Appendix A. Proof of Proposition 2.1
Given \(\alpha _r(t)\) is deterministic,
$$\begin{aligned} P_D(t,T)&=\mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}r(s)ds} \right| \mathcal {F}_t\right] = \mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}\left[ \alpha _r(s)+\zeta _r(s)\right] ds} \right| \mathcal {F}_t\right] \\&= e^{-\int _{t}^{T}\alpha _r(s)ds}\mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}\zeta _r(s)ds} \right| \mathcal {F}_t\right] = e^{-\int _{t}^{T}\alpha _r(s)ds}P_{\zeta _r}(t,T). \end{aligned}$$
By Feynman-Kac Formula, \(P_{\zeta _r}(t,T)\) satisfies the following PDE:
$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial P_{\zeta _r}}{\partial t}+\sum _{i=1}^{4}\mu _{x_i}^{\mathbb {Q}}\frac{\partial P_{\zeta _r}}{\partial x_i}+\frac{1}{2}\sum _{i,j=1}^{4}\left[ \left( CV_tR\right) \left( CV_tR\right) ^\mathbb {T}\right] _{i,j}\frac{\partial ^2 P_{\zeta _r}}{\partial x_i x_j}-\zeta _r(t)P_{\zeta _r}= 0,\\&\hbox {(Terminal condition)}\quad P_{\zeta _r}\left( T,T;\zeta _r(T),\zeta _x(T),v_1(T),v_2(T)\right) = 1, \end{aligned}\right. \nonumber \\ \end{aligned}$$
(A.1)
where \(x_1:=\zeta _r\), \(x_2:=\zeta _x\), \(x_3:=v_1\) and \(x_4:=v_2\).
We propose
$$\begin{aligned} P_{\zeta _r}(t,T)=e^{A_r(t,T)\zeta _r(t)+B_r(t,T)\zeta _x(t)+C_r(t,T)v_1(t)+D_r(t,T)v_2(t)+E_r(t,T)}. \end{aligned}$$
(A.2)
Under this affine proposition of \(P_{\zeta _r}(t,T)\), all of the partial derivatives could be obtained and substituted into (A.1). As (A.1) is true for all r(t), x(t), \(v_1(t)\), and \(v_2(t)\), the coefficients of these four stochastic random variables are zero over time, which yields \(B_r(s,T)\equiv 0\), \(D_r(s,T)\equiv 0\) for all \(0\le s \le T\), and (3). \(\square \)
Appendix B. Proof of Lemma 2.1
By Ito’s Lemma,
$$\begin{aligned}&dP_D(t,T)\nonumber \\&\quad = \frac{\partial P_D}{\partial t}\,dt+\sum _{i=1}^{4}\frac{\partial P_D}{\partial x_i}\mu _{x_i}\,dt+\frac{1}{2}\sum _{i,j=1}^{4}\frac{\partial ^2 P_D}{\partial x_ix_j}\left[ (CV_tR)(CV_tR)^\mathbb {T}\right] \,dt \end{aligned}$$
(B.1)
$$\begin{aligned}&\qquad +\sum _{i=1}^{4}\frac{\partial P_D}{\partial x_i}\sigma _{x_i}\,d\vec {W}_{x_i} \nonumber \\&\quad = r(t)P_D(t,T)\,dt+\vec {\sigma _D}(t,T)\,d\vec {W}^{\mathbb {Q}}, \end{aligned}$$
(B.2)
where \(x_1:=r\), \(x_2:=x\), \(x_3:=v_1\), \(x_4:=v_2\) and
$$\begin{aligned} \vec {\sigma _D}(t,T)&= \begin{bmatrix} A_r(t,T)&B_r(t,T)&C_r(t,T)&D_r(t,T)\end{bmatrix}CV_tR \\&= \begin{bmatrix} \sigma _{D,1}(t,T)\sqrt{v_1(t)} \\ \sigma _{D,2}(t,T)\sqrt{v_2(t)} \\ \sigma _{D,3}(t,T)\sqrt{v_1(t)} \\ \sigma _{D,4}(t,T)\sqrt{v_2(t)}. \nonumber \end{bmatrix}^T \end{aligned}$$
(B.3)
Here, \( \sigma _{D,i}\) is the coefficient of the volatilities in the ith element of
$$\begin{aligned} \begin{bmatrix} A_r(t,T)&B_r(t,T)&C_r(t,T)&D_r(t,T)\end{bmatrix}CR. \end{aligned}$$
In particular,
$$\begin{aligned} \begin{aligned} \sigma _{D,1}(t,T)&=A_r(t,T)+\rho _1\beta _1C_r(t,T),\\ \sigma _{D,2}(t,T)&=0,\\ \sigma _{D,3}(t,T)&=\sqrt{1-\rho _1^2}\beta _1C_r(t,T), \quad \text {and}\\ \sigma _{D,4}(t,T)&=0. \end{aligned} \end{aligned}$$
(B.4)
The second equation of (B.1) is due to the following PDE obtained by Feynman-Kac formula:
$$\begin{aligned} \frac{\partial P_D}{\partial t}+ \sum _{i=1}^{4}\mu _{x_i}^{\mathbb {Q}}\frac{\partial P_D}{\partial x_i}+ \frac{1}{2}\sum _{i=1}^{4}\left[ (CV_tR)(CV_tR)^{\mathbb {T}}\right] _{i,j}\frac{\partial ^2 P_D}{\partial x_i x_j}-r(t)P_D= & {} 0 ,\nonumber \\ \hbox {(Terminal condition)}\quad P_D\left( T,T;r(T),x(T),v_1(T),v_2(T)\right)= & {} 1.\nonumber \\ \end{aligned}$$
(B.5)
By Ito’s Lemma,
$$\begin{aligned} \begin{aligned} d\ln P_D(t,T)&= \frac{d\ln P_D}{dP_D}\,dP_D+\frac{1}{2}\frac{d^2\ln P_D}{dP_D^2}\,(dP_D)^2\\&=\frac{1}{P_D(t,T)}\left[ r(t)P_D(t,T)\,dt+\vec {\sigma _D}(t,T)d\vec {W}^{\mathbb {Q}}\right] \\&\quad -\frac{1}{2}\vec {\sigma _D}(t,T)\vec {\sigma _D}(t,T)^\mathbb {T}\,dt\\&=\left[ r(t)-\frac{1}{2}\vec {\sigma _D}(t,T)\vec {\sigma _D}(t,T)^\mathbb {T}\right] \,dt+\vec {\sigma _D}(t,T)\,d\vec {W}^{\mathbb {Q}}, \end{aligned} \end{aligned}$$
(B.6)
which yields (5).\(\square \)
Appendix C. Proof of Proposition 2.2
As \(\alpha _x(t)\) is deterministic,
$$\begin{aligned} \begin{aligned} P_X(t,T)&=\mathbb {E}^{\mathbb {Q}} \left[ \left. e^{-\int _{t}^{T}x(s)ds} \right| \mathcal {F}_t\right] = \mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}\left[ \alpha _x(s)+\zeta _x(s)\right] ds} \right| \mathcal {F}_t\right] \\&= e^{-\int _{t}^{T}\alpha _x(s)ds}\mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}\zeta _x(s)ds} \right| \mathcal {F}_t\right] = e^{-\int _{t}^{T}\alpha _x(s)ds}P_{\zeta _x}(t,T). \end{aligned} \end{aligned}$$
By Feynman-Kac formula, \(P_{\zeta _x}(t,T)\) satisfies the following PDE with a terminal condition:
$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial P_{\zeta _x}}{\partial t}+ \frac{\partial P_{\zeta _x}}{\partial \zeta _r}\left[ -a\zeta _r+\eta ^{\mathbb {Q}}_{r}(t,T)v_1\right] + \frac{\partial P_{\zeta _x}}{\partial \zeta _x}\left[ -b\zeta _x+\eta ^{\mathbb {Q}}_{x}(t,T)v_1\right] \\&\quad +\frac{\partial P_{\zeta _x}}{\partial v_1}\left[ \bar{V_1}-\kappa _1v_1+\eta ^{\mathbb {Q}}_{v1}(t,T)v_1\right] +\frac{\partial P_{\zeta _x}}{\partial v_2}\left[ \bar{V_2}-\kappa _2v_2+\eta ^{\mathbb {Q}}_{v2}(t,T)v_2\right] \\&\quad +\frac{1}{2}\sum _{i=1}^{4}\left[ (CV_tR)(CV_tR)^{\mathbb {T}}\right] _{i,j}\frac{\partial ^2 P_{\zeta _x}}{\partial x_i x_j}-\zeta _x(t)P_{\zeta _x}= 0,\\&\qquad \hbox {(Terminal condition)}\quad P_{\zeta _x}\left( T,T;\zeta _r(T),\zeta _x(T),v_1(T),v_2(T)\right) = 1, \end{aligned} \right. \nonumber \\ \end{aligned}$$
(C.1)
where \(x_1:=\zeta _r\), \(x_2:=\zeta _x\), \(x_3:=v_1\), \(x_4:=v_2\).
We propose
$$\begin{aligned} P_{\zeta _x}(t,T)=e^{A_x(t,T)\zeta _r(t)+B_x(t,T)\zeta _x(t)+C_x(t,T)v_1(t)+D_x(t,T)v_2(t)+E_x(t,T)}. \end{aligned}$$
(C.2)
Under this affine proposition for \(P_{\zeta _x}(t,T)\), all of the partial derivatives could be obtained and substituted into (C.1). As (C.1) should be true for \(\forall \zeta _r(t)\), \(\zeta _x(t)\), \(v_1(t)\), and \(v_2(t)\), the coefficients of these four stochastic random variables should be zero over time, which yields \(A_x(t,T)\equiv 0\) and (14).\(\square \)
Appendix D. Proof of Lemma 2.2
By Ito’s Lemma,
$$\begin{aligned} \frac{dP_X(t,T)}{P_X(t,T)}=x(t)dt+\vec {\sigma _X}(t,T) d\vec {W}^{\mathbb {Q}_1}, \end{aligned}$$
(D.1)
where
$$\begin{aligned} \begin{aligned} \vec {\sigma _X}(t,T)&= \begin{bmatrix}A_x(t,T)&B_x(t,T)&C_x(t,T)&D_x(t,T)\end{bmatrix}CV_tR\\&= \begin{bmatrix}\sigma _{X,1}(t,T)\sqrt{v_1(t)} \\ \sigma _{X,2}(t,T)\sqrt{v_2(t)} \\ \sigma _{X,3}(t,T)\sqrt{v_1(t)} \\ \sigma _{X,4}(t,T)\sqrt{v_2(t)}\end{bmatrix}^T. \end{aligned} \end{aligned}$$
Therefore,
$$\begin{aligned} P_X(t,T)=P_X(0,T)e^{\int _0^t\left[ x(s)-\frac{1}{2}\vec {\sigma _X}(s,T)\vec {\sigma _X}^\mathbb {T}(s,T)\right] \,ds+\int _0^t\vec {\sigma _X}(s,T)\,d\vec {W}^{\mathbb {Q}_1}}. \end{aligned}$$
(D.2)
Substituting (9) into (D.1), the dynamics of \(P_X(t,T)\) under \(\mathbb {Q}\) is obtained:
$$\begin{aligned} \frac{dP_X(t,T)}{P_X(t,T)}=\left[ x(t)-\vec {\sigma _X}(t,T)\vec {\sigma _D}^{\mathbb {T}}(t,T)\right] \,dt+\vec {\sigma _X}(t,T)\,d\vec {W}^{\mathbb {Q}}, \end{aligned}$$
(D.3)
which further yields (16). Thus, substituting (5) and (16) into (7), the expression of \(P_Y(t,T)\) is obtained as
$$\begin{aligned} P_Y(t,T) = P_Y(0,T)e^{\int _0^t y(s)-\frac{1}{2}\vec {\sigma _Y}(s,T)\vec {\sigma _Y}^{\mathbb {T}}(s,T)\,ds+\int _0^t \vec {\sigma _Y}(s,T)\,d\vec {W}^{\mathbb {Q}}}, \end{aligned}$$
where
$$\begin{aligned} \vec {\sigma _Y}(t,T)=\vec {\sigma _D}(t,T)+\vec {\sigma _X}(t,T). \end{aligned}$$
Let
$$\begin{aligned} \begin{aligned} A_y(t,T)&:= A_r(t,T)+A_x(t,T)=A_r(t,T),\\ B_y(t,T)&:= B_r(t,T)+B_x(t,T)=B_x(t,T),\\ C_y(t,T)&:= C_r(t,T)+C_x(t,T), \quad \text {and}\\ D_y(t,T)&:= D_r(t,T)+D_x(t,T)=D_x(t,T). \end{aligned} \end{aligned}$$
Nicely, we observe the following consistency:
$$\begin{aligned} \vec {\sigma _Y}(t,T)= \begin{bmatrix} A_y(t,T)&B_y(t,T)&C_y(t,T)&D_y(t,T) \end{bmatrix}CV_tR. \end{aligned}$$
\(\square \)
Appendix E. Solution for Riccati Equation
We introduce how to solve a Riccati-type ODE in this appendix. Take \(C_r\) as an example:
$$\begin{aligned} \dot{C_r}+\frac{1}{2}\beta _1^2C_r^2+\left( \rho _1\beta _1A_r(s,T)-\kappa _1\right) C_r+ \frac{1}{2}A_r^2(s,T)=0 \quad \text {and} \quad C_r(T,T)&=0. \end{aligned}$$
(E.1)
First, we transform the parameter s into time to maturity \(\tau =T-s\) to obtain an ODE with an initial condition, which is equivalent to (E.1). Let \(f_C(\tau )=C_r(T-\tau ,T)=C_r(s,T)\), such that
$$\begin{aligned}&\dot{f_C}(\tau )+\frac{1}{2}\beta _1^2 f_C^2(\tau )+\left( \rho _1\beta _1A_r(s,T)-\kappa _1\right) f_C(\tau )\nonumber \\&\quad +\frac{1}{2}A_r^2(s,T)=0 \quad \text {and}\quad f_C (0)=0. \end{aligned}$$
(E.2)
The Radon Lemma provided in Theorem 3.1.1 of Abou-Kandil et al. [1] suggests the solution of (E.2) is
$$\begin{aligned} f_C(\tau )= \frac{P(\tau )}{Q(\tau )}, \quad t \le \tau < T, \end{aligned}$$
where
$$\begin{aligned} \begin{aligned} \frac{d}{dt}\begin{pmatrix} Q\\ P \end{pmatrix}&=\begin{pmatrix} \kappa _1 &{} -\frac{1}{2}\beta _1^2\\ \frac{1}{2}A_r^2(\tau ,T) &{} \rho _1\beta _1A_r(\tau ,T) \end{pmatrix}\begin{pmatrix} Q\\ P \end{pmatrix},\\ Q(0)&=1, \quad \text {and}\\ P(0)&=f_C(0). \end{aligned} \end{aligned}$$
(E.3)
(E.3) can be discretized and solved explicitly as follows:
$$\begin{aligned} \begin{pmatrix} Q(T_{i+1})\\ P(T_{i+1}) \end{pmatrix}-\begin{pmatrix} Q(T_{i})\\ P(T_{i}) \end{pmatrix}=(T_{i+1}-T_i)\begin{pmatrix} \kappa _1 &{} -\frac{1}{2}\beta _1^2\\ \frac{1}{2}A_r^2(T_{i+1},T) &{} \rho _1\beta _1A_r(T_{i+1},T) \end{pmatrix}\begin{pmatrix} Q(T_i)\\ P(T_i) \end{pmatrix}. \end{aligned}$$
The unknown boundary value \(f_C(T)\) or \(C_r(t,T)\) could be determined in the final step as
$$\begin{aligned} f_C(T_n)= \frac{P(T_n)}{Q(T_n)}, \end{aligned}$$
where \(T=T_n\).
Appendix F. Proof of Theorem 4.1
By (30),
$$\begin{aligned} PC(t)\nonumber&=\mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T_1}r(\tau )\,d\tau }\max \left\{ L_Y(T,T_1)-K,0\right\} \Delta T\right| \mathcal {F}_t\right] \nonumber \\&=\mathbb {E}^{\mathbb {Q}}\left[ \left. \frac{e^{-\int _{t}^{T}r(\tau )\,d\tau }e^{-\int _{T}^{T_1}r(\tau )\,d\tau }}{P_Y(T,T_1)}\max \left\{ 1-(1+K\Delta T)P_Y(T,T_1),0\right\} \right| \mathcal {F}_t\right] \nonumber \\&=\mathbb {E}^{\mathbb {Q}}\left[ \left. \mathbb {E}^{\mathbb {Q}}\left[ \left. \frac{e^{-\int _{t}^{T}r(\tau )\,d\tau }e^{-\int _{T}^{T_1}r(\tau )\,d\tau }}{P_Y(T,T_1)}\max \left\{ 1-(1+K\Delta T)P_Y(T,T_1),0\right\} \right| \mathcal {F}_T\right] \right| \mathcal {F}_t\right] \nonumber \\&=\mathbb {E}^{\mathbb {Q}}\left[ \left. \frac{e^{-\int _{t}^{T}r(\tau )\,d\tau }\mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{T}^{T_1}r(\tau )\,d\tau }\right| \mathcal {F}_T\right] }{P_Y(T,T_1)}\max \left\{ 1-(1+K\Delta T)P_Y(T,T_1),0\right\} \right| \mathcal {F}_t\right] \nonumber \\&= \mathbb {E}^{\mathbb {Q}}\left[ \left. e^{-\int _{t}^{T}r(\tau )\,d\tau }\frac{P_D(T,T_1)}{P_Y(T,T_1)}\max \left\{ 1-(1+K\Delta T)P_Y(T,T_1),0\right\} \right| \mathcal {F}_t\right] . \end{aligned}$$
(F.1)
The following aims to simplify \(e^{-\int _{t}^{T}r(\tau )\,d\tau }\frac{P_D(T,T_1)}{P_Y(T,T_1)}\) in search of the possibility to change the measure. Under \(\mathbb {Q}\),
$$\begin{aligned} e^{-\int _{t}^{T}r(\tau )\,d\tau }&= e^{-\int _{0}^{T}r(\tau )\,d\tau }P_D(T,T)\nonumber \\&= e^{-\int _{0}^{T}r(\tau )\,d\tau }P_D(t,T)e^{\int _{0}^{T}r(\tau )\,d\tau -\frac{1}{2}\int _{t}^{T}\vec {\sigma _D}(\tau ,T)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T)\,d\tau +\int _{t}^{T}\vec {\sigma _D}(\tau ,T)\,d\vec {W}^{\mathbb {Q}}}\nonumber \\&= P_D(t,T)e^{-\frac{1}{2}\int _{t}^{T}\vec {\sigma _D}(\tau ,T)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T)\,d\tau +\int _{t}^{T}\vec {\sigma _D}(\tau ,T)\,d\vec {W}^{\mathbb {Q}}}, \end{aligned}$$
(F.2)
and
$$\begin{aligned} \frac{P_D(T,T_1)}{P_Y(T,T_1)}&= \frac{P_X(T,T)}{P_X(T,T_1)}\nonumber \\&=\frac{P_X(t,T)e^{\int _{t}^{T}\left[ x(\tau )-\vec {\sigma _X}(\tau ,T)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T)-\frac{1}{2}\vec {\sigma _X}(\tau ,T)\vec {\sigma _X}^{\mathbb {T}}(\tau ,T)\right] \,d\tau +\int _{t}^{T}\vec {\sigma _X}(\tau ,T)\,d\vec {W}^{\mathbb {Q}}}}{P_X(t,T_1)e^{\int _{t}^{T}\left[ x(\tau )-\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T_1)-\frac{1}{2}\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _X}^{\mathbb {T}}(\tau ,T_1)\right] \,d\tau +\int _{t}^{T}\vec {\sigma _X}(\tau ,T_1)\,d\vec {W}^{\mathbb {Q}}}}. \end{aligned}$$
(F.3)
Multiplying (F.2) and (F.3) enables a change of measure in (F.1):
$$\begin{aligned}&e^{-\int _{t}^{T}r(\tau )\,d\tau }\frac{P_D(T,T_1)}{P_Y(T,T_1)}\nonumber \\&=\frac{P_Y(t,T)}{P_X(t,T_1)}e^{-\frac{1}{2}\int _{t}^{T}\left[ \vec {\sigma _D}(\tau ,T)+\vec {\sigma _X}(\tau ,T)\right] \left[ \vec {\sigma _D}(\tau ,T)+\vec {\sigma _X}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau +\int _{t}^{T}\left[ \vec {\sigma _D}(\tau ,T)+\vec {\sigma _X}(\tau ,T)\right] \,d\vec {W}^{\mathbb {Q}}}\nonumber \\&\quad \times e^{\int _{t}^{T}\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T_1)\,d\tau +\frac{1}{2}\int _{t}^{T}\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _X}^\mathbb {T}(\tau ,T_1)\,d\tau -\int _{0}^{T}\vec {\sigma _X}(\tau ,T_1)\,d\vec {W}^{\mathbb {Q}}}\nonumber \\&=\frac{P_Y(t,T)}{P_X(t,T_1)}e^{-\frac{1}{2}\int _{t}^{T}\vec {\sigma _Y}(\tau ,T)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)\,d\tau +\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T)-\vec {\sigma _X}(\tau ,T_1)\right] \,d\vec {W}^{\mathbb {Q}}}\nonumber \\&\quad \times e^{\int _{t}^{T}\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _D}^{\mathbb {T}}(\tau ,T_1)\,d\tau +\frac{1}{2}\vec {\sigma _X}(\tau ,T_1)\vec {\sigma _X}^{\mathbb {T}}(\tau ,T_1)\,d\tau }\nonumber \\&= \frac{P_Y(t,T)}{P_X(t,T_1)}e^{-\frac{1}{2}\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T)-\vec {\sigma _X}(\tau ,T_1)\right] \left[ \vec {\sigma _Y}(\tau ,T)-\vec {\sigma _X}(\tau ,T_1)\right] ^{\mathbb {T}}\,d\tau +\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T)-\vec {\sigma _X}(\tau ,T_1)\right] \,d\vec {W}^{\mathbb {Q}}}\nonumber \\&\quad \times e^{\int _{t}^{T}\vec {\sigma _X}(\tau ,T_1)\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau }. \end{aligned}$$
(F.4)
By changing the measure from \(\mathbb {Q}\) to \(\mathbb {Q}_2\), where \(\mathbb {Q}_2\) is defined by (33) and (34), and the price at time t of the caplet is reduced to
$$\begin{aligned}&PC(t) \\&\quad =\frac{P_Y(t,T)}{P_X(t,T_1)}\mathbb {E}^{\mathbb {Q}_2}\left[ \left. e^{\int _t^T\vec {\sigma _X}(\tau ,T_1)\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau }\max \left\{ 1-\left( 1+K\Delta T\right) P_Y(T,T_1),0\right\} \right| \mathcal {F}_t\right] .\nonumber \end{aligned}$$
(F.5)
Under \(\mathbb {Q}_2\), the risk-bearing zero bond \(P_Y(t,T_1)\) follows these dynamics:
$$\begin{aligned} \frac{dP_Y(t,T_1)}{P_Y(t,T_1)}=\left[ y(t)+\vec {\sigma _Y}(t,T_1)\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}\right] \,dt+\vec {\sigma _Y}(t,T_1)\,d\vec {W}^{\mathbb {Q}_2}, \end{aligned}$$
which yields
$$\begin{aligned} P_Y(T,T_1)&= P_Y(t,T_1)e^{\int _{t}^{T}\left[ y(\tau )+\vec {\sigma _Y}(\tau ,T_1)\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}-\frac{1}{2}\vec {\sigma _Y}(t,T_1)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T_1)\right] \,d\tau +\int _t^T\vec {\sigma _Y}(\tau ,T_1)\,d\vec {W}^{\mathbb {Q}_2}} \quad \text {and} \end{aligned}$$
(F.6)
$$\begin{aligned} P_Y(T,T)&= P_Y(t,T)e^{\int _{t}^{T}\left[ y(\tau )+\vec {\sigma _Y}(\tau ,T)\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}-\frac{1}{2}\vec {\sigma _Y}(t,T)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)\right] \,d\tau +\int _t^T\vec {\sigma _Y}(\tau ,T)\,d\vec {W}^{\mathbb {Q}_2}}. \end{aligned}$$
(F.7)
Dividing (F.6) by (F.7) gives a simplified form of \(P_Y(T,T_1)\):
$$\begin{aligned} \begin{aligned} P_Y(T,T_1)&=\frac{P_Y(T,T_1)}{P_Y(T,T)}=\frac{P_Y(t,T_1)}{P_Y(t,T)}\\&\quad \times e^{\int _0^T\left[ \left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}-\frac{1}{2}\vec {\sigma _Y}(\tau ,T_1)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T_1)+\frac{1}{2}\vec {\sigma _Y}(\tau ,T)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)\right] \,d\tau }\\&\quad \times e^{\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \,d\vec {W}^{\mathbb {Q}_2}}\\&= \frac{P_Y(t,T_1)}{P_Y(t,T)}e^{-\frac{1}{2}\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau +\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \,d\vec {W}^{\mathbb {Q}_2}}\\ {}&\quad \times e^{-\frac{1}{2}\left[ 2\vec {\sigma _Y}(\tau ,T_1)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)-\vec {\sigma _Y}(\tau ,T)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)\right] \,d\tau +\frac{1}{2}\int _{t}^{T}\vec {\sigma _Y}(\tau ,T)\vec {\sigma _Y}^{\mathbb {T}}(\tau ,T)\,d\tau }\\ {}&\quad \times e^{\int _t^T\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \left[ \vec {\sigma _Y}(\tau ,T)-\vec {\sigma _X}(\tau ,T_1)\right] ^{\mathbb {T}}\,d\tau }\\&=\frac{P_Y(t,T_1)}{P_Y(t,T)}e^{-\frac{1}{2}\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau +\int _{t}^{T}\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \,d\vec {W}^{\mathbb {Q}_2}}\\ {}&\quad \times e^{\int _{t}^{T}-\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] \vec {\sigma _X}^{\mathbb {T}}(\tau ,T_1)\,d\tau }. \end{aligned} \end{aligned}$$
Let \(S_1(\tau )\) and \(S_2(\tau )\) be defined by (35). Then \(P_Y(T,T_1)\) is represented by \(e^{S_1(T)+S_2(T)}\) and \(e^{\int _t^T\vec {\sigma _X}(\tau ,T_1)\left[ \vec {\sigma _Y}(\tau ,T_1)-\vec {\sigma _Y}(\tau ,T)\right] ^{\mathbb {T}}\,d\tau }\) is represented by \(e^{-S_1(T)}\).
Appendix G. Proof of Lemma 4.1
Proof
By (34),
$$\begin{aligned} \begin{aligned}&\begin{bmatrix} dr(t)\\dx(t)\\dv_1(t)\\dv_2(t) \end{bmatrix}= \begin{bmatrix} \theta _r(t)-ar(t)\\ \theta _x(t)-bx(t)\\\bar{V_1}-\kappa _1 v_1(t)\\\bar{V_2}-\kappa _2 v_2(t) \end{bmatrix}\,dt+CV_t R\left[ d\vec {W}^{\mathbb {Q}_2}+\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}dt\right] \\&\quad =\left[ \begin{bmatrix} \theta _r(t)-ar(t)\\ \theta _x(t)-bx(t)\\\bar{V_1}-\kappa _1 v_1(t)\\\bar{V_2}-\kappa _2 v_2(t) \end{bmatrix}+CV_t R\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}\right] \,dt+CV_t R \,d\vec {W}^{\mathbb {Q}_2}. \end{aligned} \end{aligned}$$
We further simplify the extra drift term under \(\mathbb {Q}_2\).
$$\begin{aligned} CV_t R\left[ \vec {\sigma _Y}(t,T)-\vec {\sigma _X}(t,T_1)\right] ^{\mathbb {T}}&= CV_t R R^{\mathbb {T}}V_t^{\mathbb {T}}C^{\mathbb {T}} \begin{bmatrix} A_y(t,T)-A_x(t,T_1)\\B_y(t,T)-B_x(t,T_1)\\C_y(t,T)-C_x(t,T_1)\\D_y(t,T)-D_x(t,T_1) \end{bmatrix}\\&:= \begin{bmatrix} \eta _r^{\mathbb {Q}_2}(t)v_1(t)\\ \eta _{x,1}^{\mathbb {Q}_2}(t)v_1(t)+\eta _{x,2}^{\mathbb {Q}_2}(t)v_2(t)\\ \eta _{v_1}^{\mathbb {Q}_2}(t)v_1(t)\\ \eta _{v_2}^{\mathbb {Q}_2}(t)v_2(t), \end{bmatrix}, \end{aligned}$$
where \(\eta _r^{\mathbb {Q}_2}\), \(\eta _x^{\mathbb {Q}_2}\), \(\eta _{v_1}^{\mathbb {Q}_2}\), and \(\eta _{v_2}^{\mathbb {Q}_2}\) are defined in (36). \(\square \)
