## 2 Pricing Formula under the Time Fractional Vasicek Model  
In this section, we present some fundamental definitions for our exploration and the pricing formula of European put options under the Caputo time fractional Vasicek model, laying the foundation for the subsequent exploration and reconstruction of the volatility \(\sigma\) and the parameter \(\alpha\). **Definition 1** ([PERSON] [28]). : _Assume that the system state \(Y_{t}\) satisfies the stochastic differential equation_  
\[dY_{t}=a(Y_{t},t)dt+b(Y_{t},t)dW_{t},\]  
_where W(t) is a Wiener process (also called Brownian motion), then the conditional mathematical expectation \(u(y,t)=E(f(Y_{t}|Y_{t}=y))\) is the solution of the Cauchy problem (terminal value problem) of the following backward parabolic equation_  
\[\frac{\partial u}{\partial t}+a(y,t)\frac{\partial u}{\partial y}+\frac{1}{2} b^{2}(y,t)\frac{\partial^{2}u}{\partial y^{2}}=0.\]  
\[u(y,T)=f(y)\quad(y\in\mathbb{R}).\]  
**Definition 2** (Feynman-Kac [28]). : _Let \(v\) be the conditional mathematical expectation_  
\[v(y,t)=E\Big{(}f(Y_{t})e^{\int_{t}^{T}g(Y_{s},s)ds}|Y_{t}=y\Big{)}=E_{y,t} \Big{(}f(Y_{t})e^{\int_{t}^{T}g(Y_{s},s)ds}\Big{)},\]  
_then it is the solution of the terminal value problem of the following backward parabolic equation_  
\[\frac{\partial v}{\partial t}+a(y,t)\frac{\partial v}{\partial y}+\frac{1}{2} b^{2}(y,t)\frac{\partial^{2}v}{\partial y^{2}}+g(y,t)v=0,\quad(y\in\mathbb{R},t \in[0,T)),\]  
\[v(y,T)=f(y)\quad(y\in\mathbb{R}).\]  
**Definition 3**. : _The Caputo time fractional derivative is defined as follows_  
\[\prescript{c}{0}{D_{t}^{a}V(t)}= \begin{cases}\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t }\frac{V^{\prime}(t^{\prime})}{(t-t^{\prime})^{a}}dt^{\prime},\quad 0<\alpha<1,\\ V^{\prime}(t),\quad\alpha=1.\end{cases} \]  
The pricing problem of the European option under the Vasicek model is considered. Suppose the asset price \(S_{t}\) adheres to the following stochastic process  
\[dS_{t}=r_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{S} \tag{1}\]  
and the interest rate process \(r_{t}\) satisfies  
\[dr_{t}=(a+br_{t})dt+\omega dW_{t}^{r}, \tag{2}\]  
where \(\sigma_{t}\) is the volatility, \(W_{t}^{S}\) and \(W_{t}^{r}\) are two correlated Brown motions with \(Cov(dW_{t}^{s},dW_{t}^{r})=\rho dt\) and \(\rho\in(-1,1)\), and \(a,b,\omega\) are constants. Zero coupon bond is the carrier of interest rate, and we first consider the pricing of zero coupon bond. Let \(P_{t}=P(r,t)\) represent the value of zero-coupon bonds at time \(\mathbf{t}\). When \(r=r_{t}\) is a stochastic process, the value of zero coupon bonds \(P_{t}=P(r_{t},t)\) is  
\[P_{t}=E\Big{(}e^{-\int_{t}^{T}r(s)ds}|r(t)=r_{t}\Big{)}.\]  
Therefore, from the Kolmogorov theorem and the Feynman-Kac formula, we can obtain the backward parabolic equation that the price of the zero coupon bond satisfies under the Vasicek model  
\[\frac{\partial P}{\partial t}+\frac{\sigma^{2}}{2}\frac{\partial^{2}P}{ \partial r^{2}}+(a+br)\frac{\partial P}{\partial r}-rP=0,\quad(r\in\mathbb{ R},t\in[0,T)), \tag{3}\]  
\[P(r,T)=1.\]  
Next, the pricing formula of the European put option under the Vasicek model is derived. **Lemma 1**. : _Assuming that the stochastic interest rate \(r\) and the underlying asset price \(S\) satisfy the Formulas (1) and (2), then the price of the European put option \(V(r,S,t)\) with maturity \(T\) and strike \(K\) satisfies the following equation_  
\[\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}(t)S^{2}\frac{\partial^{2} V}{\partial S^{2}}+\rho\sigma(t)\omega S\frac{\partial^{2}V}{\partial S \partial r}+\frac{1}{2}\omega^{2}\frac{\partial^{2}V}{\partial r^{2}}+(a+br) \frac{\partial V}{\partial r}+rS\frac{\partial V}{\partial S}-rV=0, \tag{4}\]  
_with the condition_  
\[V(r,S,T)=(K-S)^{+}.\]  
Proof.