Let \(\varOmega ^{K}= \{ 0=t_{0}< t_{1}<\cdots <t_{K}=T \} \) and \(\varOmega ^{N}= \{ 0=x_{0}< x_{1}<\cdots <x_{N}=X \} \). An approximation to the time-fractional derivative on \(\varOmega ^{K}\) can be obtained by the quadrature formula,

$$\begin{aligned} \frac{\partial ^{\alpha }u(x_{i},t_{j})}{\partial t^{\alpha }} & = \frac{1}{ \varGamma (1-\alpha )}\sum _{k=1}^{j} \int _{t_{k-1}}^{t_{k}} (t_{j}-s ) ^{-\alpha }\frac{\partial u(x_{i},s)}{\partial s}\,{\mathrm{d}}s \\ & \approx \frac{1}{\varGamma (2-\alpha )}\sum_{k=1}^{j} \bigl[ (t _{j}-t_{k-1} )^{1-\alpha }- (t_{j}-t_{k} )^{1-\alpha } \bigr]D_{t}^{-}u_{i}^{k}, \end{aligned}$$

where \(D_{t}^{-}u_{i}^{k}=\frac{u^{k}_{i}-u_{i}^{k-1}}{\triangle t _{k}}\) with \(\triangle t_{k}=t_{k}-t_{k-1}\).

Since the Black–Scholes differential operator becomes a convection-dominated when the volatility or the asset price is small, a piecewise uniform mesh \(\varOmega ^{N}\) is constructed as that in [3] for the spatial discretization to ensure the stability:

$$\begin{aligned} x_{i}= \textstyle\begin{cases} h, & i=1, \\ h[1+\frac{\mu }{\beta ^{*}}(i-1)], & i=2,\ldots,N, \end{cases}\displaystyle \end{aligned}$$

(3.1)

where

$$\begin{aligned} h=\frac{X}{1+\frac{\mu }{\beta ^{*}}(N-1)}. \end{aligned}$$

Then the mesh sizes \(h_{i}=x_{i}-x_{i-1}\) satisfy

$$\begin{aligned} h_{i}=\textstyle\begin{cases} h, & i=1, \\ \frac{\mu }{\beta ^{*}} h, & i=2,\ldots,N. \end{cases}\displaystyle \end{aligned}$$

(3.2)

On the piecewise uniform mesh \(\varOmega ^{N}\) we apply a central difference scheme to approximate the spatial derivatives.

Hence, combining the time discretization scheme with spatial discretization scheme we can derive the fully discretized scheme on \(\varOmega ^{N\times K}\equiv \varOmega ^{N}\times \varOmega ^{K}\) as follows:

$$\begin{aligned} & L^{N,K}U_{i}^{j}=0,\quad 1\leq i< N,\ 1\leq j \leq K, \end{aligned}$$

(3.3)

$$\begin{aligned} & U_{i}^{0}=\max (x_{i}-E,0 ), \quad0\leq i \leq N, \end{aligned}$$

(3.4)

$$\begin{aligned} & U_{0}^{j}=0,\qquad U_{N}^{j}=X-E e^{-rt_{j}},\quad 0\leq j\leq K, \end{aligned}$$

(3.5)

where \(U_{i}^{j}\) is the approximation solution of \(u(x_{i},t_{j})\),

$$\begin{aligned} L^{N,K}U_{i}^{j} ={} & \frac{1}{\varGamma (2-\alpha )} \sum_{k=1}^{j} \bigl[ (t_{j}-t_{k-1} )^{1-\alpha }- (t_{j}-t_{k} ) ^{1-\alpha } \bigr]D_{t}^{-}U_{i}^{k} \\ &{} -\frac{1}{2} \bigl(\sigma ^{j} \bigr)^{2}x_{i}^{2} \delta _{x}^{2}U _{i}^{j}-r^{j}x_{i}D^{0}_{x}U_{i}^{j}+r^{j}U_{i}^{j}, \end{aligned}$$

(3.6)

and

$$\begin{aligned} &\delta _{x}^{2}U_{i}^{j}= \frac{2}{h_{i}+h_{i+1}} \biggl(\frac{U_{i+1} ^{j}-U_{i}^{j}}{h_{i+1}}-\frac{U_{i}^{j}-U_{i-1}^{j}}{h_{i}} \biggr),\\ & D_{x}^{0}U_{i}^{j}= \frac{U_{i+1}^{j}-U_{i-1}^{j}}{h_{i}+h_{i+1}},\qquad D_{t}^{-}U_{i}^{j}= \frac{U_{i}^{j}-U_{i}^{j-1}}{\triangle t_{j}}. \end{aligned}$$

Next we show that the matrix associated with the discrete operator \(L^{N,K}\) is an M-matrix. Hence the scheme is maximum-norm stable.

### Lemma 3.1

(Discrete maximum principle)

*The operator*
\(L^{N,K}\)*defined by* (3.6) *on the mesh*
\(\varOmega ^{N\times K}\)*satisfies a discrete maximum principle*, *i*.*e*. *if*
\(v^{j}_{i}\)*is a mesh function that satisfies*
\(v_{0}^{j}\geq 0,\ v^{j}_{N}\geq 0\ (0\leq j\leq K)\), \(v_{i}^{0}\geq 0\ (0\leq i \leq N) \)*and*
\(L^{N,K}v^{j}_{i}\geq 0 \ (1 \leq i< N,\ 0< j \leq K)\), *then*
\(v^{j}_{i}\geq 0\)*for all*
\(i,j\).

### Proof

Let

$$\begin{aligned} & a_{i,i-1}^{j}=-\frac{ (\sigma ^{j} )^{2}x_{i}^{2}}{ (h_{i}+h_{i+1} )h_{i}}+ \frac{r^{j}x_{i}}{h_{i}+h_{i+1}},\qquad a_{i,i+1}^{j}=-\frac{ (\sigma ^{j} )^{2}x_{i}^{2}}{ (h _{i}+h_{i+1} )h_{i+1}}- \frac{r^{j}x_{i}}{h_{i}+h_{i+1}}, \\ & a_{i,i}^{j}=\frac{ (\sigma ^{j} )^{2}x_{i}^{2}}{h_{i}h _{i+1}}+r^{j}+ \frac{ (\triangle t_{j} )^{-\alpha }}{\varGamma (2-\alpha )},\quad 1\leq i< N,\ 1\leq j\leq K, \end{aligned}$$

and

$$\begin{aligned} a_{i,i}^{k}={}&\frac{1}{\varGamma (2-\alpha )\triangle t_{k}} \bigl[ (t _{j}-t_{k-1} )^{1-\alpha }- (t_{j}-t_{k} )^{1-\alpha } \bigr] \\ &{} -\frac{1}{\varGamma (2-\alpha )\triangle t_{k+1}} \bigl[ (t_{j}-t _{k} )^{1-\alpha }- (t_{j}-t_{k+1} )^{1-\alpha } \bigr],\quad 1\leq k\leq j-1. \end{aligned}$$

By simple calculation we have

$$\begin{aligned} a_{i,i-1}^{j} & < -\frac{ (\sigma ^{j} )^{2}x_{1}x_{i}}{(h _{i}+h_{i+1})h_{i}} + \frac{r^{j}x_{i}}{h_{i}+h_{i+1}}= \frac{ [r ^{j}h_{i}- (\sigma ^{j} )^{2} x_{1} ]x_{i}}{(h_{i}+h _{i+1})h_{i}} \\ & = \frac{ [r^{j}\frac{\mu }{\beta ^{*}}- (\sigma ^{j} ) ^{2} ]hx_{i}}{(h_{i}+h_{i+1})h_{i}}\leq 0 \end{aligned}$$

for \(2\leq i< N\). It is easy to show

$$\begin{aligned} & a_{i,i+1}^{j}< 0,\qquad a_{i,i}^{j}>0,\quad 1 \leq i< N, \ 1\leq j\leq K, \\ & a_{i,i}^{k}=\frac{1}{\varGamma (1-\alpha )} \bigl[ (t_{j}-\xi _{k} )^{-\alpha }- (t_{j}- \xi _{k+1} )^{-\alpha } \bigr] \leq 0,\quad 1\leq k\leq j-1, \end{aligned}$$

and

$$\begin{aligned} & a_{1,1}^{j}+a_{1,2}^{j}>0,\quad 1\leq j\leq K, \\ & a_{i,i-1}^{j}+a_{i,i}^{j}+a_{i,i+1}^{j}+ \sum_{k=1}^{j-1}a_{i,i} ^{k}>0,\quad 1< i< N, \ 1\leq j\leq K, \\ & a_{N-1,N-1}^{j}+a_{N-1,N}^{j}>0,\quad 1\leq j\leq K, \end{aligned}$$

where \(\xi _{k}\in (t_{k-1},t_{k})\). Hence, it is easy to see that the matrix associated with \(L^{N,K}\) is a strictly diagonally dominant L-matrix, which means that it is an M-matrix. By applying the same argument as that in [14, Lemma 3.1], it is straightforward to obtain the result of our lemma. □

The next lemma gives us a useful formula for the truncation error.

### Lemma 3.2

*Let**U**be the solution of the difference scheme* (3.3)*–*(3.5) *and**u**be the exact solution of problem* (2.4)*–*(2.6). *Then we have the following truncation error estimates*:

$$\begin{aligned} \bigl\vert L^{N,K} \bigl(u_{i}^{j}-U_{i}^{j} \bigr) \bigr\vert \leq {}& C\max_{1\leq k \leq j} (\triangle t_{k} )^{1-\alpha } \int _{t_{k-1}}^{t _{k}} \biggl\vert \frac{\partial ^{2} u}{\partial t^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s \\ &{} +C (h_{i}+h_{i+1} ) \int _{x_{i-1}}^{x_{i+1}} \biggl(x _{i}^{2} \biggl\vert \frac{\partial ^{4}u}{\partial x^{4}}(y,t_{j}) \biggr\vert +x _{i} \biggl\vert \frac{\partial ^{3}u}{\partial x^{3}}(y,t_{j}) \biggr\vert \biggr) \,\mathrm{d}y \end{aligned}$$

*for*
\(1\leq i< N\)*and*
\(1\leq j\leq K\), *where**C**is a positive constant independent of the mesh*.

### Proof

It follows from (3.3) and (3.6) that

$$\begin{aligned} \bigl\vert L^{N,K} \bigl(u_{i}^{j}-U_{i}^{j} \bigr) \bigr\vert = {}& \bigl\vert L^{N,K}u_{i} ^{j}-Lu(x_{i},t_{j}) \bigr\vert \\ \leq {}& \frac{1}{\varGamma (1-\alpha )}\sum_{k=1}^{j} \biggl\vert \int _{t_{k-1}} ^{t_{k}} (t_{j}-s )^{-\alpha } \biggl[D_{t}^{-}u_{i}^{k}- \frac{ \partial u}{\partial s}(x_{i},s) \biggr]\,{\mathrm{d}}s \biggr\vert \\ &{} +\frac{1}{2} \bigl(\sigma ^{j} \bigr)^{2}x_{i}^{2} \biggl\vert \delta _{x}^{2}u_{i}^{j}- \frac{\partial ^{2}u}{\partial x^{2}}(x_{i},t_{j}) \biggr\vert +r ^{j}x_{i} \biggl\vert D_{x}^{0}u_{i}^{j}- \frac{\partial u}{\partial x}(x _{i},t_{j}) \biggr\vert . \end{aligned}$$

(3.7)

For \(k< j\) we use an integration by parts as that in [24] to obtain

$$\begin{aligned} & \int _{t_{k-1}}^{t_{k}} (t_{j}-s )^{-\alpha } \biggl[D _{t}^{-}u_{i}^{k}- \frac{\partial u}{\partial s}(x_{i},s) \biggr]\,{\mathrm{d}}s \\ & \quad=-\alpha \int _{t_{k-1}}^{t_{k}} (t_{j}-s )^{-\alpha -1} \bigl[ (s-t_{k-1} )D_{t}^{-}u_{i}^{k}- \bigl(u(x_{i},s)-u(x _{i},t_{k-1}) \bigr) \bigr]\,{\mathrm{d}}s \\ &\quad =-\alpha \bigl[ (\gamma _{1}-t_{k-1} )D_{t}^{-}u_{i} ^{k}- \bigl(u(x_{i},\gamma _{1})-u(x_{i},t_{k-1}) \bigr) \bigr] \int _{t_{k-1}}^{t_{k}} (t_{j}-s )^{-\alpha -1}\,{\mathrm{d}}s \\ &\quad =-\alpha (\gamma _{1}-t_{k-1} ) \biggl[ \frac{\partial u}{ \partial t}(x_{i},\gamma _{2})- \frac{\partial u}{\partial t}(x_{i}, \gamma _{3}) \biggr] \int _{t_{k-1}}^{t_{k}} (t_{j}-s )^{- \alpha -1}\,{\mathrm{d}}s, \end{aligned}$$

where we have used the mean value theorem with \(\gamma _{1},\gamma _{2}, \gamma _{3}\in (t_{k-1},t_{k} )\). Hence, applying a Taylor formula with the integral form of the remainder we can obtain

$$\begin{aligned} &\biggl\vert \int _{t_{k-1}}^{t_{k}} (t_{j}-s )^{-\alpha } \biggl[D _{t}^{-}u_{i}^{k}- \frac{\partial u}{\partial s}(x_{i},s) \biggr]\,{\mathrm{d}}s \biggr\vert \\ & \quad\leq C \triangle t_{k} \int _{t_{k-1}}^{t_{k}} \biggl\vert \frac{\partial ^{2} u}{\partial t^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s \int _{t_{k-1}} ^{t_{k}} (t_{j}-s )^{-\alpha -1}\,{\mathrm{d}}s \\ & \quad\leq C (\triangle t_{k} )^{1-\alpha } \int _{t_{k-1}} ^{t_{k}} \biggl\vert \frac{\partial ^{2} u}{\partial t^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s \end{aligned}$$

(3.8)

for \(k< j\). Similarly, we have

$$\begin{aligned} &\biggl\vert \int _{t_{j-1}}^{t_{j}} (t_{j}-s )^{-\alpha } \biggl[D _{t}^{-}u_{i}^{j}- \frac{\partial u}{\partial s}(x_{i},s) \biggr]\,{\mathrm{d}}s \biggr\vert \\ & \quad= \biggl\vert \frac{\partial u}{\partial t}(x_{i},\gamma _{4})-\frac{ \partial u}{\partial t}(x_{i},\gamma _{5}) \biggr\vert \int _{t_{j-1}}^{t _{j}} (t_{j}-s )^{-\alpha }\,{\mathrm{d}}s \\ & \quad\leq C (\triangle t_{j} )^{1-\alpha } \int _{t_{j-1}} ^{t_{j}} \biggl\vert \frac{\partial ^{2}u}{\partial t^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s, \end{aligned}$$

(3.9)

where we also have used the mean value theorem with \(\gamma _{4},\gamma _{5}\in (t_{j-1},t_{j} )\). By applying Taylor’s formulas about \(x_{i}\) we also have

$$\begin{aligned} \biggl\vert \delta _{x}^{2}u_{i}^{j}- \frac{\partial ^{2}u}{\partial x^{2}}(x _{i},t_{j}) \biggr\vert \leq C (h_{i}+h_{i+1} ) \int _{x_{i-1}} ^{x_{i+1}} \biggl\vert \frac{\partial ^{4}u}{\partial x^{4}}(y,t_{j}) \biggr\vert \,{\mathrm{d}}y \end{aligned}$$

(3.10)

and

$$\begin{aligned} \biggl\vert D_{x}^{0}u_{i}^{j}- \frac{\partial u}{\partial x}(x_{i},t_{j}) \biggr\vert \leq C (h_{i}+h_{i+1} ) \int _{x_{i-1}}^{x_{i+1}} \biggl\vert \frac{ \partial ^{3}u}{\partial x^{3}}(y,t_{j}) \biggr\vert \,{\mathrm{d}}y. \end{aligned}$$

(3.11)

Combining (3.7) with (3.8)–(3.11) we have

$$\begin{aligned} &\bigl\vert L^{N,K} \bigl(u_{i}^{j}-U_{i}^{j} \bigr) \bigr\vert \\ & \quad\leq C\max_{1\leq k \leq j} (\triangle t_{k} )^{1-\alpha } \int _{t_{k-1}}^{t _{k}} \biggl\vert \frac{\partial ^{2} u}{\partial t^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s \\ & \qquad{}+C (h_{i}+h_{i+1} ) \int _{x_{i-1}}^{x_{i+1}} \biggl(x _{i}^{2} \biggl\vert \frac{\partial ^{4}u}{\partial x^{4}}(y,t_{j}) \biggr\vert +x _{i} \biggl\vert \frac{\partial ^{3}u}{\partial x^{3}}(y,t_{j}) \biggr\vert \biggr) \,\mathrm{d}y. \end{aligned}$$

(3.12)

From this we complete the proof. □

Based on the properties of the European option [2, 3, 25] we assume that the solution *u* satisfies the following regularities:

$$\begin{aligned} \biggl\vert x^{2}\frac{\partial ^{4} u}{\partial x^{4}} \biggr\vert \leq C,\qquad \biggl\vert x\frac{\partial ^{3} u}{\partial x^{3}} \biggr\vert \leq C \quad\text{for } (x,t ) \in \varOmega. \end{aligned}$$

(3.13)

Then applying the maximum principle and the truncation error estimates we have the following bound.

### Theorem 3.3

*Let**U**be the solution of difference scheme* (3.3)*–*(3.5) *and**u**be the exact solution of problem* (2.4)*–*(2.6). *Then*, *under the assumption* (3.13) *we have the following bound*:

$$\begin{aligned} \Vert u-U \Vert _{\bar{\varOmega }^{N,K}} \leq C \max_{1\leq i\leq N, 1\leq j\leq K} ( \triangle t_{j} )^{1- \alpha } \int _{t_{j-1}}^{t_{j}} \biggl\vert \frac{\partial ^{2} u}{\partial t ^{2}}(x_{i},s) \biggr\vert \,{\mathrm{d}}s +CN^{-2}, \end{aligned}$$

(3.14)

*where**C**is a positive constant independent of the mesh*.