3.3 New algorithm
\begin{equation}
BV^{{}^{\prime}}\left(\tau\right)=\text{AV}\left(\tau\right)+\text{BW}\left(\tau\right)+\left(\tau\right),\nonumber \\
\end{equation}where
\begin{equation}
V\left(\tau\right)=\left[V_{1}\left(\tau\right),V_{2}\left(\tau\right),\cdots,V_{M-1}\left(\tau\right)\right]^{T},\omega\left(\tau\right)=\left[\omega_{1}\left(\tau\right),\omega_{2}\left(\tau\right),\cdots,\omega_{M-1}\left(\tau\right)\right]^{T},\left(\tau\right)=\left[{}_{1}\left(\tau\right),0,\cdots,0,_{M-1}\left(\tau\right)\right]^{T}\nonumber \\
\end{equation}\begin{equation}
\text{\ \ \ \ ϰ}_{1}\left(\tau\right)=-\alpha V_{0}^{{}^{\prime}}\left(\tau\right)+\alpha\omega_{0}\left(\tau\right)+\left(\delta-v\right)V_{0}\left(\tau\right),_{M-1}\left(\tau\right)=-\gamma V_{M}^{{}^{\prime}}\left(\tau\right)+\gamma\omega_{M}\left(\tau\right)+\left(\delta+v\right)V_{M}\left(\tau\right).\nonumber \\
\end{equation}Let
\({\ \delta}_{v}^{-}=\delta-v,\delta_{v}^{+}=\delta+v\),
define
\(A=\par
\begin{pmatrix}\xi&\delta_{v}^{+}&0&0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&\delta_{v}^{-}&\xi&\delta_{v}^{+}\\
0&0&0&\delta_{v}^{-}&\xi\\
\end{pmatrix},B=\par
\begin{pmatrix}\beta&\gamma&0&0&0\\
\alpha&\beta&\gamma&0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&\alpha&\beta&\gamma\\
0&0&0&\alpha&\beta\\
\end{pmatrix}\).
Let
\begin{equation}
\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }e\left(\tau\right)=\left[e_{1},e_{2},\cdots,e_{M-1}\right]^{T},\nonumber \\
\end{equation}where\(e_{m}=2\sum_{j=1}^{\frac{M}{2}-1}{V_{2j}\left(\tau\right)}f_{\text{jm}}+4\sum_{j=1}^{\frac{M}{2}}{V_{2j-1}\left(\tau\right)}f_{2j-1,m}.\)
Let
\begin{equation}
{\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }f}_{j}=f\left(-\text{jh}\right)\nonumber \\
\end{equation}then\(\ e\left(\tau\right)\) can be written as
\(e\left(\tau\right)=\text{FGV}\left(\tau\right),\)
where\(\ G\) is a diagonal matrix and can be written
as\(\ G=\text{diag}\left[4,2,4,\cdots,2,4\right]\),\(\ F\)is Toeplitz matrix and can be written as
\(F=\par
\begin{pmatrix}f_{0}&f_{1}&f_{2}&\cdots&f_{M-3}&f_{M-2}\\
f_{-1}&f_{0}&f_{1}&f_{2}&\cdots&f_{M-3}\\
f_{-2}&f_{-1}&f_{0}&f_{1}&\cdots&\vdots\\
\vdots&\ddots&\ddots&\ddots&\ddots&f_{2}\\
f_{-\left(M-3\right)}&\cdots&f_{-2}&f_{-1}&f_{0}&f_{1}\\
f_{-\left(M-2\right)}&f_{-\left(M-3\right)}&\cdots&f_{-2}&f_{-1}&f_{0}\\
\end{pmatrix}\).
Let\(\text{\ ð}\left(\tau\right)=\left[\eth_{1}\left(\tau\right),\eth_{2}\left(\tau\right),\cdots,\eth_{M-1}\left(\tau\right)\right]^{T}\),
where\(\ \eth_{m}\left(\tau\right)=\frac{h}{3}\left(g_{0m}\left(\tau\right)+g_{\text{Mm}}\left(\tau\right)\mathcal{+g}\left(x_{m}\right)\right)\),
then\(\ \omega\left(\tau\right)\ \)can be written as\(\omega\left(\tau\right)=\lambda\left(\frac{h}{3}\text{FGV}\left(\tau\right)+\eth\left(\tau\right)\right).\)
Coefficient \(B\) is a nonsingular matrix, then
\(V^{{}^{\prime}}\left(\tau\right)=\text{CV}\left(\tau\right)+\theta\left(\tau\right),\text{τϵ}\left[0,T\right],\)(3.3)
where\(\text{\ C}=B^{-1}A+\frac{\text{λh}}{3}\text{FG},\theta\left(\tau\right)=\lambda\eth\left(\tau\right)+B^{-1}d\left(\tau\right)\).
The payoff function of shout options is not smooth, so in order to
achieve the fourth-order convergence rate, grid refinement technology is
used around the strike price.