## 4 Numerical Experiments  
### Numerical Simulation  
In this part, we study the effectiveness of the reconstruction algorithm for two examples where the volatility \(\sigma(t)\) and the fractional order \(\alpha\) are known. In our numerical experiments, we utilized the parameters mentioned in [32] for consistency; let \(K=[10,12,14,16]\), \(T=[0.25,0.5,0.75,1]\), \([S_{min},S_{max}]=[0.25,40]\), \([r_{min},r_{max}]=[0.002,1.2]\), the initial value \(S_{0}=8\), \(r_{0}=0.25\), and other parameters  
\[a=0.001,\,\omega=0.3,\,b=-0.2,\,\rho=0.4.\]  
**Example 1**.: _Considering volatility function \(\sigma(\tau)=0.2-0.1\log(1.5+3\tau),\alpha=0.7\), the Lagrangian multiplier and the update step size of \(\mu_{ij}\) are respectively taken as \(\lambda=5,\,\,\mu_{ij}^{0}=0.3,\,\,\beta=5\)._  
We will use the European put option price calculated using the exact volatility \(\sigma(\tau)\) and parameter \(\alpha\) as the market price of the option. Stop iterating when \(|\sigma_{d}(\tau)-\sigma_{d-1}(\tau)|<1\times 10^{-6}\), where \(\sigma_{d}(\tau)\) represents the volatility generated by the \(d\)-th iteration, and the total number of iterations is not greater than 100. Figure 1 presents a graph of exact volatility versus reconstructed volatility, and the recover value of parameter \(\alpha\) is 0.7. We add noise with intensity \(\delta\) of 0.01, 0.03, and 0.05 to the exact volatility to verify the stability  
\[\sigma^{\delta}=0.2-0.1\log(1.5+3\tau)+\delta\times rand(-1,1).\]  
We first use perturbed volatility to calculate option prices and then use the resulting option prices to reconstruct fractional \(\alpha\) and volatility \(\sigma(\tau)\). Figure 2 shows the comparison of real volatility and reconstructed volatility under different noise intensities. Table 1  
Figure 1: Exact volatility and reconstructed volatility for Example 1.  
gives the root mean square error (RMSE), the maximum volatility error, the reconstructed fractional \(\alpha\), and the maximum option price error, where RMSE can be calculated by  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\sigma_{i}-\sigma_{i})^{2}}.\]  
**Example 2**.: _In this example, we examine the volatility function represented as \(\sigma(\tau)=0.5(\tau-0.5)^{2}+0.1\), \(\alpha=0.37\); the Lagrangian multiplier and the update step size of \(\mu_{ij}\) are respectively taken as_  
\[\lambda=1.5,\,\mu_{ij}^{0}=0.2,\,\beta=1,\]  
_the volatility curve exhibits a smile-shaped pattern._  
Figure 3 shows an image of exact volatility versus reconstructed volatility. We can obtain the precise volatility and reconstructed volatility under different noise intensities from Figure 4. Finally, Table 2 shows the RMSE, the absolute error of volatility, and the reconstructed fractional \(\alpha\) under different noise intensities.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\mathbf{max}\,|\sigma_{d}-\sigma_{d-1}|\) & **RMSE** & \(\mathbf{max}\,|\sigma-\hat{\sigma}|\) & \(\boldsymbol{\alpha}\) & \(\mathbf{max}\,|V-\hat{V}|\) \\ \hline \(\delta=0\) & \(9.9770\times 10^{-7}\) & \(0.0030\) & \(4.621\times 10^{-3}\) & \(0.364\) & \(2.855\times 10^{-3}\) \\ \(\delta=0.01\) & \(9.9975\times 10^{-7}\) & \(0.0044\) & \(7.820\times 10^{-3}\) & \(0.367\) & \(5.162\times 10^{-3}\) \\ \(\delta=0.03\) & \(9.9999\times 10^{-7}\) & \(0.0073\) & \(1.9694\times 10^{-2}\) & \(0.372\) & \(1.9478\times 10^{-2}\) \\ \(\delta=0.05\) & \(4.8906\times 10^{-7}\) & \(0.0156\) & \(3.2824\times 10^{-2}\) & \(0.359\) & \(3.4620\times 10^{-2}\) \\ \hline \end{tabular}
\end{table}
Table 2: Numerical results under different intensity disturbances.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\mathbf{max}\,|\sigma_{d}-\sigma_{d-1}|\) & **RMSE** & \(\mathbf{max}\,|\sigma-\hat{\sigma}|\) & \(\boldsymbol{\alpha}\) & \(\mathbf{max}\,|V-\hat{V}|\) \\ \hline \(\delta=0\) & \(9.9770\times 10^{-7}\) & \(0.0030\) & \(4.621\times 10^{-3}\) & \(0.364\) & \(2.855\times 10^{-3}\) \\ \(\delta=0.01\) & \(9.9975\times 10^{-7}\) & \(0.00