**Appendix B. Helpful Results**

Given the 4/2 process, the following c.f. are used in this paper:

$$\begin{aligned} dZ(t) &= \left[ L + h\left(\sqrt{v(t)} + \frac{b}{\sqrt{v(t)}}\right)^2 \right] dt + g\left(\sqrt{v(t)} + \frac{b}{\sqrt{v(t)}}\right) dW\_t \\ dv(t) &= \alpha(\theta - v(t))dt + \xi\sqrt{v(t)}dB(t) \\ \langle dB(t), dW(t) \rangle &= \rho dt \end{aligned}$$

•

Φ*G* (*<sup>T</sup>*, *u*; *L*, *h*, *g*, *α*, *θ*, *ξ*, *ρ*, *b*, *c*, *vt*, *Zt*) = E *<sup>e</sup>uZ*(*T*)|F*t*= *exp* (*uZ*(*t*) + *α*2*θ ξ*2 (*T* − *t*) + *u r* + 2(*h* − 12 )*g*2*<sup>b</sup>* − *gραθ ξ* + *gbρα ξ* (*T* − *t*) + *u*<sup>2</sup>(<sup>1</sup> − *ρ*<sup>2</sup>)*g*2*b*(*<sup>T</sup>* − *t*)\* × ⎛⎝ √*Au ξ*<sup>2</sup>*sinh* √*Au* 2 *t* ⎞⎠*mu*+1 *ν*(*t*) 12 + *mu*2 − *αθ ξ*2 − *ugbρ ξ Ku*(*T*) − *ugρξ* − 12 + *mu*2 + *αθ ξ*2 + *ugbρ ξ* × exp ( *ν*(*t*) *ξ*2 −*Au* coth √*Au*(*T* − *t*) 2 + *α* − *ugρξ* \* Γ 12 + *mu*2 + *αθξ*2 + *ugbρ ξ* <sup>Γ</sup>(*mu* + 1) ×1 *F*1 ⎛⎝12 + *mu*2 + *αθξ*2 + *ugbρ ξ* , *mu* + 1, *Auν*(*t*) *ξ*<sup>4</sup>(*Ku*(*T*) − *ugρξ* ) sinh<sup>2</sup> √*Au*(*<sup>T</sup>*−*<sup>t</sup>*) 2 ⎞⎠ ,

with

•

$$\begin{aligned} A\_{\
u} &= a^2 - 2\xi^2 \left( u \left( \frac{g\rho u}{\xi} + (h - \frac{1}{2})g^2 \right) + \frac{1}{2}u^2(1 - \rho^2)g^2 \right), \\ m\_{\rm u} &= \frac{2}{\xi^2} \sqrt{\left( a\theta - \frac{\xi^2}{2} \right)^2 - 2\xi^2 \left( u \left( \frac{g\theta\rho}{\xi} \left( \frac{\xi^2}{2} - a\theta \right) + (h - \frac{1}{2})g^2b^2 \right) + \frac{1}{2}u^2(1 - \rho^2)g^2b^2 \right)}, \\ K\_{\rm u}(T) &= \frac{1}{\xi^2} \left( \sqrt{A\_{\rm u}} \coth\left( \frac{\sqrt{A\_{\rm u}}(T - t)}{2} \right) + a \right) \end{aligned}$$

$$\begin{split} &\Phi\_{\mathbf{G},T}\left(T,u;L,h,\xi,\kappa,\theta,\xi\_{t}^{t},\rho,b,c,u\_{t},Z\_{t}\right) = \mathbb{E}\left[e^{uZ(T)}|\mathcal{F}\_{t}\cup v(T)\right] \\ &= \exp\left\{uZ(t) + u\left(r + 2(h-\frac{1}{2})g^{2}b - \frac{a\rho a\theta}{\xi} + \frac{b\rho a}{\xi}\right)(T-t) + u^{2}(1-\rho^{2})g^{2}b(T-t)\right\} \\ &\times \exp\left\{\frac{u\varrho\rho}{\xi}\left(\nu(T) - \nu(t)\right) + \frac{u\varrho b\rho}{\xi}\log\frac{\nu(T)}{\nu(t)}\right\} \\ &\times \frac{\sqrt{A\_{u}}\sinh\left(\frac{a(T-t)}{2}\right)}{a\sinh\left(\frac{\sqrt{A\_{u}}(T-t)}{2}\right)}\exp\left(\frac{\nu(T) + \nu(t)}{\xi^{2}}\left(a\coth\left(\frac{a(T-t)}{2}\right) - \sqrt{A\_{u}}\coth\left(\frac{\sqrt{A\_{u}}(T-t)}{2}\right)\right)\right) \\ &\times \frac{I\_{1}}{\frac{2}{\xi^{2}}\sqrt{\left(a^{2}-\frac{\xi^{2}}{2}\right)^{2} + 2\xi^{2}\_{\mathrm{B}\_{u}}\frac{\left(2\sqrt{A\_{u}}\nu(t)\right)}{\xi^{2}\sinh\left(\frac{\sqrt{A\_{u}}(T-t)}{2}\right)}}\,\Big{]} \\ &\times \frac{I\_{2\varrho}}{\frac{2\sigma^{2}}{\xi^{2}} - \left(\frac{2\alpha\sqrt{v(T)}\nu(t)}{\xi^{2}\sinh\left(\frac{\sqrt{A\_{u}}(T-t)}{2}\right)}\right)}\,\mathrm{Ad}\left(\frac{\$$

with

$$B\_{\mathsf{H}} = \mu \left( \frac{gb\rho}{\xi^3} \left( \frac{\xi^2}{2} - a\theta \right) + (h - \frac{1}{2})g^2b^2 \right) + \frac{1}{2}\mu^2(1 - \rho^2)g^2b^2\rho$$
