## 2 Pricing Formula under the Time Fractional Vasicek Model
: Consider a portfolio \(\Pi\), which consists of \((-\Delta_{1})\) units of the underlying asset \(S\), \((-\Delta_{2})\) units of the zero coupon bond \(P(r,t)\), and one option \(V(S,r,t)\), then the value of the portfolio at time \(t\) is  
\[\Pi=V-\Delta_{1}S-\Delta_{2}P.\]  
Applying \(It\hat{\sigma}\)'s Lemma, we obtain  
\[d\Pi =\bigg{(}\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}(t)S ^{2}\frac{\partial^{2}V}{\partial S^{2}}+\frac{1}{2}\omega^{2}\frac{\partial ^{2}V}{\partial r^{2}}+S\sigma(t)\omega\rho\frac{\partial^{2}V}{\partial S \partial r}\bigg{)}dt+\bigg{(}\frac{\partial V}{\partial S}-\Delta_{1} \bigg{)}dS\] \[+\bigg{(}\frac{\partial V}{\partial r}-\Delta_{2}\frac{\partial P }{\partial r}\bigg{)}dr+\Delta_{2}\bigg{(}\frac{\partial P}{\partial t}+ \frac{1}{2}\sigma^{2}\frac{\partial^{2}P}{\partial r^{2}}\bigg{)}dt. \tag{5}\]To eliminate risk in this portfolio, we take  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]0  
substitute it into (5), then combine Formula (3) to obtain  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]1  
The no-arbitrage principle gives us  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]2  
then  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]3  
To ensure the completeness of the narrative, we first introduce the direct problem. Replacing the term in Equation (4) that involves the integer-order derivative with the Caputo fractional-order derivative, denoted as \(\frac{\partial^{4}V}{\partial t^{2}}\), where order \(\alpha\in(0,1)\), we have  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]4  
the fractional order derivative \(\frac{\partial^{\alpha}V}{\partial t^{\alpha}}\) is defined as  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]5  
Then, we invert the time variable as \(\tau:=T-t\) and the relationship between the Caputo derivative \({}_{0}^{c}D_{\tau}^{\alpha}V\) and the time fractional derivative \(\frac{\partial^{4}V}{\partial t^{\alpha}}\) given by article [29], that is, \(\frac{\partial^{2}V}{\partial t^{\alpha}}=-_{0}^{c}D_{\tau}^{\alpha}V\). In this way, we obtain the pricing formula of European put options under the [PERSON] time fractional Vasicek model  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]6 \[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]7  
with the conditions  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]8  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]9  
\[P_{t}=E\Big{(}e^{-\int_{t}^{T}r(s)ds}|r(t)=r_{t}\Big{)}.\]0  
Direct problem: For the given initial boundary problem, the price of the European put option can be precisely computed as long as the volatility and parameter \(\alpha\) are known. Subsequently, we can use the implicit finite difference method to numerically address this problem. Inverse problem: Let \(V_{ij}\) denote the market price of an option with strike price \(K_{ij}(j=1,2,\ldots,M_{i})\) and expiration date \(T_{i}(i=1,2,\ldots,N)\), where \(K_{1}\leq K_{2}\leq\cdots\leq K_{M_{i}}\) and \(T_{1}\leq T_{2}\leq\cdots\leq T_{N}\). The inverse problem is a calibration problem, which can be formulated as finding a time-dependent volatility function and parameter \(\alpha\) such that the predictedprice \(V(\tau_{0},r_{0},S_{0},T_{i},K_{ij},\alpha,\sigma(\tau))\) falls between the corresponding bid price \(V^{b}_{ij}\) and ask price \(V^{a}_{ij}\), that is  
\[P_{t}=E\Big{(}e^{-\int_{t}^{T}r(s)ds}|r(t)=r_{t}\Big{)}.\]1  
From the above formula, the calibration problem is transformed to solve such an optimization problem:  
Minimize the following error function associated with the given set of market option prices  
\[P_{t}=E\Big{(}e^{-\int_{t}^{T}r(s)ds}|r(t)=r_{t}\Big{)}.\]2  
where \(\hat{V}_{ij}=(V^{a}_{ij}+V^{b}_{ij})/2\) is the mean of the bid and ask prices. The calibration problem can be formulated as finding the parameters that minimize the error function \(\Gamma(\alpha,\sigma)\).