Valuations of Variance and Volatility Swaps Under Double Heston Jump-Diffusion Model With Approximative Fractional Stochastic Volatility

Abstract

In this paper, we study the variance and volatility swaps pricing problem under the framework of double Heston jump diffusion model with approximative fractional stochastic volatility. The pricing formulas of discretely-sampled variance and volatility swaps are obtained by deriving the characteristic function and solving the governing partial integro-differential equations(PIDEs). We also obtain the limits of discretely-sampled variance and volatility swaps pricing formulas, which are the pricing formulas of continuously-sampled variance and volatility swaps. Finally, the effectiveness of the pricing formula is illustrated by comparing with some existing works, and the influence of approximation factor and Hurst parameter variation on the prices of swaps are studied.

Appendices

Appendix A

We know that $\int _{0}^{+\infty }\frac{sinx}{x}dx=\frac{\pi }{2}$, then we have

$$\begin{aligned} \int _{0}^{+\infty }\frac{sinax}{x}dx= \left\{ \begin{array}{lr} \frac{\pi }{2}, &{} a>0 \\ 0, &{}a=0\\ -\frac{\pi }{2}. &{} a<0 \end{array} \right. \end{aligned}$$

(A.1)

Next, we express Eq. (A.1) in terms of an indicator function sgn(a) as follows

$$\begin{aligned} sgn(a)=\frac{2}{\pi }\int _{0}^{+\infty }\frac{sinax}{x}dx=\frac{2}{\pi }\int _{0}^{+\infty }RE\left[ \frac{e^{iax}}{ix}\right] dx. \end{aligned}$$

(A.2)

Then, we have $I_{a>0}=\frac{1}{2}+\frac{1}{\pi }\int _{0}^{+\infty }RE\left[ \frac{e^{iax}}{ix}\right] dx$, where $I= \left\{ \begin{array}{lr} 1, &{}a>0\\ 0. &{}a\le 0 \end{array} \right.$.

$$\begin{aligned} P(X>0)=E[I_{X>0}]=\frac{1}{2}+\frac{1}{\pi }\int _{0}^{+\infty }RE\left[ \frac{E[e^{i\phi X}]}{i\phi }\right] d\phi . \end{aligned}$$

which $f(\phi ):=E[e^{i\phi X}]$. Furthermore, we know that $P(X>0)=\int _{0}^{+\infty }p(x)dx$. Thus, one has

$$\begin{aligned} \int _{0}^{+\infty }p(x)dx=\frac{1}{2}+\frac{1}{\pi }\int _{0}^{+\infty }RE\left[ \frac{f(\phi )}{i\phi }\right] d\phi . \end{aligned}$$

(A.3)

This completes the proof.

Appendix B

We now present a brief Proof of Theorem 3. Similar to Eq. (36), one has

$$\begin{aligned} K^1_{cvol}&=E[RV^1_{cvol}\vert S_0,v_0,\hat{v}_0]\\&=E\left[ \sqrt{\frac{1}{T}\left[ \int _{0}^{T}(v_t+\hat{v}_t)dt+\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}\right] }\Bigg \vert S_0,v_0,\hat{v}_0\right] \times 100. \end{aligned}$$

From Schürger (2002), the square root function can be expressed as follows

$$\begin{aligned} \sqrt{x}=\frac{1}{2\sqrt{\pi }}\int _{0}^{+\infty }\frac{1-e^{-sx}}{s^{\frac{3}{2}}}ds. \end{aligned}$$

(B.1)

Taking the expectations on both sides of Eq. (B.1) with $x=\frac{1}{T}\left[ \int _{0}^{T}(v_t+\hat{v}_t)dt+\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}\right]$, which can be derived as

$$\begin{aligned} K^1_{cvol}=\frac{100}{2\sqrt{\pi }}\int _{0}^{+\infty }\frac{1-E\left[ e^{-s\frac{1}{T}\left[ \int _{0}^{T}(v_t+\hat{v}_t)dt+\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}\right] }\bigg \vert S_0,v_0,\hat{v}_0\right] }{s^{\frac{3}{2}}}ds \end{aligned}$$

(B.2)

where

$$\begin{aligned}&E\left[ e^{-s\frac{1}{T}\left[ \int _{0}^{T}(v_t+\hat{v}_t)dt+\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}\right] }\bigg \vert S_0,v_0,\hat{v}_0\right] \\&\quad =E\left[ e^{-\frac{s}{T}\int _{0}^{T}(v_t+\hat{v}_t)dt}\bigg \vert v_0,\hat{v}_0\right] +E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right] . \end{aligned}$$

Firstly, we let $P(t,v,\hat{v})=E\left[ e^{-\frac{s}{T}\int _{t}^{T}(v_t+\hat{v}_t)dt}\bigg \vert v_t,\hat{v}_t\right] ,\ t \in [0,T]$. By using the Feynman–Kac theorem, the PDE governing P can be derived as

$$\begin{aligned} \frac{\partial P}{\partial \tau }= & \left[ \kappa (\theta -v)+\sigma _v\sqrt{v}\left( H-\frac{1}{2}\right) \psi \right] \frac{\partial P}{\partial v}+\frac{1}{2}\sigma _v^2\varepsilon ^{2H-1}V\frac{\partial ^2 P}{\partial v^2}\nonumber \\{} & {} +\,\hat{\kappa }(\hat{\theta }-\hat{v})\frac{\partial P}{\partial \hat{v}}+\frac{1}{2}\sigma _{\hat{v}}^2\hat{v}\frac{\partial ^2 P}{\partial \hat{v}^2} \end{aligned}$$

(B.3)

where $\tau =T-t$ and $P(\tau ,v,\hat{v})\big \vert _{\tau =0}=1$. Then, we can assume that the solution of Eq. (B.3) has the following form

$$\begin{aligned} P(\tau ,v,\hat{v})=e^{A(s,\tau )+B(s,\tau )v_t+C(s,\tau )\hat{v}_t} \end{aligned}$$

Substituting P into Eq. (B.3), since $\psi _t=\int _{0}^{t}(t-s+\varepsilon )^{H-\frac{3}{2}}dW_s^{\psi }$ is a martingle, $\psi _0=E[\psi _t]=0$. Then, the PDE is reduced to three ODEs

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial A}{\partial \tau }=\kappa \theta B+\hat{\kappa }\hat{\theta }C\\ \frac{\partial B}{\partial \tau }=-\kappa B+\frac{1}{2}\sigma _v^2\varepsilon ^{2H-1}B^2-\frac{s}{T}\\ \frac{\partial C}{\partial \tau }=-\hat{\kappa }C+\frac{1}{2}\sigma _{\hat{v}}^2C^2-\frac{s}{T}, \end{array} \right. \end{aligned}$$

(B.4)

with $A(s,0)=B(s,0)=C(s,0)=0$.

Clearly, the ODE governing $B(s,\tau )$ and $C(s,\tau )$ are Riccati equations with constant coefficients, which have the following solution form

$$\begin{aligned} B(s,T)=\frac{\kappa +d_1}{\sigma ^2_v\varepsilon ^{2H-1}}\frac{1-e^{d_1 \tau }}{1-g_1e^{d_1 \tau }}, \quad C(s,T)=\frac{\hat{\kappa }+\hat{d}_1}{\sigma ^2_{\hat{v}}}\frac{1-e^{\hat{d}_1 \tau }}{1-\hat{g}_1e^{\hat{d}_1 \tau }} \end{aligned}$$

where

$$\begin{aligned} d_1=\sqrt{\kappa ^2+2\sigma ^2_v\varepsilon ^{2H-1}\frac{s}{T}},\quad \hat{d}_1=\sqrt{\hat{\kappa }^2+2\sigma ^2_{\hat{v}}\frac{s}{T}},\quad g_1=\frac{-\kappa -d_1}{\kappa +d_1},\quad \hat{g}_1=\frac{\hat{\kappa }-\hat{d}_1}{\hat{\kappa }+\hat{d}_1}. \end{aligned}$$

Moreover, we can arrive at a closed form solution of $A(s,\tau )$ by integrating both sides of the first ODE

$$\begin{aligned} A(s,T)&=\frac{\kappa \theta (\kappa +d_1)}{\sigma ^2_v\varepsilon ^{2H-1}}\left[ \tau -\frac{g_1-1}{d_1g_1}\ln (1-g_1e^{d_1\tau })\right] \\&\quad +\,\frac{\hat{\kappa }\hat{\theta }(\hat{\kappa }+\hat{d}_1)}{\sigma ^2_{\hat{v}}}\left[ \tau -\frac{\hat{g}_1-1}{\hat{d}_1\hat{g}_1}\ln (1-\hat{g}_1e^{\hat{d}_1\tau })\right] . \end{aligned}$$

We know that $E\left[ e^{-\frac{s}{T}\int _{0}^{T}(v_t+\hat{v}_t)dt}\bigg \vert v_0,\hat{v}_0\right]$ can be derived by setting $\tau =T$,

$$\begin{aligned} E\left[ e^{-\frac{s}{T}\int _{0}^{T}(v_t+\hat{v}_t)dt}\bigg \vert v_0,\hat{v}_0\right] =e^{A(s,T)+B(s,T)v_0+C(s,T)\hat{v}_0}. \end{aligned}$$

Then, we calculate $E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right]$. Using the property of expectation, $E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right]$ can be written as

$$\begin{aligned} E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right] =E\left[ E\left[ e^{-\frac{s}{T}\sum _{i=1}^{n}{(e^{Y_i}-1)^2}}\bigg \vert N_T=n\right] \bigg \vert S_0,v_0,\hat{v}_0\right] . \end{aligned}$$

Let $Z=e^Y-1$, the density function of Z is given by

$$\begin{aligned} f_Z(z)=\left\{ \begin{array}{lr} 0, &{}z\le 1 \\ q\eta _2(1+z)^{\eta _2-1}, &{} -1< z \le 0\\ p\eta _1(1+z)^{-\eta _1-1}. &{} z>0 \end{array} \right. \end{aligned}$$

The inner expectation can be computed as

$$\begin{aligned} E\left[ e^{-\frac{s}{T}\sum _{i=1}^{n}{(e^{Y_i}-1)^2}}\bigg \vert N_T=n\right] =(D(s,T))^n \end{aligned}$$

where $D(s,T)=q\eta _2\int _{-1}^{0}e^{-\frac{s}{T}z^2}(1+z)^{\eta _2-1}dz+p\eta _1\int _{0}^{+\infty }e^{-\frac{s}{T}z^2}(1+z)^{-\eta _1-1}dz$.

We can compute the outer expectation as follows

$$\begin{aligned} E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right] =\sum _{n=0}^{+\infty }\frac{e^{-\lambda T}(\lambda T)^n}{n!}(D(s,T))^n. \end{aligned}$$

(B.5)

Simplifying the infinite sum gives

$$\begin{aligned} E\left[ e^{-\frac{s}{T}\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}}\bigg \vert S_0,v_0,\hat{v}_0\right] =e^{\lambda T(D(s,T)-1)}. \end{aligned}$$

(B.6)

Summarizing above discussions, we have the results as follows

$$\begin{aligned}{} & {} E\left[ e^{-s\frac{1}{T}\left[ \int _{0}^{T}(v_t+\hat{v}_t)dt+\sum _{i=1}^{N_T}{(e^{Y_i}-1)^2}\right] }\bigg \vert S_0,v_0,\hat{v}_0\right] \nonumber \\{} & {} \quad =e^{A(s,T)+B(s,T)v_0+C(s,T)\hat{v}_0+\lambda T(D(s,T)-1)}. \end{aligned}$$

(B.7)

Thus, $K^1_{cvol}$ is obtained.

Appendix C

If the parameters $\lambda =0$ and $H=\frac{1}{2}$, our model degenerates as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{dS_t}{S_t}=rdt+\sqrt{v_{1,t}}dW_t^S+\sqrt{v_{2,t}}d\hat{W}_t^S\\ dv_t=k1(\theta _1-v_{1,t})dt+\sigma _1\sqrt{v_{1,t}}dW_t\\ d\hat{v}_t=k_2(\theta _2-v_{2,t})dt+\sigma _2\sqrt{v_{2,t}}d\hat{W}_t^v, \end{array} \right. \end{aligned}$$

(C.1)

According to the Recchioni et al. (2021), the marginal probability density of the log-price variable conditioned on $v=(v_{1,t_{i-1}},v_{2,t_{i-1}})$ is given by

$$\begin{aligned} M(x_{t_{i-1}},v,t_{i-1},x_{t_{i}},t_{i})= & \frac{1}{2\pi }\int _{-\infty }^{+\infty }e^{\iota m\left[ (t_{i}-t_{i-1})-r(t_{i}-t_{i-1})+\frac{1}{2}\Gamma _0(t_{i-1},t_{i})\right] -\frac{1}{2}\Gamma _0(t_{i-1},t_{i})m^2}\nonumber \\{} & {} \times \,e^{\sum _{j=1}^{2}\int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})\left[ \frac{\sigma _j^2}{2}B_j^2(m,s,t_{i})+\iota m\rho _j\sigma _jB_j(m,s,t_{i})\right] ds}dm \end{aligned}$$

(C.2)

where $E(v_{j,t_{i}}\vert {\mathcal {F}}_{t_{i-1}})=v_{j,t_{i-1}}e^{-k_j(t_{i}-t_{i-1})}+\sigma _j(1-e^{-k_j(t_{i}-t_{i-1})})$, $x_{t_i}=\ln S_{t_i}$ and $\iota$ is an imaginary unit. Here, $B_j$ is given by

$$\begin{aligned} B_j(m,t_{i-1},t_{i})=\frac{1}{2}(m^2-\iota m)\frac{1-e^{-2\zeta _j(t_{i}-t_{i-1})}}{(\zeta _j+\xi _j)+(\zeta _j-\xi _j)e^{-2\zeta _j(t_{i}-t_{i-1})}} \end{aligned}$$

(C.3)

where $\zeta _j$ and $\xi _j$ are the following quantities

$$\begin{aligned} \zeta _j(m)=\frac{1}{2}(4\xi _j^2+\sigma _j^2(m^2-\iota m))^\frac{1}{2}),\quad \xi _j(m)=1/2(\iota m\rho _j\sigma _j+k_j). \end{aligned}$$

(C.4)

And $\Gamma _0(t_{i},t_{i-1})$ is given by

$$\begin{aligned} \Gamma _0(t_{i},t_{i-1})=\sum _{j=1}^{2}\int _{t_{i-1}}^{t_i}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})ds. \end{aligned}$$

(C.5)

Then, we calculate the fair strike price of variance swap through the conditional marginal probability density function

$$\begin{aligned} K_{var}= \; \frac{AF}{N}\sum _{i=1}^{N}E\left[ \left( \frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}\right) ^2\Bigg \vert {\mathcal {F}}_0\right] \times 100^2\nonumber \\= \; \frac{AF}{N}\sum _{i=1}^{N}E\left[ \left( e^{x_{t_i}-x_{t_{i-1}}}-1\right) ^2\Bigg \vert {\mathcal {F}}_0\right] \times 100^2\nonumber \\= \; \frac{AF}{N}\sum _{i=1}^{N}E\left[ E\left[ \left( e^{x_{t_i}-x_{t_{i-1}}}-1\right) ^2\Bigg \vert {\mathcal {F}}_{t_{i-1}}\right] \Bigg \vert {\mathcal {F}}_0\right] \times 100^2\nonumber \\= \; \frac{AF}{N}\sum _{i=1}^{N}E\left[ E\left[ \left( e^{2(x_{t_i}-x_{t_{i-1}})}-2e^{(x_{t_i}-x_{t_{i-1}})}+1\right) \Bigg \vert {\mathcal {F}}_{t_{i-1}}\right] \Bigg \vert {\mathcal {F}}_0\right] \times 100^2 \end{aligned}$$

(C.6)

Further, we calculate Eq. (C.6) in two steps: ($\text {i}$) $E[ E[e^{(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$; ($\text {ii}$) $E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$.

• (i) $E[ E[e^{(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$

  $$\begin{aligned}{} & {} E[e^{(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}]\nonumber \\{} & {} \quad =e^{-x_{t_{i-1}}}\int _{-\infty }^{+\infty }e^{x_{t_i}}M(x_{t_{i-1}},v,t_{i-1},x_{t_{i}},t_{i})dx_{t_i}\nonumber \\{} & {} \quad =\int _{-\infty }^{+\infty }\frac{1}{2\pi }\int _{-\infty }^{+\infty }e^{\iota (m-\iota ) (x_{t_i}-x_{t_{i-1}})}dx_{t_i}e^{\iota m[-r(t_{i}-t_{i-1})+\frac{1}{2}\Gamma _0(t_{i-1},t_{i})]-\frac{1}{2}\Gamma _0(t_{i-1},t_{i})m^2}\nonumber \\{} & {} \qquad \times e^{\sum _{j=1}^{2}\int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})[\frac{\sigma _j^2}{2}B_j^2(m,s,t_{i})+\iota m\rho _j\sigma _jB_j(m,s,t_{i})]ds}dm\nonumber \\{} & {} \quad =\int _{-\infty }^{+\infty }\delta (m-\iota )e^{\iota m[-r(t_{i}-t_{i-1})+\frac{1}{2}\Gamma _0(t_{i-1},t_{i})]-\frac{1}{2}\Gamma _0(t_{i-1},t_{i})m^2}\nonumber \\{} & {} \qquad \times e^{\sum _{j=1}^{2}\int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})[\frac{\sigma _j^2}{2}B_j^2(m,s,t_{i})+\iota m\rho _j\sigma _jB_j(m,s,t_{i})]ds}dm\nonumber \\{} & {} \quad =e^{r(t_{i}-t_{i-1})} \end{aligned}$$

  (C.7)

  Thus, $E[ E[e^{(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]=E[e^{r(t_{i}-t_{i-1})}\vert {\mathcal {F}}_0]=e^{r(t_{i}-t_{i-1})}$.

• (ii) $E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$. Similar to $E[e^{(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}]$, the $E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}]$ can be expressed as

  $$\begin{aligned} E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}]= & \int _{-\infty }^{+\infty }\delta (m-2\iota )e^{\iota m[-r(t_{i}-t_{i-1})+\frac{1}{2}\Gamma _0(t_{i-1},t_{i})]-\frac{1}{2}\Gamma _0(t_{i-1},t_{i})m^2}\nonumber \\{} & {} \times \,e^{\sum _{j=1}^{2}\int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})[\frac{\sigma _j^2}{2}B_j^2(m,s,t_{i})+\iota m\rho _j\sigma _jB_j(m,s,t_{i})]ds}dm\nonumber \\= & e^{2r(t_{i}-t_{i-1})}e^{\sum _{j=1}^{2}\int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})[H_j(2\iota ,s,t_{i})+1]ds} \end{aligned}$$

  (C.8)

  where $H_j(m,s,t_{i})=\frac{\sigma _j^2}{2}B_j^2(m,s,t_{i})+\iota m\rho _j\sigma _jB_j(m,s,t_{i})$. Recchioni et al. (2021), the $H_j(m,s,t_{i})$ is given by

  $$\begin{aligned} H_j(m,s,t_{i})= & \frac{\sigma _j^2(m^2-\iota m)^2}{2}\frac{\psi _j^2(s,t_i)}{4}+\iota m\rho _j\sigma _j\left( \frac{\psi _j(s,t_i)}{2}(m^2-\iota m)\right. \nonumber \\{} & {} \left. +\,\frac{\sigma _j\rho _j}{2k_j}f_j(s,t_i)(-\iota m^3-m^2)\right) +o(\sigma _j^2),\ \sigma _j\rightarrow 0^{+}. \end{aligned}$$

  (C.9)

  where $\psi _j(t_{i-1},t_i)=\frac{(1-e^{-k_j(t_i-t_{i-1})})}{k_j}$ and $f_j(t_{i-1},t_{i})=\psi _j(t_{i-1},t_i)-(t_i-t_{i-1})e^{-k_j(t_i-t_{i-1})}$. Next,

  $$\begin{aligned}{} & {} \int _{t_{i-1}}^{t_{i}}E(v_{j,s}\vert {\mathcal {F}}_{t_{i-1}})[H_j(m,s,t_{i})+1]ds\nonumber \\{} & {} \quad =\int _{t_{i-1}}^{t_{i}}[v_{j,t_{i-1}}e^{-k_j(s-t_{i-1})} +\theta _j(1-e^{-k_j(s-t_{i-1})})][H_j(m,s,t_{i})+1]ds\nonumber \\{} & {} \quad =v_{j,t_{i-1}}A_j+\theta _jC_j \end{aligned}$$

  (C.10)

  where $A_j(m,\tau )=\int _{t_{i-1}}^{t_{i}}e^{-k_j(s-t_{i-1})}[H_j(m,s,t_{i})+1]ds$ and $C_j(m,\tau )=\int _{t_{i-1}}^{t_{i}}(1-e^{-k_j(s-t_{i-1})})[H_j(m,s,t_{i})+1]ds$. We let $\tau =t_{i}-t_{i-1}$, $A_j$ can be derived as

  $$\begin{aligned} A_j(m,\tau )= & \frac{\sigma _j^2(m^2-\iota m)^2[1-e^{-2k_j\tau }]}{8k_j^2}-\frac{\sigma _j^2(m^2-\iota m)^2\tau e^{-k_j\tau }}{4k_j}\nonumber \\{} & {} +\,\frac{\iota m\rho _j\sigma _j(m^2-\iota m)}{2k_j^2}(1-e^{-k_j\tau }-k_j\tau e^{-k_j\tau })+\frac{1-e^{-k_j\tau }}{k_j}\nonumber \\{} & {} +\,\frac{\iota m(-\iota m^3-m^2)\sigma _j^2\rho _j^2}{2k_j}[\frac{1-e^{-k_j\tau }-k_j\tau e^{-k_j\tau }}{k_j^2}-\frac{e^{-k_j\tau }\tau ^2}{2}] \end{aligned}$$

  (C.11)

  The $C_j(m,\tau )$ is then given by

  $$\begin{aligned} C_j(m,\tau )= & \frac{\sigma _j^2(m^2-\iota m)^2}{8}\left[ \frac{\tau }{k_j^2}+\frac{1-e^{-2k_j\tau }}{2k_j^3}-\frac{2(1-e^{-k_j\tau })}{k_j^3}\right] \nonumber \\{} & {} +\,\frac{\iota m(-\iota m^3-m^2)\sigma _j^2\rho _j^2}{2k_j^2}\left[ \tau -\frac{2-2e^{-k_j\tau }}{k_j}+\tau e^{-k_j\tau }\right] \nonumber \\{} & {} +\,\frac{\iota m\rho _j\sigma _j(m^2-\iota m)}{2k_j}\left[ \tau -\frac{1-e^{-k_j\tau }}{k_j}\right] +\tau -A_j \end{aligned}$$

  (C.12)

  Thus, $E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}]=e^{2r\tau +\sum _{j=1}^{2}\theta _jC_j(2\iota ,\tau )}e^{\sum _{j=1}^{2}A_j(2\iota ,\tau )v_{j,t_{i-1}}}$. Further, $E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$ can be expressed as

  $$\begin{aligned} E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]=e^{2r\tau +\sum _{j=1}^{2}\theta _jC_j(2\iota ,\tau )}E\left[ e^{\sum _{j=1}^{2}A_j(2\iota ,\tau )v_{j,t_{i-1}}}\vert {\mathcal {F}}_0\right] \end{aligned}$$

  (C.13)

  Similar to Eq. (29), we obtain

  $$\begin{aligned} E[e^{\sum _{j=1}^{2}A_j(2\iota ,\tau )v_{j,t_{i-1}}}\vert {\mathcal {F}}_0]=e^{\hat{C}(2\iota ,t_{i-1})+\hat{D}(2\iota ,t_{i-1})v_{1,0}+\hat{E}(2\iota ,t_{i-1})v_{2,0}} \end{aligned}$$

  (C.14)

  where

  $$\begin{aligned} \hat{C}(2\iota ,t_{i-1})&=\frac{2k_1\theta _1}{\sigma _1^2}\left[ k_1 t_{i-1}+\ln \frac{2k_1}{\sigma _1^2A_1(2\iota ,\tau )+(2k_1-\sigma _1^2A_1(2\iota ,\tau ))e^{k_1 t_{i-1}}}\right] \\&\quad +\frac{2k_2\theta _2}{\sigma _2^2}\left[ k_2t_{i-1}+\ln \frac{2k_2}{\sigma _2^2A_2(2\iota ,\tau )+(2k_2-\sigma _2^2A_2(2\iota ,\tau ))e^{k_2t_{i-1}}}\right] ,\\ \hat{D}_1(2\iota ,t_{i-1})&=\frac{2k_1 A_1(2\iota ,\tau )}{\sigma _1^2A_1(2\iota ,\tau )+(2k_1-\sigma _1^2A_1(2\iota ,\tau ))e^{k_1 t_{i-1}}},\\ \hat{D}_2(2\iota ,t_{i-1})&=\frac{2k_2A_2(2\iota ,\tau )}{\sigma _2^2A_2(2\iota ,\tau )+(2k_2-\sigma _2^2A_2(2\iota ,\tau ))e^{k_2 t_{i-1}}}. \end{aligned}$$

  Finally, $E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]$ is given by

  $$\begin{aligned} E[ E[e^{2(x_{t_i}-x_{t_{i-1}})} \vert {\mathcal {F}}_{t_{i-1}}] \vert {\mathcal {F}}_0]=e^{2r\tau +\hat{C}(2\iota ,t_{i-1})+\sum _{j=1}^{2}C_j(2\iota ,t_{i-1})+\hat{D}_j(2\iota ,t_{i-1})v_{j,0}} \end{aligned}$$

  (C.15)

  Above all, the fair strike price of variance swap can be expressed as

  $$\begin{aligned} K_{var}=\frac{100^2}{T}\sum _{i=1}^{N}\left[ e^{2r\tau +\hat{C}(2\iota ,t_{i-1})+\sum _{j=1}^{2}C_j(2\iota ,\tau )+\hat{D}_j(2\iota ,t_{i-1})v_{j,0}}-2e^{r\tau }+1\right] . \end{aligned}$$

  (C.16)

  An appendix contains supplementary information that is not an essential part of the text itself but which may be helpful in providing a more comprehensive understanding of the research problem or it is information that is too cumbersome to be included in the body of the paper.