Pricing Vulnerable Options with Stochastic Volatility and Stochastic Interest Rate

Abstract

This paper considers the pricing issue of vulnerable European options when the price process of the underlying asset follows the GARCH diffusion model with stochastic interest rate. Based on the proposed model, we obtain an approximate solution for the vulnerable European option price via means of Fourier transform. In addition, the Greeks of vulnerable option price are derived explicitly. Besides, the approximate solution of vulnerable option price can be quickly computed by using the fast Fourier transform (FFT) algorithm. The results of Monte Carlo simulations indicate that FFT is accurate, fast and easy to implement. More important, the pricing model also reveals that: (i) a negative correlation of volatility with the spot return creates a fat left tail and thin right tail in the distribution of continuously compounded spot returns. Thus, for in-the-money options, the vulnerable option prices of the proposed model are higher than those of Klein (J Bank Finance 20(7):1211–1229, 1996). While for deep-out-of-the-money options, the vulnerable option prices of the proposed model are smaller; (ii) the higher long-run mean of the underlying asset price’s instantaneous variance, the higher vulnerable option price; (iii) the long-run mean of the stochastic interest rate exerts a positive effect on the value of vulnerable European option.

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Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (Grant Nos. 71431008, 71521061, 71601075, 71850012, and 71790593), Major special Projects of the Department of Science and Technology of Hunan province (Grant No. 2018GK1020) and the China Postdoctoral Science Foundation (Grant No. 2017M612768).

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Correspondence to Shengjie Yue.

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Appendices

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

$$\begin{aligned} {\widetilde{k}}_{0}= & {} -\frac{1}{2}\left( i\varphi _{3}+\varphi _{3}^{2}\right) , \\ {\widetilde{k}}_{1}= & {} \left( \frac{3}{2}i\varphi _{3}\theta ^{\frac{1}{2}}\sigma _{z}\rho _{1}-\kappa _{z}\right) , \\ {\widetilde{k}}_{2}= & {} \sigma _{z}^{2}\theta , \\ {\widetilde{k}}_{3}= & {} \left( \frac{1}{4}i\varphi _{3}\theta ^{\frac{3}{2}}\sigma _{z}\rho _{1} +\frac{1}{2}\kappa _{z}\theta \right) ,\\ {\widetilde{\omega }}_{\pm }= & {} \frac{{\widetilde{k}}_{1} \pm {\widetilde{\zeta }}}{2},\\ {\widetilde{\zeta }}= & {} \sqrt{{\widetilde{k}}_{1}^{2} -4{\widetilde{k}}_{0}{\widetilde{k}}_{2}},\\ {\widetilde{\varLambda }}(\tau )= & {} \left( \frac{i\varphi _{3}b}{a}-\frac{1}{a^{2}} \left( i\varphi _{3}\sigma _{r}^{2}+\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) \right) \tau +\frac{i\varphi _{3}b}{a^{2}}e^{-a\tau } +\frac{1}{a^{3}}\left( i\varphi _{3}\sigma _{r}^{2}\right. \\&\left. +\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) \left( \frac{1}{2}e^{-2a\tau }-2e^{-a\tau }\right) +\frac{3}{2a^{3}} \left( i\varphi _{3}\sigma _{r}^{2}+\frac{1}{2}\varphi _{3} ^{2}\sigma _{r}^{2}\right) -\frac{i\varphi _{3}b}{a^{2}}. \end{aligned}$$

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Ma, C., Yue, S., Wu, H. et al. Pricing Vulnerable Options with Stochastic Volatility and Stochastic Interest Rate. Comput Econ 56, 391–429 (2020). https://doi.org/10.1007/s10614-019-09929-4

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Keywords

  • Vulnerable European options
  • Characteristic function
  • GARCH diffusion model
  • Stochastic interest rate
  • Fast Fourier transform