Introduction

Most of the phenomena that arise in the real world are described by nonlinear differential equations (ordinary and partial) and in some case by nonlinear integral equations. However, most of the methods developed so far in mathematics are valid only for solving linear differential equations. The decomposition method as developed by the mathematician George Adomian (1922–1996) has been very useful in applied mathematics, the applied sciences and engineering [32, 33]. The Adomian decomposition method (ADM) has the advantage that its sequence of approximate solutions converges to the exact solution in the vast majority of important cases for very few prerequisites which are naturally satisfied when modeling physical phenomena, and applications can be easily handled for a wide class of ordinary and partial differential equations encompassing both linear and nonlinear cases [18]. In this paper, we will use the ADM specifically to solve the nonlinear Black-Scholes equation that arises when considering the market model in which volatility is not constant as a result of considering liquidity and transaction costs as set out in the work cited in [7, 9, 11, 22]. We will see that the nonlinear term can be easily handled with the help of the Adomian polynomials.

In “Analysis of the Method” section, we present, in a brief and self-contained way, the ADM and several references are presented to delve deeper into the subject and to study its mathematical foundation that is beyond the scope of this present work. In “The Black-Scholes Equation (The Standard Case)” section, the Black-Scholes model is addressed in its standard linear version; by contrast, in “A Nonlinear Case” section, we show how the ADM can be implemented in order to solve a nonlinear model involving a modified volatility due to the market illiquidity. We also show, through two numerical examples, the efficacy and accuracy of the method by contrasting the results with the exact solution found indirectly in [20]. Finally, in “Conclusion and Summary” section, we present our conclusions.

Analysis of the Method

The ADM is a technique to solve ordinary and nonlinear differential equations. Using this method, it is possible to express analytic solutions in terms of a rapdily converging series [4]. In a nutshell, the method identifies and separates the linear and nonlinear parts of a differential equation. By inverting and applying the highest order differential operator that is contained in the linear part of the equation, it is possible to express the solution in terms of the rest of the equation affected by the inverse operator. At this point, the solution is proposed by means of a decomposition series with terms that will be determined by recursion and that gives rise to the solution components [34]. The nonlinear part is expressed in terms of the Adomian polynomials. The initial or the boundary conditions and the terms that contain the independent variables will be considered as the initial approximation. In this way and by means of a recurrence relation, it is possible to calculate the terms of the series by recursion that give the approximate solution of the differential equation. A reference to the topic of the general progress of the Adomian decomposition method can be found in [29].

Given a partial (or ordinary) differential equation

$$\begin{aligned} Fu(x,t)=g(x,t) \end{aligned}$$
(1)

with the initial condition

$$\begin{aligned} u(x,0)=f(x), \end{aligned}$$
(2)

where F is a differential operator that could itself, in general, be nonlinear and therefore includes linear and non-linear terms.

In general, Eq. (1) is be written as

$$\begin{aligned} L_{t}u(x,t)+Ru(x,t)+Nu(x,t)=g(x,t) \end{aligned}$$
(3)

where \(L_{t}=\frac{\partial }{\partial t}\), R is the linear remainder operator that could include partial derivatives with respect to x, N is a nonlinear operator which is presumed to be analytic and g is a non-homogeneous term that is independent of the solution u.

Solving for \(L_{t}u(x,t)\), we have

$$\begin{aligned} L_{t}u(x,t)= g(x,t)-Ru(x,t)-Nu(x,t). \end{aligned}$$
(4)

As L is presumed to be invertible, we can apply \(L_{t}^{-1}(\cdot )=\int _{0}^{t}(\cdot )dr\) to both sides of Eq. (4), obtaining

$$\begin{aligned} L_{t}^{-1}L_{t}u(x,t)= L_{t}^{-1}g(x,t)-L_{t}^{-1}Ru(x,t)-L_{t}^{-1}Nu(x,t). \end{aligned}$$
(5)

An equivalent expression to (5) is

$$\begin{aligned} u(x,t)= f(x)+L_{t}^{-1}g(x,t)-L_{t}^{-1}Ru(x,t)-L_{t}^{-1}Nu(x,t). \end{aligned}$$
(6)

where f(x) is the constant of integration with respect to t that satisfies \(L_{t}f=0\). In equations where the initial value \(t = t_0\), we can conveniently define \(L^{-1}\).

The ADM proposes a decomposition series solution u(x, t) given as

$$\begin{aligned} u(x,t)= \sum _{n=0}^{\infty }u_{n}(x,t). \end{aligned}$$
(7)

The nonlinear term Nu(x, t) is given as

$$\begin{aligned} Nu(x,t)= \sum _{n=0}^{\infty }A_{n}(u_{0},u_{1},\ldots , u_{n}) \end{aligned}$$
(8)

where \(\{A_{n}\}_{n=0}^{\infty }\) is the Adomian polynomials sequence established in [6, 34].

The Adomian polynomials have been studied in a formal manner in [16].

Substituting (7) and (8) into Eq. (6), we obtain

$$\begin{aligned} \sum _{n=0}^{\infty }u_{n}(x,t)= f(x)+L_{t}^{-1}g(x,t)-L_{t}^{-1}R\sum _{n=0}^{\infty }u_{n}(x,t)-L_{t}^{-1}\sum _{n=0}^{\infty }A_{n}(u_{0},u_{1},\ldots , u_{n}),\nonumber \\ \end{aligned}$$
(9)

with \(u_0\) identified as \(f(x)+L_{t}^{-1}g(x,t)\), and therefore, we can write

$$\begin{aligned} u_{0}(x,t)= & {} f(x)+L_{t}^{-1}g(x,t),\\ u_{1}(x,t)= & {} -L_{t}^{-1}Ru_{0}(x,t)-L_{t}^{-1}A_{0}(u_{0}),\\&\vdots \\ u_{n+1}(x,t)= & {} -L_{t}^{-1}Ru_{n}(x,t)-L_{t}^{-1}A_{n}(u_{0},\ldots ,u_{n}). \end{aligned}$$

From which we can establish the following recurrence relation, that is obtained in a explicit way for instance in reference [35],

$$\begin{aligned} \left\{ \begin{array}{ll} u_{0}(x,t)=f(x)+L_{t}^{-1}g(x,t),\\ u_{n+1}(x,t)=-L_{t}^{-1}Ru_{n}(x,t)-L_{t}^{-1}A_{n}(u_{0},u_{1},\ldots , u_{n}),\quad n=0,1,2,\ldots . \end{array} \right. \end{aligned}$$
(10)

Using (10), we can obtain an approximate solution of (1), (2) as

$$\begin{aligned} u(x,t)\approx \sum _{n=0}^{k}u_{n}(x,t),\quad \text{ where } \; \lim _{k\rightarrow \infty }\sum _{n=0}^{k}u_{n}(x,t)=u(x,t). \end{aligned}$$
(11)

This method has been successfully applied to a large class of both linear and nonlinear problems [17]. The Adomian decomposition method requires far less work in comparison with traditional methods [1]. This method considerably decreases the volume of calculations. The decomposition procedure of Adomian easily obtains the solution without linearizing the problem by implementing the decomposition method rather than the standard methods. In this approach, the solution is found in the form of a convergent series with easily computed components; in many cases, the convergence of this series is extremely fast and consequently only a few terms are needed in order to have an idea of how the solutions behave. Convergence conditions of this series have been investigated by several authors, e.g., [2, 3, 12, 13].

The Black-Scholes Equation (the Standard Case)

This model is a partial differential equation whose solution describes the value of an European Option, see [8, 28]. Nowadays, it is widely used to estimate the pricing of options other than the European ones. For an European call or put on an underlying stock paying no dividends, the equation is:

$$\begin{aligned} V_\tau (S,\tau )+\frac{1}{2}\sigma ^{2}S^{2}V_{SS}(S,\tau )+rSV_S(S,\tau )-rV(S,\tau )= 0, \, \end{aligned}$$
(12)

where V is the price of the option as a function of underlying price S and time \(\tau \); with \(0\le \tau < T,\) and \(S\ge 0;\) here the risk-free interest rate r and the volatility \(\sigma \) are assumed to be constant.

There are many varieties of options. European options may only be exercised on the maturity date. American options may be exercised any time up to and including the maturity date. The Asian option is an option whose payoff depends on the average price of the underlying asset during the period since the issue of the option until its expiration date, e.g., see [26] for details.

In the case of a European option, the value of the like-call option can be obtained from (12) with the boundary conditions:

$$\begin{aligned} \left\{ \begin{array}{llr} V_{c}(S,T)= \max (S-K,0) &{} : \tau =T\\ V_{c}(0,\tau )=0 &{} :S=0\\ V_{c}(S,\tau )\rightarrow S-Ke^{-r(T-\tau )} &{} : S\rightarrow \infty . \end{array} \right. \end{aligned}$$

We have that

$$\begin{aligned}&V_c(S,\tau )=SN\left( \frac{1}{\sigma \sqrt{T-\tau }}\left[ \ln \left( \frac{S}{K}\right) +\left( r+\frac{1}{2}\sigma ^{2}\right) (T-\tau )\right] \right) \\&\quad -e^{-r(T-\tau )}KN\left( \frac{1}{\sigma \sqrt{T-\tau }}\left[ \ln \left( \frac{S}{K}\right) +\left( r-\frac{1}{2}\sigma ^{2}\right) (T-\tau )\right] \right) , \end{aligned}$$

where \(N(\eta )=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{\eta }e^{-\frac{S^{2}}{2}}dx\) and \(K>0\) is the strike price.

Here \(S=S(\tau )\) for \(\tau \in [0,T]\); the solution of (12) provides both an option pricing formula for a European option and a hedging portfolio that replicates the contingent claim assuming that:

  1. 1.

    The asset price S or the value of the underlying asset follows a geometric Brownian motion.

  2. 2.

    The trend or drift \(\mu \) which measures the average rate of growth of the asset price, the volatility \(\sigma \) which measures the standard deviation of the returns and the riskless interest rate r are all constant for \(0\le \tau \le T\), and no dividends are paid in that time period.

  3. 3.

    The market is assumed to be frictionless, thus there are no transaction costs such as fees or taxes, the interest rates for borrowing and lending money are equal, all parties have immediate access to any information, and all securities and credits are available at any time and any size. That is, all variables are perfectly divisible and may assume any real number. Moreover, individual trading is assumed to not influence the price.

  4. 4.

    There are no arbitrage opportunities, meaning that there are no opportunities for instantly extracting a risk-free profit.

Under these assumptions the market is complete, which means that any derivative and any asset can be replicated or hedged with a portfolio of other assets in the market.

The linear equation (12) has been studied using various mathematical techniques, for example, from the point of view of the operator theory in [25]; from the point of view of mathematical physics in [21, 30], and through the ADM as a particular case study in [19]. In the next section, we study the Black-Scholes model for a nonlinear case.

A Nonlinear Case

It is easy to conceive that the restrictive assumptions mentioned at the end of the previous section are never actually fulfilled in real life as stated in [5]. Due to transaction costs, see for example, [7, 11], large investor preferences, see [23, 24], and for incomplete markets [31], such assumptions turn unrealistic and the classical model becomes a strongly nonlinear, possibly degenerate, parabolic convection-diffusion equation, where both the volatility \(\sigma \) and the drift \(\mu \) can depend on the time \(\tau \), the stock price S or the derivatives of the option price V itself.

There are several transaction cost models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a constant drift \(\mu \) and a nonconstant modified volatility function

$$\begin{aligned} \hat{\sigma }^{2}=\hat{\sigma }^{2}(\tau ,S,V_{S},V_{SS}). \end{aligned}$$
(13)

There have been many approaches to improve the aforementioned model by treating the volatility in different ways, e.g., using a modified volatility function \(\hat{\sigma }\) to model the effects of transaction costs, illiquid markets and large traders, which is the reason for the nonlinearity of (12). Illiquid markets and large trader effects have been modeled by several authors. In [24], Frey and Stremme and later Frey and Patie [23] consider these effects on the price and deduced the result

$$\begin{aligned} \hat{\sigma }(\tau ,S,V_{S},V_{SS})=\frac{\sigma }{1-\rho S\lambda (S)u_{SS}}, \end{aligned}$$
(14)

where \(\sigma \) the traditional volatility, \(\rho \) constant and \(\lambda \) (price of the risk) is a continuous positive function describing the liquidity profile of the market. Bordag and Chmakova in [10] assume that \(\lambda \) is a constant in order to solve a nonlinear Black-Scholes equation with the modified volatility (14).

In [22, 23], the authors derived a generalized Black-Scholes pricing PDE that, for the case where the interest rate \(r \ge 0\) and the reference volatility \(\sigma >0\) are constant, assumes the form

$$\begin{aligned} \left\{ \begin{array}{ll} V_\tau (S,\tau )+\frac{1}{2}\sigma ^{2}S^{2}\cdot \frac{V_{SS}(S,\tau )}{(1-\rho S\lambda (S)V_{SS}(S,\tau ))^{2}}+rSV_{S}(S,\tau )-rV(S,\tau )=0,\\ V(S,T)=f(S),\;\; 0<S<\infty . \end{array} \right. \end{aligned}$$
(15)

In [27], the authors establish the existence and uniqueness of a classical solution to this nonlinear PDE, for the case where the payoff function f satisfies: \(f(e^{x})\) is Lipschitz-continuous and \(e^{-a\sqrt{x^{2}+1}}f(e^{x})\) is bounded for some \(a\ge 0\). Recently two numerical studies of (15) have been completed in [14] and [15].

If we consider the special case where the price of risk is one unit, \(\lambda (S)=1\), in this case, and if we further assume that \(\vert \vert \rho SV_{SS}\vert \vert <\varepsilon \) for some \(\varepsilon >0\) that is small enough (low impact of hedging, assumption A4 of [23]) and using the functional approximation \(\frac{1}{(1-F)^{2}}\approx 1+2F+\fancyscript{O}(F)^{3}\), we find that Eq. (15) becomes

$$\begin{aligned} \left\{ \begin{array}{ll} V_\tau (S,\tau )+\frac{1}{2}\sigma ^{2}S^{2} V_{SS}(S,\tau )(1+2\rho S V_{SS}(S,\tau ))+rSV_{S}(S,\tau )-rV(S,\tau )=0,\\ V(S,T)=f(S),\;\; S\in [0,\infty ). \end{array} \right. \end{aligned}$$
(16)

Note by considering the translation \(t = T-\tau \), and denoting \(V(S, \tau )=u(S, t)\), problem (16) assumes the form

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}(S,t)+\frac{1}{2}\sigma ^{2}S^{2}u_{SS}(S,t)(1+2\rho S u_{SS}(S,t))+rSu_{S}(S,t)-ru(S,t)=0,\\ u(S,0)=f(S),\;\; S\in [0,\infty ). \end{array} \right. \end{aligned}$$
(17)

An exact solution of Eq. (17) has been obtained in [20], as

$$\begin{aligned} u(S,t)=S-\frac{\sqrt{S_{0}}}{\rho }\Big ( \sqrt{S}\exp \left( \frac{r+\frac{\sigma ^{2}}{4}}{2}\right) t+\frac{\sqrt{S_{0}}}{4}\exp \left( r+\frac{\sigma ^{2}}{4}\right) t\Big ), \end{aligned}$$
(18)

where \(S_0\) is the initial stock price.

The Nonlinear Case Solved by the ADM

Comparing (17) with Eq. (3), we have that \(g(S,t)=0\), and

$$\begin{aligned} L_{t}=\frac{\partial (\cdot )}{\partial t},\;\; R= \frac{1}{2}\sigma ^{2}S^{2}\frac{\partial ^{2}(\cdot )}{\partial S^{2}}+rS\frac{\partial (\cdot )}{\partial S}-r,\quad N=\rho \sigma ^{2}S^{3}\Big (\frac{\partial ^{2}(\cdot )}{\partial S^{2}}\Big )^{2}. \end{aligned}$$
(19)

Referring to Eq. (10), establish the nonlinear recursion relation

$$\begin{aligned} \left\{ \begin{array}{ll} u_{0}(S,t)=f(S),\\ u_{n+1}(S,t)=-L_{t}^{-1}Ru_{n}(S,t)-\rho \sigma ^{2}S^{3}L_{t}^{-1}A_{n}(u_{0},u_{1},\ldots , u_{n}),\quad n=0,1,2,\ldots , \end{array} \right. \end{aligned}$$
(20)

where \(\Big (\frac{\partial ^{2}u}{\partial S^{2}}\Big )^{2},\) the nonlinear term,

$$\begin{aligned} F(u)=u_{ss}^{2}=\sum _{n=0}^{\infty }A_{n}(u_{0},u_{1},\ldots , u_{n}). \end{aligned}$$
(21)

Referring to Eq. (7), the solution comprises its solution components

$$\begin{aligned} u= \sum _{n=0}^{\infty }u_{n}, \end{aligned}$$

and the nonlinearity comprises the Adomian polynomials

$$\begin{aligned} F(u)= & {} u_{ss}^{2}=(u_{0,ss}+u_{1,ss}+u_{2,ss}+u_{3,ss}+u_{4,ss}+u_{5,ss}+\cdots )^{2}=u_{0,ss}^{2}+2u_{0,ss}u_{1,ss}\nonumber \\&+2u_{0,ss}u_{2,ss}+2u_{0,ss}u_{3,ss}+2u_{0,ss}u_{4,ss}+2u_{0,ss}u_{5,ss}+u_{1,ss}^{2}+2u_{1,ss}u_{2,ss}\nonumber \\&+2u_{1,ss}u_{3,ss}+2u_{1,ss}u_{4,ss}+2u_{1,ss}u_{5,ss}+u_{2,ss}^{2}+2u_{2,ss}u_{3,ss}+2u_{2,ss}u_{4,ss}\nonumber \\&+2u_{2,ss}u_{5,ss}\!+\!u_{3,ss}^{2}\!+\!2u_{3,ss}u_{4,ss}\!+\!2u_{3,ss}u_{5,ss}\!+\!u_{4,ss}^{2}\!+\!2u_{4,ss}u_{5,ss}+u_{5,ss}^{2}+\cdots .\nonumber \\ \end{aligned}$$
(22)

The above expression can be rearranged by grouping all terms such that the sum of subscripts of \(u_n\) are equal. This procedure gives the Adomian polynomials [34] for the quadratic differential nonlinearity \(u_{ss}^{2}\):

$$\begin{aligned} A_{0}= & {} u_{0,ss}^{2}\\ A_{1}= & {} 2u_{0,ss}u_{1,ss} \\ A_{2}= & {} 2u_{0,ss}u_{2,ss}+u_{1,ss}^{2}\\ A_{3}= & {} 2u_{0,ss}u_{3,ss}+2u_{1,ss}u_{2,ss}\\ A_{4}= & {} 2u_{0,ss}u_{4,ss}+2u_{1,ss}u_{3,ss}+u_{2,ss}^{2}\\ A_{5}= & {} 2u_{0,ss}u_{5,ss}+2u_{1,ss}u_{4,ss}+2u_{2,ss}u_{3,ss}\\&\vdots&. \end{aligned}$$

Further study on the convergence of the ADM, which is beyond the scope of application of this work, can be found in references [1, 36].

Example 1

In this first example, we consider the particular case of (17) such that \(r=0\); this case was proposed in [23] in which the authors studied a model of an illiquid market where the implementation of a dynamic hedging strategy has a feedback effect on the price process of the underlying asset. Here we propose \(f(S)=S+10(\sqrt{S}+\frac{1}{4})\), \(\sigma =0.2\), \(\lambda (S)=1\) and \(\vert \rho \vert =0.01\), noting that the prerequisite in [27] is satisfied. That is, for \(a=2\), it holds that

$$\begin{aligned} \vert e^{-4\sqrt{S^{2}+1}}f(e^{S})\vert \le 2\;\; \text{ for } \text{ every } \; S\in [0,\infty ). \end{aligned}$$

In this example, we denote the operators as

$$\begin{aligned} L_{t}=\frac{\partial (\cdot )}{\partial t},\;\; R= \frac{1}{2}\sigma ^{2}S^{2}\frac{\partial ^{2}(\cdot )}{\partial S^{2}},\;\; N=\rho \sigma ^{2}S^{3}\Big (\frac{\partial ^{2}(\cdot )}{\partial S^{2}}\Big )^{2}. \end{aligned}$$

In the ADM framework, we choose \(u_{0}(S,t)=S+10(\sqrt{S}+\frac{1}{4})\) and therefore, the first several Adomian polynomials are given as

$$\begin{aligned} A_{0}= & {} u_{0,ss}^{2}=\frac{25}{4S^{3}},\\ A_{1}= & {} 2u_{0,ss}u_{1,ss}=-6.25\times 10^{-2}tS^{-3}, \\ A_{2}= & {} 2u_{0,ss}u_{2,ss}+u_{1,ss}^{2}= 1.7188\times 10^{-4}t^{2}S^{-3}. \end{aligned}$$

With these results and using (19) and (20) and considering the change of variable \(t=T-\tau \), implying \(L_{t}^{-1}=-\int _{0}^{t}\), and if \(t\in [0,T]\), we calculate

$$\begin{aligned} u_{1}(S,t)= & {} -L_{t}^{-1}Ru_{0}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{0}(u_{0})\\= & {} -5\times 10^{-2}t\sqrt{S}-2.5\times 10^{-3}t,\\ u_{2}(S,t)= & {} -L_{t}^{-1}Ru_{1}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{1}(u_{0},u_1)\\= & {} 1.25\times 10^{-5}t^{2}\sqrt{S}+1.25\times 10^{-5}t^{2},\\ u_{3}(S,t)= & {} -L_{t}^{-1}Ru_{2}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{2}(u_{0},u_1, u_2)\\= & {} -2.0833\times 10^{-8}t^{3}\sqrt{S}-2.2917\times 10^{-8}t^{3}. \end{aligned}$$

Thus the solution of (17) is

$$\begin{aligned} u_{ADM}(S,t)= u_{0}(S,t)+u_1(S,t)+u_2(S,t)+u_3(S,t) +\cdots \end{aligned}$$
(23)

In Table 1, we compare the solution of the nonlinear Black-Scholes equation obtained from (23) with the exact solution obtained in [20], in which a solution of (17) is A nonlinear casewaves in porous media, Eq. (18). In Figs. , 2 and 3, we display the values of the option against the underlying asset prices obtained from ADM and the exact values for \(t=0.5 \), \(t=1.0 \) and \( t=1.5\) years. In addition, in Fig. 4, we show that the first terms values of the partial sum are not distinguish of the exact solution. All the numerical work was accomplished with the Mathematica software package.

Fig. 1
figure 1

Graph of the values of \(u_{ADM}\) and \(u_{ex}\) for \(t = 0.5\) years

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Fig. 2
figure 2

Graph of the values of \(u_{ADM}\) and \(u_{ex}\) for \(t = 1.0\) years

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Fig. 3
figure 3

Graph of the values of \(u_{ADM}\) and \(u_{ex}\) for \(t = 1.5\) years

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Fig. 4
figure 4

Graph of the values of \(u_{ADM}\) for \(k=0,1,2\) and \(u_{ex}\) for \(t = 1.0\) years

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Table 1 The values of \(u_{ADM}\) and \(u_{ex}\) and the relative difference between them for \(t\,=\,0.5\), 1.0 and 1.5 years
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Example 2

In the second example, assume \(f(S)=S+200\sqrt{S}+100\), \(r=0.06\), \(\sigma =0.4\) and \(\rho =-0.01\). Notice as well that the prerequisite in [27] is fulfilled, that is, for \(a=5\) we have that

$$\begin{aligned} \vert e^{-5\sqrt{S^{2}+1}}f\left( e^{S}\right) \vert \le 3\quad \text{ for } \text{ every } \quad S\in [0,\infty ). \end{aligned}$$

In the ADM framework, we have \(u_{0}(S,t)=S+200\sqrt{S}+100\) thus

$$\begin{aligned} A_{0}= & {} u_{0,ss}^{2}=\frac{2500}{S^{3}}, \\ A_{1}= & {} 2u_{0,ss}u_{1,ss}=-\frac{250t}{S^{3}}, \\ A_{2}= & {} 2u_{0,ss}u_{2,ss}+u_{1,ss}^{2}= \frac{12.5t^{2}}{S^{3}}, \\ A_{3}= & {} 2u_{0,ss}u_{3,ss}+2u_{1,ss}u_{2,ss}= \frac{5t}{\sqrt{S^{3}}}-\frac{0.3125t^{3}}{S^{3}},\\ A_{4}= & {} 2u_{0,ss}u_{4,ss}+2u_{1,ss}u_{3,ss}+u_{2,ss}^{2}= \frac{1.2136\times 10^{-2}t^{4}-0.3St^{2}}{S^{3}}. \end{aligned}$$

As before, using (19) and (20) and considering the translation \(t=T-\tau \), implying \(L_{t}^{-1}=-\int _{0}^{t}\), and if \(t\in [0,T]\), we calculate

$$\begin{aligned} u_{1}(S,t)= & {} -L_{t}^{-1}Ru_{0}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{0}(u_{0})\\= & {} 2t+10t\sqrt{S}+0.06St,\\ u_{2}(S,t)= & {} -L_{t}^{-1}Ru_{1}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{1}(u_{0},u_1)\\= & {} -0.25t^{2}\sqrt{S}+0.14t^{2},\\ u_{3}(S,t)= & {} -L_{t}^{-1}Ru_{2}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{2}(u_{0},u_1, u_2)\\= & {} 4.1667\times 10^{-3}t^{3}\sqrt{S}-9.4667\times 10^{-3}t^{3},\\ u_{4}(S,t)= & {} -L_{t}^{-1}Ru_{3}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{3}(u_{0},u_1, u_2,u_3)\\= & {} -5.2084\times 10^{-5}\sqrt{S}t^{4}+2.67\times 10^{-4}t^{4}-0.004t^{2}S^{\frac{3}{2}},\\ u_{5}(S,t)= & {} -L_{t}^{-1}Ru_{4}(S,t)-\rho \sigma ^{2}L_{t}^{-1}S^{3}A_{4}(u_{0},\ldots ,u_4)\\= & {} -5.2085\times 10^{-5}\sqrt{S}t^{5}-6.5374\times 10^{-6}t^{5}-1.2\times 10^{-4}t^{3}S^{\frac{3}{2}}+1.6\times 10^{-4}St^{3},\\ \end{aligned}$$

Thus the solution of (17) is

$$\begin{aligned} u_{ADM}(S,t)= u_{0}(S,t)+u_1(S,t)+u_2(S,t)+u_3(S,t)+u_4(S,t)+u_5(S,t) +\cdots . \end{aligned}$$
(24)

As in the first example, we display the corresponding results in Table 2 and Figs. 5 and 6 below. In addition, in Fig. 7, we show that the first terms values of the partial sum are not distinguish of the exact solution.

Table 2 The values of \(u_{ADM}\) and \(u_{ex}\) and the relative difference between them for \(t = 0.0\), 0.5 and 1.0 years
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We remark that, as these examples demonstrate, the Adomian decomposition method avoids several difficulties in the calculation including massive computational work are required, e.g., by discretization techniques, in determining the approximate analytic solution.

Fig. 5
figure 5

Graph of \(u_{ADM}\) and \(u_{ex}\) for \(t = 0.5\) years

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Fig. 6
figure 6

Graph of \(u_{ADM}\) and \(u_{ex}\) for \(t = 1.0\) years

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Fig. 7
figure 7

Graph of the values of \(u_{ADM}\) for \(k=0,1,2\) and \(u_{ex}\) for \(t = 1.0\) years

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Conclusion and Summary

In this paper, we briefly discussed the Adomian decomposition method for approximate analytic solutions of both nonlinear ordinary and partial differential equations, and applied this method for the solution of the nonlinear Black- Scholes equation. We have also studied two hypothetical examples through the ADM. In the first example, we consider the case for which \( r = 0 \). As shown in Table 1, the percentage error is not greater than 1.63  % despite that we have only considered an ADM approximation of third degree. In the second example, we considered a case in which \( r \ne 0\) and as shown in Table 2, the error percentage is not exceeding 7.58  % by considering an ADM approximation of fifth degree; in both examples, we have graphically compared our approximate analytic solutions by the ADM with the exact result found indirectly in [20] in the framework of a fluid traveling through porous media. It is also observed from the Figs. 1, 2, 3, 5 and 6 that the option always exceeds the price of the underlying asset as a result of variable volatility. The results presented here provide further evidence of the practicality and efficiency of the Adomain decomposition method to find solutions of nonlinear partial differential equations such as the nonlinear Black-Scholes equation. In this paper, the MATHEMATICA software package was used to calculate the decomposition series.