3.1 Handling integral term
Integral term is defined on\(\left(-\infty,+\infty\right)\),then the region outside \(\Omega_{1}\) also needs to be calculated
We set
\(\ z=logy,\tilde{H}\left(z,\tilde{y},U,\tau\right)=H\left(e^{z},\tilde{y},U,\tau\right)\ \)and\(\tilde{f}\left(z\right)=e^{z}f\left(e^{z}\right)\)
so the integral term of (3.1)
\begin{equation} \lambda\int_{0}^{+\infty}{V\left(T-\tau,ye^{x}\right)f\left(y\right)}\text{dy}\nonumber \\ \end{equation}\begin{equation} =\lambda\int_{-\infty}^{+\infty}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}\text{dz}\nonumber \\ \end{equation}\begin{equation} =\lambda\int_{-\infty}^{-\varphi}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}dz+\lambda\int_{-\varphi}^{+\varphi}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}dz+\lambda\int_{+\varphi}^{+\infty}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}\text{dz}\nonumber \\ \end{equation}
By the composite trapezoidal rule in the region\(\ \Omega\), we have the following approximation
\begin{equation} \lambda\int_{-\varphi}^{+\varphi}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}\text{dz}\nonumber \\ \end{equation}\begin{equation} \approx\frac{h}{3}\sum_{m=1}^{M}\left[V\left(T-\tau,e^{x+z_{m-1}}\right)f\left(e^{z_{m-1}}\right)e^{z_{m-1}}+4V\left(T-\tau,ye^{x+z_{m}}\right)f\left(e^{z_{m}}\right)e^{z_{m}}+V\left(T-\tau,e^{x+z_{m+1}}\right)f\left(e^{z_{m+1}}\right)e^{z_{m+1}}\right]\nonumber \\ \end{equation}
Integral term is defined in the region\(\left(-\infty,+\infty\right)\), then the region outside\(\Omega\) also needs to be calculated.
For shout put options, when \(\varphi\) is sufficiently large, we obtain
\begin{equation} \lambda\int_{+\varphi}^{+\infty}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}dz\approx\lambda\int_{+\varphi}^{+\infty}{\max\left(1-e^{\xi},0\right)}f\left(\xi\right)d\xi=0.\nonumber \\ \end{equation}
For the integral term of shout put options defined in the region\(\left(-\infty,-\varphi\right)\), we estimate its value by use of the composite trapezoidal rule
\begin{equation} \lambda\int_{-\infty}^{-\varphi}{V\left(T-\tau,e^{x+z}\right)f\left(e^{z}\right)e^{z}}dz\approx\lambda\int_{-\infty}^{-\varphi}{\max\left(1-e^{\xi},0\right)}f\left(\xi\right)\text{dξ.}\nonumber \\ \end{equation}