Keywords

1 Introduction

Asian options are path-dependent exotic options. The payoffs of Asian options depend on some sort of average of the underlying asset price over a specified time interval. The averaging can be arithmetic (mean of the asset prices) or geometric (exponential of the mean of the logarithm of the asset prices). The price of geometric average Asian option are analytically evaluated as the geometric average of the asset prices follows the lognormal distribution [9, 16] whereas the distribution of arithmetic average of the asset prices is not explicitly known. Hence the analytical solution of arithmetic average Asian option is not known. The averaging can be weighted or unweighted. These options are among the most popular derivative securities as they reduce the risk of market manipulation by big and influential traders of the market near the maturity time. Also the average-value options are always economical than the standard European options [16].

Asian option in a complete and arbitrage free financial market consisting of a risky asset, with asset price \(S_{\tau }\), as the underlying security and a risk free asset with interest rate r, follows the following stochastic differential equation

$$\begin{aligned} dS_{\tau }=(r-D)S_{\tau }d\tau +\sigma dW_{\tau }, \end{aligned}$$
(1)

where \(S_{\tau }\) is the asset price at time \(\tau \), r is the risk free interest rate, D is the dividend yield, \(\sigma \) is the market volatility and \(W_{\tau }\) is the standard Wiener process under the risk-neutral measure.

The payoff of European call option is

$$\begin{aligned} \max (S-K,0). \end{aligned}$$
(2)

The payoff of the “fixed strike” Asian call option is obtained by replacing \(S_{\tau }\) in the payoff of European call option by the average of asset prices, keeping the strike price K fixed and is given as

$$\begin{aligned} \max \left( \frac{1}{T}\int _0^{T} S_{z}dz-K,0\right) . \end{aligned}$$
(3)

Whereas the payoff of the “floating strike” Asian call option is obtained by replacing the strike price K in the payoff of European call option by the average of asset prices and given as

$$\begin{aligned} \max \left( S-\frac{1}{T}\int _0^{T} S_{z}dz,0\right) . \end{aligned}$$
(4)

The closed form solution of Asian option pricing model is not known.

Asian options were first explained in [13]. We consider the arithmetic average fixed strike Asian option. The existence, uniqueness and regularity of solution of arithmetic and geometric average Asian options was discussed in [2]. The partial differential equation for pricing Asian options consists of Black-Scholes equation along with an advection term, which can not be reduced to heat equation and its closed form solution is not known. Different approaches were used to solve various models for pricing Asian options such as applied statistics [19, 20, 26], applied probability [24, 25] and mathematical analysis [14, 32]. Various numerical schemes are also applied to approximate the option price. The work [3] described reducing the dimension of the partial differential equation governing the option price and then solving it using finite difference scheme. An explicit and an implicit finite difference methods were described in [21] for pricing Asian options. A hybrid finite difference along with Crank-Nicolson method was used in [6] to get a second order convergent scheme for evaluating the option price. A radial basis function based finite difference approximation for spacial operator along with \(\theta \)-method for temporal approximation was applied in [17] to obtain a second order convergent scheme for pricing the Asian option. In all these works where a numerical approximation was given for pricing option value, only non-dividend paying contract was considered. Hence after the dimension reducing transformation, the reaction term was not there in the partial differential equation to be solved numerically. Also the parameters \(\sigma \) and r were taken as constants. In the works [22, 23], these parameters along with dividend yield were taken to be functions of asset price and time variables and numerical solution of such generalized Black-Scholes model in dividend paying market was computed for different European style options. The generalizing of Black-Scholes model for pricing European call option by making the parameters \(r,~\sigma \) and D variables was also done in [4, 8, 15, 27, 30] and then the option price was numerically approximated. The other related works were [11, 29].

In the present work we have developed a numerical scheme for a generalized Asian option pricing model, transformed from two dimensional partial differential equation to one dimensional partial differential equation. Two step backward differentiation formula for temporal discretization and the High Order Difference approximation with Identity Expansion (HODIE) scheme for spacial discretization is applied simultaneously yielding second order accuracy in both time and space.

Rest of the paper is organized as follows: Sect. 2 introduces the Asian Option pricing model and the applied modifications for solving it numerically. In Sect. 3, the discrete scheme is presented. In Sect. 4, the convergence of the approximate solution is proved. Section 5 gives the numerical experiments based on our scheme and Sect. 6 concludes the paper.

2 Asian Option Pricing Model

The European style Asian arithmetic average fixed strike call option price \(V(S,A,\tau )\) is determined by [13, 31]

$$\begin{aligned} V(S,A,\tau )=\exp \left( -\int _{\tau }^T r(z)dz\right) v(S,A,\tau ), \end{aligned}$$
(5)

where the risk free interest rate \(r(\tau )\) is taken as function of time variable, T is the maturity time of the contract, A is the continuous arithmetic running sum defined as

$$\begin{aligned} A(\tau )=\int _0^{\tau }S(z)dz \end{aligned}$$
(6)

and \(v(S,A,\tau )\) is governed by the following partial differential equation

$$\begin{aligned} \frac{\partial v}{\partial \tau }+\frac{1}{2}\sigma ^2(\tau )S^2\frac{\partial ^2 v}{\partial S^2}+(r(\tau )-D(\tau ))S\frac{\partial v}{\partial S}+S\frac{\partial v}{\partial A}-r(\tau )v=0, \end{aligned}$$
(7)

along with the terminal condition

$$\begin{aligned} v(S,A,T)=\max \left( \frac{A}{T}-K,0 \right) , \end{aligned}$$
(8)

where \(\sigma (\tau )\) and \(D(\tau )\) are taken as functions of time, and K is the strike price. The current asset price S and the history of asset prices are independent. Hence the variables S, A and \(\tau \) are independent variables. The Eq. (7) is a two dimensional ultra-parabolic partial differential equation which is numerically expensive to approximate. Inspired from Rogers and Shi [24] and Alziary, Décamps and Koehl [1], we use the following change of variables to transform the problem (7)–(8) from final value problem to initial value problem and to reduce the dimension of the problem

$$\begin{aligned} \tilde{x}=\frac{K-A/T}{S},~\tau = T-t \text { and } v(S,A,\tau )=S\tilde{u}(\tilde{x},t), \end{aligned}$$
(9)

where \(\tilde{x}\) and t are the new space and time variables respectively. Introducing the new variables

$$\begin{aligned} \begin{array}{l} \displaystyle \hat{\sigma }(t) = \sigma (\tau ),\\ \displaystyle \hat{r}(t) = r(\tau ),\\ \displaystyle \hat{D}(t) = D(\tau ). \end{array} \end{aligned}$$

The transformed problem is as follows

(10)

along with the terminal condition

$$\begin{aligned} \tilde{u}(\tilde{x},0)=\max (-\tilde{x},0). \end{aligned}$$
(11)

The boundary conditions for the problem (7)–(8) are well established in [12]. The left boundary condition is

$$\begin{aligned} v(0,A,\tau )=\exp \left( -\int _{\tau }^T r(z)dz\right) \max \left( \frac{A}{T}-K,0 \right) . \end{aligned}$$
(12)

The value of Asian call and put options are interrelated by the put-call parity. A closed form solution for the Asian option price for the case \(A>KT\) is derived using the put-call parity [1, 10, 16], which is as follows

(13)

Hence we will solve the problem (7)–(8) only for the case \(A\le KT\). Applying the transformations (9) to the formula (13), we get

(14)

The left boundary condition of the problem (10)–(11) is obtained from (14)

(15)

The right boundary condition of the problem (10)–(11) for large \(\tilde{x}\) is obtained from (12)

$$\begin{aligned} \tilde{u}(\infty ,t)=0,~\text {for }A\le KT,~t\in [0,T]. \end{aligned}$$
(16)

Let us introduce another change of variable

$$\begin{aligned} x=\exp (-\tilde{x}), \end{aligned}$$
(17)

which converts the semi-infinite space domain \((0,\infty )\) into a finite space domain (0, 1) [1]. Accordingly the partial differential equation (10) is transformed as

(18)

and the initial and boundary conditions (11), (15) and (16) are transformed as

$$\begin{aligned} u(x,0)=0,~x\in [0,1], \end{aligned}$$
(19)
(20)

and

$$\begin{aligned} u(0,t)=0,~\text {for }A\le KT,~t\in [0,T] \end{aligned}$$
(21)

respectively. Let \(\varOmega _x=(0,1)\) be the space domain, \(\varOmega _t=(0,T)\) be the time domain and \(\varOmega = \varOmega _x \times \varOmega _t\). The final problem to be solved numerically is

$$\begin{aligned} Lu(x,t)\equiv \frac{\partial u}{\partial t}-a_2(x,t)\frac{\partial ^2 u}{\partial x^2}-a_1(x,t)\frac{\partial u}{\partial x}-a_0(t)u=f(x,t),~ (x,t)\in \varOmega , \end{aligned}$$
(22a)

where

$$\begin{aligned} \begin{array}{l} \displaystyle a_2(x,t) = \frac{1}{2}\hat{\sigma }^2(t)\left( \ln (x)\right) ^2x^2,\\ \displaystyle a_1(x,t) = \left( \frac{1}{2}\hat{\sigma }^2(t) \left( \ln (x)\right) ^2 +\frac{1}{T}-\left( \hat{r}(t)-\hat{D}(t) \right) \ln (x) \right) x,\\ \displaystyle a_0(t) = -\hat{D}(t),\\ \displaystyle f(x,t) = 0, \end{array} \end{aligned}$$

with the initial condition

$$\begin{aligned} u(x,0)=0,~x\in \bar{\varOmega }_x, \end{aligned}$$
(22b)

and the boundary conditions

$$\begin{aligned} u(0,t)=0,~t\in \bar{\varOmega }_t \end{aligned}$$
(22c)

and

$$\begin{aligned} u(1,t)=\frac{1}{T}\int _0^t \exp \left( -\left( \int _0^y\hat{r}(z)dz+\int _y^t\hat{D}(z)dz\right) \right) dy,~t\in \bar{\varOmega }_t. \end{aligned}$$
(22d)

In a financial market, it is reasonable to assume that \(\hat{r}\ge \hat{D}\) [28]. Also assume that

$$\begin{aligned} \left| \frac{\partial ^{m+n}u}{\partial x^m \partial t^n}(x,t)\right| \le \mathcal {C}~\text {on}~\bar{\varOmega };~0 \le n \le 3 \text { and } 0 \le m + n \le 4. \end{aligned}$$
(23)

Note that \(\mathcal {C}\) is a generic constant.

3 Discretization

Discretize the domain \(\bar{\varOmega }\) uniformly in both the space and time directions. Let the discrete space direction be \(\bar{\varOmega }_h=\lbrace x_m|m=0,1,...M\rbrace \) where M is the total number of grid intervals in space direction with uniform spacing \(h=x_m-x_{m-1},~m=1,2,...M\). Let the discrete time direction be \(\bar{\varOmega }^k=\lbrace t_n|n=0,1,...N\rbrace \) where N is the total number of grid intervals in time direction with uniform spacing \(k=t_n-t_{n-1},~n=1,2,...N\). Now the problem (22a–22d) is discretized simultaneously in both the space and time directions using two-step backward differentiation formula for temporal discretization and the High Order Difference approximation with Identity Expansion (HODIE) scheme with the stencil points \(\lbrace x_{m-1},x_{m},x_{m+1},~m=1,2,...M-1 \rbrace \) and two nodal auxiliary points \(\lbrace x_{m},x_{m+1},~m=1,2,...M-1 \rbrace \), for the spacial discretization. The discretization of the partial differential equation (22a) is as follows

(24)

where \(\alpha 's\) and \(\beta 's\) are the HODIE coefficients to be computed, \(U_m^n,~m=0,1,...M,~n=0,1,...N\) is the numerical approximation of the solution u(x, t) of the problem (22a–22d) at the grid point \((x_m,t_n),~m=0,1,...M,~n=0,1,...N\) and the operator \(\delta _t\) is defined as follows

$$\begin{aligned} \delta _t U_m^n = (U_m^n - U_m^{n-1})/k,~ n=1, \end{aligned}$$
$$\begin{aligned} \delta _t U_m^n = \left( \frac{3}{2}U_m^n - 2U_m^{n-1}+\frac{1}{2}U_m^{n-2}\right) /k,~ n=2,3,...,N, \end{aligned}$$

We make the Eq. (24) exact on \(P_3\), the space of polynomials with degree less than or equal to 3 and use the following normalization condition

$$\begin{aligned} \beta _{m,c}^n + \beta _{m,+}^n=1,~m=1,2,...M-1,~n=1,2,...,N, \end{aligned}$$
(25)

to compute the HODIE coefficients uniquely, which are as follows

$$\begin{aligned} \alpha _{m,-}^n = \frac{\beta _{m,c}^n(-2a_{2,m}^n + ha_{1,m}^n) + \beta _{m,+}^n(-2a_{2,m+1}^n - ha_{1,m+1}^n)}{2h^2}, \end{aligned}$$
(26)
$$\begin{aligned} \alpha _{m,+}^n = \frac{\beta _{m,c}^n(-2a_{2,m}^n - ha_{1,m}^n) + \beta _{m,+}^n(-2a_{2,m+1}^n - 3ha_{1,m+1}^n - 2h^2a_{0}^n)}{2h^2}, \end{aligned}$$
(27)
$$\begin{aligned} \alpha _{m,c}^n = \frac{\beta _{m,c}^n(4a_{2,m}^n - 2h^2a_{0}^n) + \beta _{m,+}^n(4a_{2,m+1}^n + 4ha_{1,m+1}^n)}{2h^2}, \end{aligned}$$
(28)
$$\begin{aligned} \beta _{m,c}^n = \frac{6ha_{2,m+1}^n+2h^2a_{1,m+1}^n}{6ha_{2,m+1}^n+2h^2a_{1,m+1}^n+h^2a_{1,m}^n} \end{aligned}$$
(29)

and

$$\begin{aligned} \beta _{m,+}^n = \frac{h^2a_{1,m}^n}{6ha_{2,m+1}^n+2h^2a_{1,m+1}^n+h^2a_{1,m}^n}. \end{aligned}$$
(30)

Now using these coefficients, the final scheme to compute the numerical approximation to the solution of the problem (22a–22d) is as follows

(31a)
(31b)
$$\begin{aligned} U_m^0=0,~m=0,1,...,M, \end{aligned}$$
(31c)
$$\begin{aligned} U_0^n=0,~n=0,1,...N. \end{aligned}$$
(31d)
$$\begin{aligned} U_M^n=\frac{1}{T}\int _0^{t_n} \exp \left( -\left( \int _0^y\hat{r}(z)dz+\int _y^{t_n}\hat{D}(z)dz\right) \right) dy,~n=0,1,...N, \end{aligned}$$
(31e)

Now putting \(m=1,2,...M\) in (31a) and (31b) and shifting the boundary components to the right side of the equation, we can write the scheme in the matrix form as

$$\begin{aligned} A^nU^n=b^n,\quad n=1,2,...,N, \end{aligned}$$
(32)

where \(U^n=[U_1^n,U_2^n,...,U_{M-1}^n]^T\).

4 Error Analysis

Lemma 1

Assume that

(33)

and

$$\begin{aligned} \beta ^n_{m,+}\left( -a^n_{0}+\frac{3}{2k}\right) \le \frac{2(\beta ^n_{m,c}a^n_{2,m}+\beta ^n_{m,+}a^n_{2,m+1})+h(\beta ^n_{m,c}a^n_{1,m}+3\beta ^n_{m,+}a^n_{1,m+1})}{2h^2},\\~m=1,2,...,M-1,~n=2,3,...,N. \end{aligned}$$
(34)

then

$$\begin{aligned} \alpha _{m,-}^n&\le 0,~m=1,2,...,M-1,~n=1,2,...,N, \end{aligned}$$
(35)
$$\begin{aligned} \alpha _{m,+}^n + \frac{3}{2k} \beta _{m,+}^n&\le 0,~m=1,2,...,M-1,~n=2,3,...,N. \end{aligned}$$
(36)

Proof

The inequalities (35) and (36) can be easily obtained from (26), (27), (33) and (34).

Remark 1

Similarly assume that

$$\begin{aligned} \beta ^n_{m,+}\left( -a^n_{0}+\frac{1}{k}\right) \le \frac{2(\beta ^n_{m,c}a^n_{2,m}+\beta ^n_{m,+}a^n_{2,m+1})+h(\beta ^n_{m,c}a^n_{1,m}+3\beta ^n_{m,+}a^n_{1,m+1})}{2h^2},\\~m=1,2,...,M-1,~n=1, \end{aligned}$$
(37)

then

$$\begin{aligned} \alpha _{m,+}^n + \frac{1}{k} \beta _{m,+}^n \le 0,~m=1,2,...,M-1,~n=1. \end{aligned}$$
(38)

Lemma 2

(Discrete Maximum Principle). Under the assumptions of the Lemma 1, the operator \(L_h^k\) defined by (31a)–(31b) satisfies discrete maximum principle, that is, if \({v_m^n}\) and \({w_m^n}\) are mesh functions that satisfy \(v_0^n \le w_0^n,~ v_M^n \le w_M^n~ (n=0,1,...,N),~v_m^0 \le w_m^0,~(m=0,1,...,M)\) and \(L_h^kv_m^n \le L_h^kw_m^n ~(m=1,2,...,M-1,~n=1,2,...,N)\), then \(v_m^n \le w_m^n\) for all m, n.

Proof

The row sum of the matrices \(A^n,n=1,2,...N\), are

$$\begin{aligned} \alpha _{m,-}^n + \left( \alpha _{m,c}^n+\frac{3}{2k}\beta _{m,c}^n \right) + \left( \alpha _{m,+}^n+\frac{3}{2k}\beta _{m,+}^n \right)&= -a_0^n + \frac{3}{2k}\end{aligned}$$
(39)
$$\begin{aligned}&> 0, \end{aligned}$$
(40)

\(m=1,2,...,M-1,~n=1,2,...,N\). \(\alpha _{m,-}^n,~m=1,2,...,M-1,~n=1,2,...,N\) are the sub-diagonal elements and \(\alpha _{m,+}^n+\frac{3}{2k}\beta _{m,+}^n,~m=1,2,...,M-1,~n=1,2,...,N\) are the super-diagonal elements of the tridiagonal matrices \(A^n,n=1,2,...N\). Under the hypothesis of the Lemma 1, the inequalities (35)–(36) and (39) show that the coefficient matrices corresponding to the discrete operator \(L_h^k\) are irreducible M-matrices which preserve the positivity. Hence the solution to the linear system of equations (32) exists and if \({v_m^n}\) and \({w_m^n}\) are mesh functions as defined in Lemma, then \(v_m^n \le w_m^n\) for all m, n.

The operator \(L_h^k\), satisfying the discrete maximum principle, establishes the stability of the scheme.

Theorem 1

Let u be the solution of the continuous problem (22a–22d) and \(U_m^n\) be the solution of the discrete problem (31a–31e). Then under the assumptions (23),

$$\begin{aligned} \left| u_m^n-U_m^n\right| \le \mathcal {C}(k^2 + h^2),~~m=0,1,...,M,~n=0,1,...,N, \end{aligned}$$
(41)

where \(\mathcal {C}\) is a positive constant independent of k and h.

Proof

From (24) we have

(42)

Applying the Taylor’s expansion in two variables and taking modulus, we have

$$\begin{aligned} \left| L_h^k(u_m^n - U_m^n)\right|&= \left| L_h^k u_m^n - (Lu)_m^n\right| \nonumber \\&\le \mathcal {C}_1 k^2 \int _0^1 \left[ \left| \frac{\partial ^3 u}{\partial t^3}(x_m,t_n-ky)\right| + \left| \frac{\partial ^3 u}{\partial t^3}(x_m,t_n-2ky)\right| \right. \nonumber \\&\qquad \left. {} + \left| \frac{\partial ^3 u}{\partial t^3}(x_m+hy,t_n-ky)\right| +\left| \frac{\partial ^3 u}{\partial t^3}(x_m+hy,t_n-2ky)\right| \right] dy \nonumber \\&\ + \mathcal {C}_2 h^2 \int _0^1 \left[ \left| \frac{\partial ^3 u}{\partial x^2 \partial t}(x_m+hy,t_n-ky)\right| + \left| \frac{\partial ^3 u}{\partial x^2 \partial t}(x_m+hy,t_n-2ky)\right| \right. \nonumber \\&\qquad \left. {} + \left| \frac{\partial ^3 u}{\partial x^2 \partial t}(x_m+hy,t_n)\right| + x_{m+1}^2 \left| \frac{\partial ^4 u}{\partial x^4}(x_m+hy,t_n)\right| \right] dy \nonumber \\&\le \mathcal {C}_3(k^2 + h^2),~m=1,2,...,M-1,~ n=2,3,...,N. \end{aligned}$$
(43)

For \(n=1\), backward Euler method is used for the computation of the discrete solution, and the one-step error of the backward Euler method is \(O(k^2)\) [18]. Hence we have

$$\begin{aligned} \left| L_h^k(u_m^n - U_m^n)\right| \le \mathcal {C}_4(k^2 + h^2),~m=1,2,...,M-1,~ n=0,1,...,N. \end{aligned}$$
(44)

We construct the following barrier function

$$\begin{aligned} \varphi _m^n = \mathcal {C}(k^2+h^2)(1+t_n)\pm (u_m^n-U_m^n), \end{aligned}$$

and apply the Lemma 2 on \(\varphi _m^n\), to show

$$\begin{aligned} |u_m^n-U_m^n| \le \mathcal {C}(k^2+h^2) ~ \text {for every}~m=1,2,...,M-1,~n=1,2,...N. \end{aligned}$$
(45)

5 Numerical Experiments

Since the closed form solution of the considered Asian option pricing model is not known, we use double mesh principle to find the maximum error (\(E_{max}\)), the root mean square error (\(E_{rms}\)) and the corresponding order of convergence \(p_{max}\) and \(p_{rms}\) as follows

$$\begin{aligned} \begin{array}{l} \displaystyle E_{\text {max}}^{M,N} = \displaystyle \max _{0\le m\le M} \left| U^{M,N}(x_m,t_N)-U^{2M,2N}(x_{2m},t_{2N}) \right| ,\\ \\ \displaystyle E_{\text {rms}}^{M,N} = \sqrt{\frac{\displaystyle \sum _{m=0}^M\left[ \left( U^{M,N}(x_m,t_N) - U^{2M,2N}(x_{2m},t_{2N})\right) ^2 \right] }{M+1}},\\ \\ \displaystyle p_{\text {max}}^{M,N} = \log _2\left( \frac{E_{\text {max}}^{M,N}}{E_{\text {max}}^{2M,2N}} \right) ~\text {and}~p_{\text {rms}}^{M,N} = \log _2\left( \frac{E_{\text {rms}}^{M,N}}{E_{\text {rms}}^{2M,2N}} \right) . \\ \end{array} \end{aligned}$$

Example 1

Consider the initial boundary value problem (22a–22d) with \(\hat{\sigma }=0.5\), \(\hat{r}=0.09\), \(\hat{D}=0\). Take \(T=3\) and \(K=40\). The computed solution is shown in Fig. 1 and the convergence results are given in Table 1.

Example 2

Consider the initial boundary value problem (22a–22d) with \(\hat{\sigma }(t)=0.4(2+(T-t))\), \(\hat{r}(t)=0.06(1+t)\), \(\hat{D}(t)=0.02\exp (-t)\). Take \(T=1\) and \(K=40\). The computed solution is shown in Fig. 2 and the convergence results are given in Table 2.

Example 3

Consider the initial boundary value problem (22a–22d) with \(\hat{\sigma }(t)=0.4(2+\sin (T-t))\), \(\hat{r}(t)=0.06\exp (t)\), \(\hat{D}(t)=0.02\sin (t)\). Take \(T=1\) and \(K=40\). The computed solution is shown in Fig. 3 and the convergence results are given in Table 3.

Table 1. Maximum absolute error (\({E}_{\text {max}}\)), root mean square error (\({E}_{\text {rms}}\)) and corresponding orders of convergence \({p}_{\text {max}}\) and \({p}_{\text {rms}}\) for Example 1
Full size table
Table 2. Maximum absolute error (\({E}_{\text {max}}\)), root mean square error (\({E}_{\text {rms}}\)) and corresponding orders of convergence \({p}_{\text {max}}\) and \({p}_{\text {rms}}\) for Example 2
Full size table
Table 3. Maximum absolute error (\({E}_{\text {max}}\)), root mean square error (\({E}_{\text {rms}}\)) and corresponding orders of convergence \({p}_{\text {max}}\) and \({p}_{\text {rms}}\) for Example 3
Full size table
Fig. 1.
figure 1

Computed solution of Asian option price for Example 1

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Fig. 2.
figure 2

Computed solution of Asian option price for Example 2

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Fig. 3.
figure 3

Computed solution of Asian option price for Example 3

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6 Conclusions

In the present work, HODIE scheme with two auxiliary points is used to achieve second order accuracy in space and two-step backward differentiation formula is applied for temporal approximation yielding second order accuracy in time also. The main features of our scheme is that it easily deals with variable coefficients occurring in the partial differential equation as we have taken volatility (\(\sigma \)), risk-free interest rate (r) and dividend yield (D) not as constants but as function of the time variable, which is more likely to happen in real financial market. Also this scheme can be applied simultaneously in space and time directions which leads to simpler analysis of the convergence of the solution [7]. It deals with the degeneracy issue of this problem easily without any extra efforts unlike in other works done in this field [5, 6, 33]. The numerical results are in accordance with the theoretical results.