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Vol. 4, No. 2 | Jul – Dec 2020 
 

 

 

 

SJCMS | P-ISSN: 2520-0755| E-ISSN: 2522-3003 © 2020 Sukkur IBA University 

22 

On Study Of Generalized Nonlinear Black Scholes  Equation 

By Reduced Differential Transform Algorithm 
 

Naresh Kumar Solanki1 , Syed Feroz Shah1 

Abstract: 

The objective of the work is essential to construct an approximate solution of the generalization 

of nonlinear Black-Scholes partial differential equation, modeling price slippage impact of 

transaction coast option, through promising computational algorithm called Reduced 

Differential Transform Algorithm. This work also shows that the algorithm can be efficiently 

employed to construct explicit solutions highly nonlinear equations arising in the financial 

market. We have also shown a graphical behavior of the constructed solutions. 

Keywords: Nonlinear Black-Scholes model, PDE, differential transform algorithm 

1. Introduction  

This paper is intended to construct an 

approximate and closed-form solution to the 

following initial value problem,  

𝜕𝑢

𝜕𝑡
+

1

2
𝜎 2𝑠2

𝜕2𝑢

𝜕𝑠2
(1 + 2𝜌𝑠

𝜕2𝑢

𝜕𝑠2
) + 𝑟𝑠

𝜕𝑢

𝜕𝑠

− 𝑟𝑢 = 0                    (1) 
subject to the initial condition, 

 
Where 𝑆 represents the stock price, 𝜌 ≥ 0 it 
shows the measure of liquidity of the market, 

and also represents volatility as well. 𝑢(𝑠, 𝑡) 
represents the option price and it’s a measure 

of the price of slippage impact of a trade felt 

by all participants of a market. 

                                                           
1Basic Science and related science MUET, Jamshoro, Pakistan 

Corresponding Author: feroz.shah@faculty.muet.edu.pk 

2. Description of Differential 
Transform Algorithm 

This section has been dedicated to give a 

precise description of the Reduced 

Differential Transform algorithm and how it 

works. Assume that we have a function

( , )u x t with arguments x and t, that can 

express as the product two functions of 𝑥 and 
𝑡 i.e. i.e., 𝑢(𝑥, 𝑡) = 𝑓(𝑥)𝑔(𝑡). Then 
differential transform of the function 𝑢(𝑥, 𝑡) 
can be explicitly written as, 

 

 

where ( )
k

U x  is transformed function in x. 

The more careful and precise definitions of 

transform of function 𝑢(𝑥, 𝑡) is following, 
(cf. 14-16). 

Definition: Consider a function ( , )u x t  is 

call option where u represents the price of 

option and t represents the time 𝑡 ≥ 0 and 



 
Naresh Kumar (et al.), On Study Of Generalized Nonlinear Black Scholes Equation By Reduced Differential Transform 
Algorithm                                                               (pp. 22 - 27) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 4 No. 2 July - December 2020 © Sukkur IBA University 

23 

space 𝑥 ∈ ℝ. Then define the transform of 
𝑢(𝑥, 𝑡) as, 

0

.
1

( ) ( , )
!

k

k k

t

U x u x t
k t

=

 
=  

 
 

where the ( )
k

U x  can be treated as 

transformed 𝑢(𝑥, 𝑡), and is essentially 
analogous to Taylor’s coefficient in the 2D 

Taylor expansion. 

To recover the function 𝑢(𝑥, 𝑡) from 
transformed functions 𝑈𝑘 (𝑥), we define the 
following inverse of differential transform in 

the following manner.  

Definition: Consider a function ( , )u x t  is 

call option where u represents the price of 

option and t represents the time 𝑡 ≥ 0,  Then 
we can define the transform of 𝑢(𝑥, 𝑡) as, 
Then define the transform of 𝑈𝑘 (𝑥), as,  

𝑢(𝑥, 𝑡) = ∑  𝑈𝑘 (𝑥)𝑡
𝑘                           (2)

∞

𝑘=0

 

Or more explicitly,  

0 0

1
( , ) ( , ) .

!

k
k

k
k t

u x t u x t t
k t



= =

 
=  

 
  

Next, we discuss how the above-described 

transformations can be implemented to solve 

the concrete nonlinear partial differential 

equations. Consider a nonlinear PDE in its 

generalized form,  

𝐿𝑢(𝑥, 𝑡) + 𝑅𝑢(𝑥, 𝑡) + 𝑁𝑢(𝑥, 𝑡)

= 𝑔(𝑥, 𝑡)                     (3) 

Subject to the initial condition 𝑢(𝑥, 0) =

𝑓(𝑥). Here 𝐿 denotes an operator 
𝜕

𝜕𝑡
, 𝑅𝑢(𝑥, 𝑡) 

denotes the linear part of PDE that contains 

the linear expressions of 𝑢 and its derivatives, 

( , )Nu x t denotes the operator/expression 

containing the nonlinear terms involving u, 

and its derivatives operator, ( , )g x t  stands 

for an in-homogeneous term that can be 

treated a forcing factor in the model. Taking 

the differential transform of the equation (3) 

leads to the following recursive relation, 

 

(𝑘 + 1)𝑈𝑘+1(𝑥)
= 𝐺𝑘 (𝑥) − 𝑅𝑈𝑘 (𝑥)
− 𝑁𝑈𝑘 (𝑥)                    (4) 

 

where ( ), ( ), ( )
k k k

U x RU x NU x  and ( )kG x  

denotes the differential transformation of  

( , ), ( , ), ( , )Lu x t Ru x t Nu x t  and ( , )g x t  

respectively. Hence the key computation that 

one need to is the computation of functions 

𝑈1, 𝑈2, 𝑈3 … through recursive relation (4), 

by choosing 
0
( ) ( ).U x f x=  

Once 𝑈1, 𝑈2, 𝑈3 … 𝑈𝑛  are found then we can 
write n-term approximate solution of PDE as 

follows:  

              
Thus, by increasing n more and more we get 

an exact solution of nonlinear PDE (4), 

 

    �̃�(𝑥, 𝑡) = lim
𝑛⇢∞

∑ 𝑈𝑘 (𝑥)𝑡
𝑘𝑛

𝑘=0           (6) 

 

Based on the definition of the reduced 

differential transform algorithm following 

table of transformations (see next page) can be 

proved. For the readers interested in the proofs 

we refer to [14], [15], and [16]. 



 
Naresh Kumar (et al.), On Study Of Generalized Nonlinear Black Scholes Equation By Reduced Differential Transform 
Algorithm                                                               (pp. 22 - 27) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 4 No. 2 July - December 2020 © Sukkur IBA University 

24 

 

3. The solution of Generalized form of 
nonlinear Black-Scholes equation 

by Reduced Differential 

Transform Algorithm 

For following the Nonlinear Black-Scholes 

equation, we are applying the RDTM method 

to get an approximate solution. 

Let us restart by rewriting the equation (1) in 

the following form, 

𝜕𝑢

𝜕𝑡
= − [

𝜎 2𝑠2

2

𝜕2𝑢

𝜕𝑠2
(1 + 2𝜌𝑠

𝜕2𝑢

𝜕𝑠2
) + 𝑟𝑠

𝜕𝑢

𝜕𝑠

− 𝑟𝑢]                              (7) 

On the application of RDTM on the above 

last equation, using Table 1of transforms, we 

get 

(𝑘 + 1)𝑢𝑘+1 = − [
1

2
𝜎 2𝑠2

𝜕2𝑢𝑘
𝜕𝑠2

+ 𝜌𝜎 2𝑠3 (
𝜕2𝑢𝑘
𝜕𝑠2

)

2

+ 𝑟𝑠
𝜕𝑢𝑘
𝜕𝑠

− 𝑟𝑢𝑘 ]       (8) 

where ( )
k

u x is transformed function and 

dimensional spectrum function is ,t                 

𝐴𝑘 = ∑
𝜕2𝑢(𝑠, 𝑘)

𝜕𝑠2
𝜕2𝑢(𝑠, ℎ − 𝑘)

𝜕𝑠2

ℎ

𝑘=0

        (9) 

 

Let us start by computing the 𝑢1. The 
explicit expression for it can be obtained 

by setting 0k = in (9), as follows,  

𝑢1 = − [
1

2
𝜎2𝑠2

𝜕2𝑢0
𝜕𝑠2

+ 𝜌𝜎2𝑠3𝐴0

+ 𝑟𝑠
𝜕𝑢0
𝜕𝑠

− 𝑟𝑢0] 

It is clear that to compute the above 

expression we need values of  𝐴0, 
𝜕𝑢0

𝜕𝑠
 

and 
𝜕2𝑢0

𝜕𝑠2
. From the initial condition 

equation (9), we solve the first and 

second partial derivatives of equation (9), 

we have 

𝜕𝑢0
𝜕𝑠

=
1

𝜌
[
−2

𝜎 2
(𝑟 − 𝑐)

√𝐾 

√𝑠
𝑒

−𝑐
2

𝑇

− lns (
(𝑟 − 𝑐)

𝜎 2

+
1

4
)]                         (10) 



 
Naresh Kumar (et al.), On Study Of Generalized Nonlinear Black Scholes Equation By Reduced Differential Transform 
Algorithm                                                               (pp. 22 - 27) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 4 No. 2 July - December 2020 © Sukkur IBA University 

25 

𝜕2𝑢0
𝜕𝑠2

=
1

𝜌
[
(𝑟 − 𝑐)

𝜎 2
√𝐾 

𝑠
3
2

𝑒
−𝑐
2 

𝑇

−
1

s
 (

(𝑟 − 𝑐)

𝜎 2

+
1

4
)]                         (11) 

Now taking 0k =  in Equation (9) and 
plugging partial derivatives from (10) and 

(11), we get the following value of 
0

A  

𝐴0 = (
𝜕2𝑢0
𝜕𝑠2

)

2

=
1

𝜌2
[
(𝑟 − 𝑐)2

𝜎 4
K

𝑠3
𝑒 −c𝑇

− 2 (
(𝑟 − 𝑐)

𝜎 2
√𝐾 

𝑠
3
2

𝑒
−𝑐𝑇

2 ) (
1

s
 (

(𝑟 − 𝑐)

𝜎 2
+

1

4
))

+
1

s2
 (

(𝑟 − 𝑐)

𝜎 2
+

1

4
)

2

]                               (12) 

On substituting values 𝐴0, 
𝜕𝑢0

𝜕𝑠
 and 

𝜕2𝑢0

𝜕𝑠2
 

in expression for 𝑢1 and simplifying we 

obtain  

 

Next, let us move towards computing the 

valued u2. To do so let us substitute k=1 

in recursive relation (8),  

 
Therefore, to compute 𝑢2, we need to 

explicitly find  𝐴1, 
𝜕𝑢1

𝜕𝑠
 and 

𝜕2𝑢1

𝜕𝑠2
. Using 

equation (13), we may find the partial 

derivatives as following,  

  
𝜕𝑢1
𝜕𝑠

=
1

𝜌
[
−c(𝑟 − 𝑐)

𝜎 2
√𝐾 

𝑠
1
2

𝑒
−𝑐
2

𝑇

− 𝜎 2 (
(𝑟 − 𝑐)2

𝜎 4
−

1

16
)

+ 𝑟 (
𝑟 − 𝑐

𝜎 2
+

1

4
)]          (14) 

𝜕2𝑢1
𝜕𝑠2

=
1

𝜌
[
c(𝑟 − 𝑐)

2𝜎 2
√𝐾 

𝑠
3
2

𝑒
−𝑐
2

𝑇
]                   (15) 

On employing (9), (14), and (15), we can 

explicitly compute  𝐴1 as,  

 𝐴1 =
1

𝜌2
[
𝑐(𝑟 − 𝑐)2

𝜎 4
K

𝑠3
𝑒 −c𝑇

−
c(𝑟 − 𝑐)

𝜎 2
(

(𝑟 − 𝑐)

𝜎 2

+
1

4
)

√𝐾 

𝑠
5
2

𝑒
−𝑐
2

𝑇
 ]          (16) 

𝐴1 = 2
𝜕2𝑢0
𝜕𝑠2

𝜕2𝑢1
𝜕𝑠2

 

Now we are in a position to compute 𝑢2 

using the values of  𝐴1, 
𝜕𝑢1

𝜕𝑠
 and 

𝜕2𝑢1

𝜕𝑠2
 

from (16), (14), (15) respectively, 

Finally, we aim to compute 𝑢3. To do, by 
substituting k=2 in recursive relation (8) we 

get,  

 
Hence to compute 𝑢3, we need to compute 

𝐴2,
𝜕𝑢2

𝜕𝑠
 and 

𝜕2𝑢2

𝜕𝑠2
. The partial derivatives 

of 𝑢2 can be computed using equation (17),  

 

 



 
Naresh Kumar (et al.), On Study Of Generalized Nonlinear Black Scholes Equation By Reduced Differential Transform 
Algorithm                                                               (pp. 22 - 27) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 4 No. 2 July - December 2020 © Sukkur IBA University 

26 

 

Next, an explicit expression for 𝐴2 can be 
obtained by putting k=2 in (9),   

 
Using (11), (15), and (17) in the above, and 

we get, 

 

Now employing the 𝐴2,
𝜕𝑢2

𝜕𝑠
 and 

𝜕2𝑢2

𝜕𝑠2
 

from (20), (18), and (19), in 𝑢3, we get, 

 

   Finally, the approximate solution to our 

main problem can be given as,  

 

On substituting, values of 

𝑢0, 𝑢1, 𝑢2 𝑎𝑛𝑑 𝑢3  from (13), (17), and (20) 

in the last equation, we get the following 

closed-form approximate solution of   

problem (1),  

 

The 3D graph of the solution has been 

computed using Mathematica,  

 

4. Conclusion 

Transaction cost Black-Scholes model is 

highly nonlinear in its form. In this article, we 

have computed an explicit approximation 

solution to a highly nonlinear version of the 

generalized nonlinear Black-Scholes equation 

through a highly efficient algorithm known as 

the reduced differential transform method. 

This solution obtained can be used for pricing 

European call and put option at time t≥0, and 
when c≠r. Our work shows that Reduced 

differential transform methods are very 

convenient for constructing the explicit 

scheme of higher nonlinear problems arising 

in the financial world.  

 

References 

[1]  B. Lin and Z. Yin, "The Cauchy 

problem for a generalized Camassa--

Holm equation with velocity potential.," 

Applicable Analysis, vol. 97, no. 3, pp. 

354-367, 2018.  

[2]  R. Camassa and a. D. Holm, "An 

integrable shallow water equation with 

peaked solitons," Physical review 

letters, vol. 71, pp. 1661-1664, 1993.  

[3]  C. A, "On the scattering problem for the 

Camassa--Holm equation," 

Proceedings of the Royal Society of 

London. Series A: Mathematical, 

Physical and Engineering Sciences, vol. 

457, pp. 953-970, 2001.  

[4]  Y. Keskin and G. Oturanc, "Reduced 

differential transform method for 



 
Naresh Kumar (et al.), On Study Of Generalized Nonlinear Black Scholes Equation By Reduced Differential Transform 
Algorithm                                                               (pp. 22 - 27) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 4 No. 2 July - December 2020 © Sukkur IBA University 

27 

generalized KDV equations," 

Mathematical and Computational 

applications, vol. 15, pp. 382-393, 

2010.  

[5]  Y. Keskin and G. Oturan, "Application 

of reduced differential transformation 

method for solving gas dynamics 

equation," International Journal of 

Contemporary Mathematical Sciences, 

vol. 22, pp. 1091-1096, 2010.  

[6]  Y. Keskin and G. Oturanc, "Reduced 

differential transform method for 

solving linear and nonlinear wave 

equations," 2010.  

[7]  Zheng, R. a. Yin and Zhaoyang, "Global 

weak solutions for a generalized 

Novikov equation," Monatshefte fur 

Mathematik, vol. 188, pp. 387-400, 

2019.  

[8]  Novikov and Vladimir, 

"Generalizations of the Camassa–Holm 

equation.," Journal of Physics A: 

Mathematical and Theoretical, vol. 42, 

p. 342002, 2009.  

[9]  Henry and D., "Infinite propagation 

speed for the Degasperis-Procesi 

equation," Journal of mathematical 

analysis and applications, vol. 311, pp. 

755-759, 2005.  

[10]  Lenells and Jonatan, "Travelling wave 

solutions of the Degas," Journal of 

mathematical analysis and 

applications, vol. 306, pp. 72-82, 2005.  

[11]  Oclite, G. M. a. Karlsen and K. H, "On 

the well posedness of the Degasperis-

Procesi equation," Journal of 

Functional Analysis, vol. 233, pp. 60-

91, 2006.  

[12]  Degasperis, A. a. Procesi and Michela, 

"Asymptotic Integrability, in symmetry 

and Perturbation Theory," Symmetry 

and perturbation theory, vol. 1, pp. 23-

37, 1999.  

[13]  Tu, X. a. Yin and Zhaoyang, "Global 

weak solutions for a generalized 

Casmassa-Holm equation," 

Mathematische Nachrichten, vol. 291, 

2018.  

[14]  Blaszak and Maciej, The Theory of 

Hamiltonian and Bi-Hamiltonian 

Systems. In: Multi-Hamiltonian Theory 

of Dynamical Systems., Berlin: 

Springer, 1998, pp. 41-85.