}(\tau ,\varphi _{3})r+{\widetilde{E}}(\tau ,\varphi _{3})z+{\widetilde{F}}(\tau ,\varphi _{3})}, \end{aligned}$$
(94)
with the boundary conditions
$$\begin{aligned} {\widetilde{D}}(0,\varphi _{3}) ={\widetilde{E}}(0,\varphi _{3})={\widetilde{F}}(0,\varphi _{3})=0. \end{aligned}$$
(95)
Substituting Eq. (94) into PDE in Eq. (93) yields
$$\begin{aligned}&-({\widetilde{D}}_{\tau } (\tau ,\varphi _{3})r+{\widetilde{E}}_{\tau } (\tau ,\varphi _{3})z+{\widetilde{F}}_{\tau }(\tau ,\varphi _{3})) \nonumber \\&\qquad +\left( r-\frac{1}{2}z\right) i\varphi _{3} -\frac{1}{2}\varphi _{3}^{2}z+\kappa _{z}(\theta -z){\widetilde{E}} (\tau ,\varphi _{3}) +\frac{1}{2}\sigma _{z}^{2}(2\theta z-\theta ^{2}){\widetilde{E}}^{2}(\tau ,\varphi _{3})\nonumber \\&\qquad + (b-\sigma _{r}^{2}B_{r}(\tau )-ar){\widetilde{D}}(\tau ,\varphi _{3})\nonumber \\&\qquad +\frac{1}{2}\sigma _{r}^{2}{\widetilde{D}}^{2}(\tau ,\varphi _{3})+i\varphi _{3} {\widetilde{E}}(\tau ,\varphi _{3}) \left( \frac{3}{2}\theta ^{\frac{1}{2}}z -\frac{1}{2}\theta ^{\frac{3}{2}}\right) \sigma _{z}\rho _{1}=0. \end{aligned}$$
(96)
This leads to the following three ordinary differential equations (ODEs)
$$\begin{aligned} {\widetilde{D}}_{\tau }(\tau ,\varphi _{3})= & {} -a {\widetilde{D}}(\tau ,\varphi _{3})+i\varphi _{3}, \end{aligned}$$
(97)
$$\begin{aligned} {\widetilde{E}}_{\tau }(\tau ,\varphi _{3})= & {} \sigma _{z}^{2}\theta {\widetilde{E}}^{2}(\tau ,\varphi _{3}) +\left( \frac{3}{2}i\varphi _{3}\theta ^{\frac{1}{2}} \sigma _{z}\rho _{1}-\kappa _{z}\right) E(\tau ,\varphi _{3}) -\frac{1}{2}(i\varphi _{3}+\varphi _{3}^{2}),\nonumber \\ \end{aligned}$$
(98)
$$\begin{aligned} {\widetilde{F}}_{\tau }(\tau ,\varphi _{3})= & {} -\frac{1}{2}\sigma _{z}^{2} \theta ^{2}{\widetilde{E}}(\tau ,\varphi _{3})^{2} +\left( \kappa _{z}\theta -\frac{1}{2}i\varphi _{3}\sigma _{z} \theta ^{\frac{3}{2}}\rho _{1}\right) {\widetilde{E}}(\tau ,\varphi _{3})\nonumber \\&+ (b-\sigma _{r}^{2}B_{r}(\tau )){\widetilde{D}} (\tau ,\varphi _{3}) +\frac{1}{2}\sigma _{r}^{2}{\widetilde{D}}^{2}(\tau ,\varphi _{3}). \end{aligned}$$
(99)
Solving these linear ODEs, we can obtain
$$\begin{aligned} {\widetilde{D}}(\tau ,\varphi _{3})= & {} \frac{i\varphi _{3}}{a}(1-\exp \{-a\tau \}), \end{aligned}$$
(100)
$$\begin{aligned} {\widetilde{E}}(\tau ,\varphi _{3})= & {} {\widetilde{k}}_{0}\frac{1-e^{-{\widetilde{\zeta }}\tau }}{-{\widetilde{\omega }}_{-}+{\widetilde{\omega }}_{+}e^{-{\widetilde{\zeta }}\tau }}, \end{aligned}$$
(101)
$$\begin{aligned} {\widetilde{F}}(\tau ,\varphi _{3})= & {} -\frac{1}{2}\theta {\widetilde{E}}(\tau ,\varphi _{3})- \frac{{\widetilde{k}}_{3}}{{\widetilde{k}}_{2}} \left[ {\widetilde{\omega }}_{+}\tau +\ln \left( \frac{-{\widetilde{\omega }}_{-} +{\widetilde{\omega }}_{+}e^{-{\widetilde{\zeta }}\tau }}{{\widetilde{\zeta }}}\right) \right] \nonumber \\&\quad -\frac{1}{4}\left( i\varphi _{3}\theta +\varphi _{3}^{2}\theta \right) \tau +{\widetilde{\varLambda }}(\tau ), \end{aligned}$$
(102)
and
$$\begin{aligned} {\widetilde{k}}_{0}= & {} -\frac{1}{2}\left( i\varphi _{3}+\varphi _{3}^{2}\right) , \\ {\widetilde{k}}_{1}= & {} \left( \frac{3}{2}i\varphi _{3}\theta ^{\frac{1}{2}}\sigma _{z}\rho _{1}-\kappa _{z}\right) , \\ {\widetilde{k}}_{2}= & {} \sigma _{z}^{2}\theta , \\ {\widetilde{k}}_{3}= & {} \left( \frac{1}{4}i\varphi _{3}\theta ^{\frac{3}{2}}\sigma _{z}\rho _{1} +\frac{1}{2}\kappa _{z}\theta \right) ,\\ {\widetilde{\omega }}_{\pm }= & {} \frac{{\widetilde{k}}_{1} \pm {\widetilde{\zeta }}}{2},\\ {\widetilde{\zeta }}= & {} \sqrt{{\widetilde{k}}_{1}^{2} -4{\widetilde{k}}_{0}{\widetilde{k}}_{2}},\\ {\widetilde{\varLambda }}(\tau )= & {} \left( \frac{i\varphi _{3}b}{a}-\frac{1}{a^{2}} \left( i\varphi _{3}\sigma _{r}^{2}+\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) \right) \tau +\frac{i\varphi _{3}b}{a^{2}}e^{-a\tau } +\frac{1}{a^{3}}\left( i\varphi _{3}\sigma _{r}^{2}\right. \\&\left. +\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) \left( \frac{1}{2}e^{-2a\tau }-2e^{-a\tau }\right) +\frac{3}{2a^{3}} \left( i\varphi _{3}\sigma _{r}^{2}+\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) -\frac{i\varphi _{3}b}{a^{2}}. \end{aligned}$$
