Pricing Fade-in Options Under GARCH-Jump Processes

Abstract

In this paper, we investigate fade-in options under GARCH-jump processes. Specifically, we adopt NIG distributions to capture jump risk, and both market and individual jumps are considered. In the pricing model driven by GARCH-jump processes, we obtain the prices of fade-in options using the Fourier transform methods. Finally, we use the derived pricing formulae to illustrate the effects of fade-in sets and the parameters in the jump processes.

Appendix

Reform of Underlying Assets and Market Dynamics:

$$\begin{aligned} \left\{ \begin{array}{ll} \ln S(t+1) = \ln S(t)+r-\frac{1}{2}\beta ^2_{s,z}h_{m,z}(t+1)-\Pi _m(\beta _{s,y})h_{m,y}(t+1)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{1}{2}h_{s,z}(t+1)-\Pi _s(1)h_{s,y}(t+1)+\beta _{s,z}\sqrt{h_{m,z}(t+1)}Z_m(t+1)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\beta _{s,y}y_m(t+1)+\sqrt{h_{s,z}(t+1)}Z_s(t+1)+y_s(t+1),\\ \ln M(t+1) = \ln M(t)+r-\frac{1}{2}h_{m,z}(t+1)-\Pi _m(1)h_{m,y}(t+1)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\sqrt{h_{m,z}(t+1)}Z_m(t+1)+y_m(t+1),\\ h_{m,z}(t+1)=\pi _{m,z,1}+\pi _{m,z,2}h_{m,z}(t)+\pi _{m,z,6}\Big (Z_m(t)\Big )^2+\pi _{m,z,7}\sqrt{h_{m,z}(t)}Z_m(t),\\ h_{s,z}(t+1)=\pi _{s,z,1}+\pi _{s,z,2}h_{s,z}(t)+\pi _{s,z,4}h_{s,z}(t)+\pi _{s,z,6}\Big (Z_m(t)\Big )^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\pi _{s,z,7}\sqrt{h_{m,z}(t)}Z_m(t) +\pi _{s,z,8}\Big (Z_s(t)\Big )^2+\pi _{s,z,9}\sqrt{h_{s,z}(t)}Z_s(t),\\ h_{m,z}(t+1)=\pi _{m,y,1}+\pi _{m,y,2}h_{m,z}(t)+\pi _{m,y,3}h_{m,y}(t)+\pi _{m,y,6}\Big (Z_m(t)\Big )^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\pi _{m,y,7}\sqrt{h_{m,z}(t)}Z_m(t),\\ h_{s,y}(t+1)=\pi _{s,y,1}+\pi _{s,y,2}h_{m,z}(t)+\pi _{s,y,3}h_{m,y}(t)+\pi _{s,y,4}h_{s,z}(t)+\pi _{s,y,5}h_{s,y}(t)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\pi _{s,y,6}\Big (Z_m(t)\Big )^2+\pi _{s,y,7}\sqrt{h_{m,z}(t)}Z_m(t)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\pi _{s,y,8}\Big (Z_s(t)\Big )^2+\pi _{s,y,9}\sqrt{h_{s,z}(t)}Z_s(t), \end{array}\right. \end{aligned}$$

(5.1)

where $\pi _{u,v,w}\ u\in \{m,s\},v\in \{z,y\},w\in \{1,2,\ldots ,9\}$ are the reformed parameters. They have the same forms as those in Bégin et al. (2020). Table 3 lists the results.

Table 4 Parameters relationships

$\square$

Proof of Proposition 2.1: First, we discuss the moment generating function $f(t;t_i,\phi _1,\phi _2)$ when $t_i\le t\le T$,

$$\begin{aligned} f(t;t_i,\phi _1,\phi _2)= & {} \exp \Big \{\phi _1\ln S(t)+\phi _2\ln S(t_i)+A_0(t;t_i,\phi _1,\phi _2)\\{} & {} \ \ \ \ \ \ \ \ +A_1(t;t_i,\phi _1,\phi _2)h_{m,z} (t+1)+A_2(t;t_i,\phi _1,\phi _2)h_{m,y} (t+1)\\{} & {} \ \ \ \ \ \ \ \ +A_3(t;t_i,\phi _1,\phi _2)h_{s,z} (t+1)+A_4(t;t_i,\phi _1,\phi _2)h_{s,y} (t+1)\Big \}. \end{aligned}$$

In the following deduction, we use f(t), $A_0(t)$, $A_1(t)$, $A_2(t)$, $A_3(t)$, $A_4(t)$ for simplicity. From the definition of $f(t;t_i,\phi _1,\phi _2)$, we can get that

$$\begin{aligned} f(T;t_i,\phi _1,\phi _2)=e^{\phi _1\ln S(T)+\phi _2\ln S(t_i)}, \end{aligned}$$

which means that

$$\begin{aligned} A_0(T)=A_1(T)=A_2(T)=A_3(T)=A_4(T)=0. \end{aligned}$$

According to the law of iterated expectations, dynamics of the underlying asset, and the intensity process, one gets the following result,

$$\begin{aligned} f(t)= & {} E_t\Big [f(t+1)\Big ]\\= & {} E_t\Big [\exp \Big \{\phi _1\ln S(t+1)+\phi _2\ln S(t_i)+A_0(t+1)+A_1(t+1)h_{m,z} (t+2)\\{} & {} \ \ \ \ \ \ \ \ \ \ \ \ \ +A_2(t+1)h_{m,y} (t+2)+A_3(t+1)h_{s,z}(t+2)+A_4(t+1)h_{s,y}(t+2)\Big \}\Big ]. \end{aligned}$$

There are so many terms and parameters in the expression, but they have similar forms. Thus, we use substitute terms according to Bégin et al. (2020) for a clearer result. For $w \in \{1,2,\ldots ,9\}$, define

$$\begin{aligned} D_w(t;t_i,\phi _1,\phi _2)= & {} A_1(t;t_i,\phi _1,\phi _2)\pi _{m,z,w}+ A_2(t;t_i,\phi _1,\phi _2)\pi _{m,y,w}\\{} & {} +A_3(t;t_i,\phi _1,\phi _2)\pi _{s,z,w}+A_4(t;t_i,\phi _1,\phi _2)\pi _{s,y,w}. \end{aligned}$$

Then, the moment generating function takes the following form,

$$\begin{aligned} f(t)= & {} \exp \Big \{\phi _1\ln S(t)+\phi _2\ln S(t_i)+\phi _1 r+A_0(t+1)+D_1(t+1)\\{} & {} +\big (D_2(t+1)-\frac{\phi _1}{2}\beta _{s,z}^2\big )h_{m,z}(t+1)\\{} & {} +\big (D_3(t+1)-\phi _1\pi _m(\beta _{s,y})\big )h_{m,y}(t+1)+\big (D_4(t+1)-\frac{1}{2}\big )h_{s,z}(t+1)\\{} & {} +\big (D_5(t+1)-\phi _1\pi _s(1)\big )h_{s,y}(t+1)\Big \}\\{} & {} \times E_t\Big [\exp \Big \{ D_6(t+1)\big (Z_m(t+1)\big )^2+\big (D_7(t+1)+\phi _1\beta _{s,z}\big )\sqrt{h_{m,z}(t+1)}Z_m(t+1) \\{} & {} +\phi _1\beta _{s,y}y_m(t+1) \quad D_8(t+1)\big (Z_s(t+1)\big )^2+\big (D_9(t+1)+\phi _1\big )\sqrt{h_{s,z}(t+1)}Z_s(t+1)\\{} & {} +\phi _1y_s(t+1) \Big \}\Big ]. \end{aligned}$$

Moreover, the following conclusions are proposed for further derivation,

$$\begin{aligned} E_t\Big [e^{a(Z(t+1))^2+bZ(t+1)}\Big ]= & {} \exp \Big \{-\frac{1}{2}\ln (1-2a)+\frac{b^2}{2(1-2a)}\Big \},\\ E_t\Big [e^{ay(t+1)}\Big ]= & {} \exp \Big \{\Pi (a)h_y(t+1)\Big \}. \end{aligned}$$

where $Z(t+1)$ is a standard normal sequence, $y(t+1)$ is the NIG process with intensity $h_y (t+1)$ (Table 4).

Calculating conditional expectation terms in the polynomial above gives the following results,

$$\begin{aligned}{} & {} E_t\Big [\exp \Big \{ D_6(t+1)\big (Z_m(t+1)\big )^2+\big (D_7(t+1)+\phi _1\beta _{s,z}\big )\sqrt{h_{m,z}(t+1)}Z_m(t+1)\\{} & {} \quad \quad +\phi _1\beta _{s,y}y_m(t+1) D_8(t+1)\big (Z_s(t+1)\big )^2 \\{} & {} \quad \quad +\big (D_9(t+1)+\phi _1\big )\sqrt{h_{s,z}(t+1)}Z_s(t+1)+\phi _1y_s(t+1) \Big \}\Big ] \\{} & {} \quad = \exp \left\{ -\frac{1}{2}\big (1-2D_6(t+1)\big )+\frac{1}{2}\frac{\big (D_7(t+1)+\phi _1\beta _{s,z}\big )^2}{1-2D_6(t+1)}h_{m,z}(t+1) \right. \\{} & {} \quad \quad +\Pi _m(\phi _1\beta _{s,y})h_{m,y}(t+1) -\frac{1}{2}\big (1-2D_8(t+1)\big )\\{} & {} \quad \quad \left. +\frac{1}{2}\frac{\big (D_9(t+1)+\phi _1\big )^2}{1-2D_8(t+1)}h_{m,z}(t+1)+\Pi _s(\phi _1)h_{s,y}(t+1)\right\} . \end{aligned}$$

Above all, we can derive the explicit forms of $A_0(t),A_1(t),A_2(t),A_3(t),A_4(t)$ when $t_i\le t\le T$, i.e.,

$$\begin{aligned} A_0(t)= & {} \phi _1 r+A_0(t+1)+D_1(t+1)-\frac{1}{2}\ln (1-2D_6(t+1))\\{} & {} -\frac{1}{2}\ln (1-2D_8(t+1)),\\ A_1(t)= & {} D_2(t+1)-\frac{1}{2}\phi _1\beta _{s,z}^2+\frac{1}{2}\frac{\big (D_7(t+1)+\phi _1\beta _{s,z}\big )^2}{1-2D_6(t+1)},\\ A_2(t)= & {} D_3(t+1)-\phi _1\Pi _m(\beta _{s,y})+\Pi _m(\phi _1\beta _{s,y}),\\ A_3(t)= & {} D_4(t+1)-\frac{1}{2}\phi _1+\frac{1}{2}\frac{\big (D_9(t+1)+\phi _1\big )^2}{1-2D_8(t+1)},\\ A_4(t)= & {} D_5(t+1)-\phi _1\Pi _s(1)+\Pi _s(\phi _1). \end{aligned}$$

Next, we discuss the moment generating function $f(t;t_i,\phi _1,\phi _2)$ when $0\le t \le t_i$,

$$\begin{aligned} f(t;t_i,\phi _1,\phi _2)= & {} \exp \Big \{(\phi _1+\phi _2)\ln S(t)+B_0(t;t_i,\phi _1,\phi _2)\\{} & {} \ \ \ \ \ \ \ \ +B_1(t;t_i,\phi _1,\phi _2)h_{m,z} (t+1)+B_2(t;t_i,\phi _1,\phi _2)h_{m,y} (t+1)\\{} & {} \ \ \ \ \ \ \ \ +B_3(t;t_i,\phi _1,\phi _2)h_{s,z} (t+1)+B_4(t;t_i,\phi _1,\phi _2)h_{s,y} (t+1)\Big \}. \end{aligned}$$

Similarly, we use $B_0(t)$, $B_1(t)$, $B_2(t)$, $B_3(t)$, $B_4(t)$ for simplicity, and we can obtain the expression of $f(t_i;t_i,\phi _1,\phi _2)$, i.e.,

$$\begin{aligned} f(t_i;t_i,\phi _1,\phi _2)= & {} \exp \Big \{(\phi _1+\phi _2)\ln S(t_i)+A_0(t_i) +A_1(t_i)h_{m,z}(t_i+1)\\{} & {} \ \ \ \ \ \ \ +A_2(t_i)h_{m,y}(t_i+1)+A_3(t_i)h_{s,z}(t_i+1)+A_4(t_i)h_{s,y}(t+1)\Big \}, \end{aligned}$$

which means that

$$\begin{aligned} B_0(t_i)=A_0(t_i),\ \ B_1(t_i)=A_1(t_i),\ \ B_2(t_i)=A_2(t_i),\ \ B_3(t_i)=A_3(t_i),\ \ B_4(t_i)=A_4(t_i). \end{aligned}$$

We can see that the only difference in the forms of the moment generating functions between $0\le t \le t_i$ and $t_i\le t\le T$ is the term $(\phi _1+\phi _2)\ln S(t)$ and $\phi _1\ln S(t)+\phi _2\ln S(t_i)$. Thus, we can directly obtain the following results for $0\le t \le t_i$,

$$\begin{aligned} B_0(t)= & {} (\phi _1+\phi _2) r+B_0(t+1)+D_1(t+1)-\frac{1}{2}\ln (1-2D_6(t+1))\\{} & {} -\frac{1}{2}\ln (1-2D_8(t+1)),\\ B_1(t)= & {} D_2(t+1)-\frac{1}{2}(\phi _1+\phi _2)\beta _{s,z}^2+\frac{1}{2}\frac{\big (D_7(t+1)+(\phi _1+\phi _2)\beta _{s,z}\big )^2}{1-2D_6(t+1)},\\ B_2(t)= & {} D_3(t+1)-(\phi _1+\phi _2)\Pi _m(\beta _{s,y})+\Pi _m\big ((\phi _1+\phi _2)\beta _{s,y}\big ),\\ B_3(t)= & {} D_4(t+1)-\frac{1}{2}(\phi _1+\phi _2)+\frac{1}{2}\frac{\big (D_9(t+1)+(\phi _1+\phi _2)\big )^2}{1-2D_8(t+1)},\\ B_4(t)= & {} D_5(t+1)-(\phi _1+\phi _2)\Pi _s(1)+\Pi _s\big (\phi _1+\phi _2\big ). \end{aligned}$$

$\square$

Proof of Proposition 2.2: For any $\alpha ,\ \beta >0$ and $x\in \{k,l\}$, define the two-dimensional Fourier transform with respect to k and x of $C_j(k,x)$ as follows,

$$\begin{aligned} \hat{C_j}(\upsilon ,\nu ):= & {} \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{\alpha k +\beta x}C_j(k,x)e^{2\pi i(\upsilon k+\nu x)}\,\textrm{d}k\textrm{d}x. \end{aligned}$$

Simple calculations imply that

$$\begin{aligned} \hat{C_j}(\upsilon ,\nu )= & {} E\left[ \int _{-\infty }^{\infty }e^{(2\pi i\nu +\beta )x}I\big (S(t_j)\ge e^{x}\big )\left( \int _{-\infty }^{\ln S(T)}e^{(2\pi i\upsilon +\alpha )k}\left( S(T)-e^{k}\right) \,\textrm{d}k\right) \,\textrm{d}x\right] ,\\= & {} E\left[ \frac{e^{(\alpha +2\pi i\upsilon +1)\ln S(T)}}{(\alpha +2\pi i\upsilon )(\alpha +2\pi i\upsilon +1)}\int _{-\infty }^{\ln S(t_j)}e^{(2\pi i\nu +\beta )x}\,\textrm{d}x\right] ,\\= & {} \frac{1}{(\alpha +2\pi i\upsilon )(\beta +2\pi i\nu )(\alpha +2\pi i\upsilon +1)}E\Big [e^{(\alpha +2\pi i\upsilon +1)\ln S(T)+(\beta +2\pi i\nu )\ln S(t_j)}\Big ],\\= & {} \frac{1}{(\alpha +2\pi i\upsilon )(\beta +2\pi i\nu )(\alpha +2\pi i\upsilon +1)}f(0;t_j,\alpha +2\pi i\upsilon +1,\beta +2\pi i\nu ). \end{aligned}$$

$\square$