International Journal of Analysis and Applications

Volume 18, Number 3 (2020), 513-530

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-513

ON THE PRICING OF CALL-PUT PARITIES OF ASIAN OPTIONS BY REDUCED

DIFFERENTIAL TRANSFORM ALGORITHM

JAVED HUSSAIN∗, MUHAMMAD SHOAIB KHAN

Sukkur IBA University, Pakistan

∗Corresponding author: javed.brohi@iba-suk.edu.pk

Abstract. The key aim of the paper is to show that how the efficiently the Reduced Differential Transform

Algorithm (RDTA) can be employed to price the exotic financial options. In this paper we have computed

the exact solution of the parabolic partial differential equation governing the dynamics of put-call parity in

the mathematical theory of Asian options, by means of RDTA.

1. Introduction

An option is an agreement that allows the holder to buy (call option) or sell (put option) at a specified

future time (expiration or maturity time) an underlying asset at a specified price (strike or exercise price).

Pricing of an Asian option is always complicated due to its path dependents derivatives whose payoffs

depend on some form of averaged prices of the underlying asset and no closed form solution exist in general.

Pricing them efficiently and accurately is very important both in theory and practice. Asian option can

avoid manipulation of the stock near expiration time. Asian options are popular in the financial community

as well as in the over-the-counter (OTC) market because they are often cheaper than the equivalent classical

European options.

Several analytical approaches have been proposed to address the problem. In [?] a closed form solution of

the no-arbitrage price of arithmetic averaged fixed strike price is obtained through the inversion of a Laplace

Transform.

Received December 12th, 2019; accepted January 14th, 2020; published May 1st, 2020.

2010 Mathematics Subject Classification. 35Q91, 91G80.

Key words and phrases. Asian option; Put-Call Parity; Reduced Differential Transform Algorithm.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

513

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-513


Int. J. Anal. Appl. 18 (3) (2020) 514

The standard approximation methods based on partial differential equations require some regularity con-

ditions of the solution of the no-arbitrage PDE. Considering path dependent contingent claims in general

setting, the PDE is a strongly degenerate parabolic equation in three dimensions (time, the underlying asset

price, and the path-dependent variable). In this setting, the needed regularity seemed out of reach. To

avoid this difficulty, many authors considered a two-dimensional second-order PDE which is obtained from

the original one through a change of variable (similarity reduction method) when the contingent claim final

payoff has a particular form (see [2, 6, 7, 11, 20, 24, 25]). This method covers a large set of contingent claim

contracts, including arithmetic Asin options, but not a contingent claim characterized by a general final

payoff.

The Black-Scholes PDE for the fixed strike arithmetic Asian options, considering dividend (see, [17], pp.

277-278) is

∂V

∂t
+
S −J
t

∂V

∂J
+
σ2

2
S2
∂2V

∂S2
+ (r −q) S

∂V

∂S
−rV = 0 (1.1)

V (S,J,T) =


 (J −K)

+, (call option with fixed strike price)

(K −J)+, (put option with fixed strike price)
. (1.2)

V denote the price of an Asian option, S is the value of the underlying asset, K the exercise price, T the

expiration date, r is the interest rate, q is the continuously paid dividend, σ is the asset volatility, and

J = 1
t

t∫
0

Sτdτ be the path variable, denote the average of asset price for the period up to t.

It is well known that the difference between the prices of European vanilla call and put options is equal to a

European forward contract. Similarly, we have put-call parity relations for European style Asian options [16].

Let Cfix (S,J,t) and Pfix (S,J,t) denote the price of the fixed strike arithmetic averaging Asian call

option and put option, respectively. Their terminal payoff functions are given by

Cfix (S,J,T) = (J −K)+ = max (J −K, 0) (1.3)

Pfix (S,J,T) = (K −J)+ = max (K −J, 0) (1.4)

Let W (S,J,t) denote the difference of Cfix and Pfix . Since both Cfix and Pfix are governed by the

same differential equation [see (1.1)], so does W (S,J,t) .The terminal condition of their difference W (S,J,t)

is given by

W (S,J,T) = (J −K)+ − (K −J)+ = J −K.



Int. J. Anal. Appl. 18 (3) (2020) 515

Hence the problem we have to investigate is

∂W

∂t
+
S −J
t

∂W

∂J
+
σ2

2
S2
∂2W

∂S2
+ (r −q) S

∂W

∂S
−rW = 0, (1.5)

W |t=T = (J −K)
+ − (K −J)+ = J −K.

In a similar manner, we have other PDEs of put-call parities of other types of Asian options (floating/fixed

and arithmetic/geometric averaging) for the the investigation will see in section 3.

In this article, the fundamental objective is to investigate all four versions of PDEs of put-call parities

of Asian options using Reduced Differential Transform Method(RDTM) to validate the method. RDTM

proposed by Keskin [12] and successfully employed to solve many types of linear and nonlinear PDEs.

RDTM is a reliable semi-analytic method subject to appropriate initial condition. Taking into consideration

of this method, it is possible to find an exact solution or a closed approximate solution of a differential

equation.

2. Reduced Differential Transform Method

Consider a function of two variables u(x,t) and suppose that it can be represented as a product of two

single-variable functions, i.e., u(x,t) = f(x)g(t). Based on the properties of one dimensional differential

transform, the function u(x,t) can be represented as follows:

u(x,t) =

(
∞∑
i=0

F(i)xi

)(
∞∑
i=0

G(j)tj

)
=

∞∑
k=0

Uk(x)t
k, (2.1)

where Uk(x) is called t-dimensional spectrum function of u(x,t). The basic definitions of RDTM are intro-

duced as follows (cf. [12–15]):

Definition 2.1. If function u(x,t) is analytic and differentiated continuously with respect to time t and

space x in the domain of interest, then let

Uk(x) =
1

k!

[
∂k

∂tk
u(x,t)

]
t=0

(2.2)

where the t-dimensional spectrum function Uk(x) is the transformed function. In this paper, the lowercase

u(x,t) represents the original function, while the uppercase Uk(x) stands for the transformed function.

Definition 2.2. The differential inverse transform of Uk(x) is defined as follows:

u(x,t) =

∞∑
k=0

Uk(x)t
k (2.3)

Then, combining Eqs. (2.2) and (2.3) we write

u(x,t) =

∞∑
k=0

1

k!

[
∂k

∂tk
u(x,t)

]
t=0

tk (2.4)



Int. J. Anal. Appl. 18 (3) (2020) 516

from the above definitions, it can be found that the concept of the RDTM is derived from the power series

expansion.

To illustrate the basic concepts of the RDTM, consider the following nonlinear partial differential equation

written in an operator form

Lu(x,t) + Ru(x,t) + Nu(x,t) = g(x,t), (2.5)

with initial condition u(x, 0) = f(x),where L = ∂
∂t

, R is a linear operator which has partial derivatives,

Nu(x,t) is a nonlinear operator and g(x,t) is an in-homogeneous term. According to the RDTM, we can

construct the following iteration formula:

(k + 1) Uk+1 (x) = Gk (x) −RUk (x) −NUk (x) , (2.6)

where Uk (x) , RUk (x) , NUk (x) and Gk (x) are the transformations of the functions Lu (x,t) , Ru (x,t) ,

Nu (x,t) and g (x,t) respectively.

From the initial condition, we write

U0 (x) = f (x) . (2.7)

Substituting (2.7) into (2.6) and by a straightforward iterative calculation, we get the following Uk (x)

values. Then the inverse transformation of the set of values {Uk (x)}
n
k=0 gives approximation solutions as,

ũn (x,t) =

n∑
k=0

Uk (x) t
k, (2.8)

where n is the order of approximation solution.

Therefore, the exact solution of problems given by

u (x,t) = lim
n→∞

ũn (x,t) (2.9)

The fundamental mathematical operations performed by RDTM can be readily obtained and are listed

in following table.

Table 1. Reduced Differential Transformation




Functional Form Transformed Form

u (x,t) Uk (x) =
1
k!

[
∂k

∂tk
u (x,t)

]
t=0

w (x,t) = u (x,t) ± v (x,t) Wk (x) = Uk (x) ± Vk (x)

w (x,t) = αu (x,t) Wk (x) = αUk (x) , α is a constant

w (x,t) = xmtn Wk (x) = x
mδ (k − n) , the Kronecker delta

w (x,t) = xmtnu (x,t) Wk (x) = x
mUk−n (x) , when k ≥ n else 0.

w (x,t) = u (x,t) v (x,t) Wk (x) =
k∑

r=0
Vr (x) Uk−r (x) =

k∑
r=0

Ur (x) Vk−r (x)

w (x,t) = ∂
r

∂tr
u (x,t) Wk (x) = (k + 1) . . . (k + r) Uk+r (x) =

(k+r)!
k!

Uk+r (x)

w (x,t) = ∂
∂x
u (x,t) Wk (x) =

∂
∂x
Uk (x)






Int. J. Anal. Appl. 18 (3) (2020) 517

3. Pricing of Four Versions of Call-Put Parities of Asian Options

In this section, we will price call-put parities standard PDEs of four version of Asian options [17], that

are:

1) Arithmetic average Asian option with fixed strike price.

2) Geometric average Asian option with fixed strike price.

3) Arithmetic average Asian option with floating strike price.

4) Geometric average Asian option with floating strike price.

by Reduced Differential Transform Method to validate the efficiency of the RDTM.

3.1. Call-Put Parity for Arithmetic Average Asian Option with Fixed Strike Price.

Assume that C (S,J,t) and P (S,J,t)denote the valuation of an Asian call and put option, respectively.

define

W (S,J,t) = C (S,J,t) −P (S,J,t)

Then in {0 ≤ S < ∞, 0 ≤ J < ∞, 0 ≤ t ≤ T} ,W satisfies

∂W

∂t
+
S −J
t

∂W

∂J
+
σ2

2
S2
∂2W

∂S2
+ (r −q) S

∂W

∂S
−rW = 0, (3.1)

W |t=T = (J −K)
+ − (K −J)+ = J −K.

By the use of change of variable,

ξ =
TK − tJ

S

the function

w =
T

S
W

satisfies the Cauchy problem [17] in the domain {ξ ∈ R, 0 ≤ t ≤ T} :

∂w

∂t
+
σ2

2
ξ2
∂2w

∂ξ2
− [(r −q) ξ + 1]

∂w

∂ξ
−qw = 0, (3.2)

w (ξ,T) = −ξ.

where

w = w (ξ,t)

For convenience to apply RDTM, we are assuming that w (ξ,t) = u (ξ,t) ,Thus

∂u

∂t
+
σ2

2
ξ2
∂2u

∂ξ2
− [(r −q) ξ + 1]

∂u

∂ξ
−qu = 0,

withu (ξ,T) = −ξ.

We take the change of variable τ = T − t, to convert terminal condition into initial condition, so

∂u

∂t
= −

∂u

∂τ



Int. J. Anal. Appl. 18 (3) (2020) 518

or

∂u

∂τ
=
σ2

2
ξ2
∂2u

∂ξ2
− [(r −q) ξ + 1]

∂u

∂ξ
−qu, (3.3)

withu |τ=0= −ξ.

where

u = u (ξ,τ)

According to the RDTM, we construct the following recurrence relation for the Eq. (3.3)

(m + 1) Um+1 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
Um (ξ) − [(r −q) ξ + 1]

∂

∂ξ
Um (ξ) −qUm (ξ)

Note: u = u (ξ,τ) → functional form and Um = Um (ξ) → transformed form.

For m = 0

U1 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U0 (ξ) − [(r −q) ξ + 1]

∂

∂ξ
U0 (ξ) −qU0 (ξ)

From the initial condition, we write

u (ξ, 0) = U0 (ξ) = −ξ

So

∂

∂ξ
U0 (ξ) = −1 ,

∂2

∂ξ2
U0 (ξ) = 0

Thus

U1 (ξ) = 1 + rξ

For m = 1

2U2 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U1 (ξ) − [(r −q) ξ + 1]

∂

∂ξ
U1 (ξ) −qU1 (ξ)

we get,

U2 (ξ) = −
1

2

[
r2ξ +

r2 −q2

r −q

]
For m = 2

3U3 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U2 (ξ) − [(r −q) ξ + 1]

∂

∂ξ
U2 (ξ) −qU2 (ξ)

we have,

U3 (ξ) =
1

3!

[
r3ξ +

r3 −q3

r −q

]
By an inductive argument we have following,

Um (ξ) =
(−1)m+1

m!

[
rmξ +

rm −qm

r −q

]
, wherem ≥ 0.



Int. J. Anal. Appl. 18 (3) (2020) 519

For the solution, differential inverse transform of Um (ξ) is defined as below:

u (ξ,τ) =

∞∑
m=0

Um (ξ) τ
m

u (ξ,τ) =

∞∑
m=0

(−1)m+1 τm

m!

[
rmξ +

rm −qm

r −q

]

u (ξ,τ) =

∞∑
m=0

(−1)m+1 τm

m!
rmξ +

∞∑
m=0

(−1)m+1 τm

m!

rm −qm

r −q

u (ξ,τ) = −ξ
∞∑
m=0

(−1)m τmrm

m!
−

1

r −q

∞∑
m=0

(−1)m τm

m!
(rm −qm)

u (ξ,τ) = −ξ
∞∑
m=0

(−rτ)m

m!
−

1

r −q

[
∞∑
m=0

(−rτ)m

m!
−
∞∑
m=0

(−qτ)m

m!

]

u (ξ,τ) = −ξe−rτ −
1

r −q
[
e−rτ −e−qτ

]
Here

ξ =
TK − tJ

S
,u (ξ,t) = w (ξ,t)

where τ in terms of t so above equation can be written as

w (ξ,t) = −
TK − tJ

S
e−r(T−t) −

1

r −q

[
e−r(T−t) −e−q(T−t)

]
∵ w (ξ,t) =

T

S
W ⇒ W =

S

T
w (ξ,t)

The exact solution, in closed form, is given by

C (S,J,t) −P (S,J,t) = W (S,J,t)

W (S,J,t) =

[
t

T
J −

S

(r −q) T
−K

]
e−r(T−t) −

S

(r −q) T
e−q(T−t)

which is same solution as obtained in [17]. Consider a six-month call option on stock. If S0 = K =

$145,r = 6,q = 3 and σ = 29.5 then graphs of W (S,J,t), for average stock price J ranging from $140 to

$150, can be found in Figure 1 on next page.

3.2. Call-Put Parity for Arithmetic Average Asian Option with Floating Strike Price.

Consider

W (S,J,t) = C (S,J,t) −P (S,J,t)

then under appropriate transformations, standard call-put parities satisfies the Cauchy problem [17] in the

domain {0 ≤ ξ < ∞, 0 ≤ t ≤ T}:

∂w

∂t
+
σ2

2
ξ2
∂2w

∂ξ2
+ [1 − (r −q) ξ]

∂w

∂ξ
−qw = 0, (3.4)

w |t=T = 1 −
ξ

T



Int. J. Anal. Appl. 18 (3) (2020) 520

Figure 1. Plot of W(S,J,t) when S0 = K = $145,r = 6, q = 3 and σ = 29.5

where

w = w (ξ,t)

Taking into consideration w (ξ,t) = u (ξ,t), for the sake of easy utility of RDTM, we have

∂u

∂t
+
σ2

2
ξ2
∂2u

∂ξ2
+ [1 − (r −q) ξ]

∂u

∂ξ
−qu = 0,

u (ξ,T) = 1 −
ξ

T

We take the change of variable τ = T − t, to convert terminal condition into initial condition, we have

∂u

∂τ
=
σ2

2
ξ2
∂2u

∂ξ2
+ [1 − (r −q) ξ]

∂u

∂ξ
−qu, (3.5)

u |τ=0= 1 −
ξ

T

Applying the RDTM to Eq. (3.5), we obtain the following recurrence equation

(m + 1) Um+1 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
Um (ξ) + [1 − (r −q) ξ]

∂

∂ξ
Um (ξ) −qUm (ξ)



Int. J. Anal. Appl. 18 (3) (2020) 521

For m = 0

U1 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U0 (ξ) + [1 − (r −q) ξ]

∂

∂ξ
U0 (ξ) −qU0 (ξ)

According to the initial condition, we can write

u (ξ, 0) = U0 (ξ) = 1 −
ξ

T

So

U1 (ξ) =
1

T
(rξ − 1) −q

For m = 1

2U2 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U1 (ξ) + [1 − (r −q) ξ]

∂

∂ξ
U1 (ξ) −qU1 (ξ)

having

U2 (ξ) = −
1

2!

[
1

T

{
r2ξ −

r2 −q2

r −q

}
−q2

]
For m = 2

3U3 (ξ) =
σ2

2
ξ2
∂2

∂ξ2
U2 (ξ) + [1 − (r −q) ξ]

∂

∂ξ
U2 (ξ) −qU2 (ξ)

we will obtained,

U3 (ξ) =
1

3!

[
1

T

{
r3ξ −

r3 −q3

r −q

}
−q3

]
By an inductive argument we have following,

Um (ξ) =
(−1)m+1

m!

[
1

T

{
rmξ −

rm −qm

r −q

}
−qm

]
, Form ≥ 0

For the solution, inverse differential transform of Um (ξ) is defined as below:

u (ξ,τ) =

∞∑
m=0

Um (ξ) τ
m

u (ξ,τ) =

∞∑
m=0

(−1)m+1

m!

[
1

T

{
rmξ −

rm −qm

r −q

}
−qm

]
τm

u (ξ,τ) = −
ξ

T

∞∑
m=0

(−rτ)m

m!
+

1

T (r −q)

∞∑
m=0

(−rτ)m − (−qτ)m

m!
+

∞∑
m=0

(−qτ)m

m!

u (ξ,τ) = −
ξ

T
e−rτ +

1

T (r −q)
(
e−rτ −e−qτ

)
+ e−qτ

u (ξ,τ) = −
ξ

T
e−rτ +

1

T (r −q)
e−rτ

(
1 −

1

T (r −q)

)
e−qτ

Here

ξ =
tJ

S
,u (ξ,t) = w (ξ,t)

where τ in terms of t so above equation can be written as

w (ξ,t) = −
tJ

ST
e−r(T−t) +

1

T (r −q)
e−r(T−t)

(
1 −

1

T (r −q)

)
e−q(T−t)

∵ w =
W

S
⇒ W (S,J,t) = Sw (ξ,t)



Int. J. Anal. Appl. 18 (3) (2020) 522

In closed form, the exact solution is given by

C (S,J,t) −P (S,J,t) = −
t

T
Je−r(T−t) +

S

T (r −q)
e−r(T−t)

(
1 −

1

T (r −q)

)
Se−q(T−t)

which is same as obtained in [17]. Consider a six-month call option on stock. If S0 = K = $145,r =

6,q = 3 and σ = 29.5 then graphs of W (S,J,t), for average stock price J ranging from $140 to $150, can

be found in Figure 2.

Figure 2. Plot of w(S,J,t) when S0 = K = $145,r = 6, q = 3 and σ = 29.5

3.3. Call-Put Parity for Geometric Average Asian Option with Fixed Strike Price.

Suppose that

W (S,J,t) = C (S,J,t) −P (S,J,t)

Thus in {0 ≤ S < ∞, 0 ≤ J < ∞, 0 ≤ t ≤ T} ,W satisfies the problem:

∂W

∂t
+ J

ln S − ln J
t

∂W

∂J
+
σ2

2
S2
∂2W

∂S2
+ (r −q) S

∂W

∂S
−rW = 0, (3.6)

W |t=T = (J −K)
+ − (K −J)+ = J −K.



Int. J. Anal. Appl. 18 (3) (2020) 523

By the transformation,

ξ =
t ln J + (T − t) ln S

T

in {ξ ∈ R, 0 ≤ t ≤ T} ,W satisfies [17]:

∂W

∂t
+
σ2

2

(
T − t
T

)2
∂2W

∂ξ2
+

(
r −q −

σ2

2

)(
T − t
T

)
∂W

∂ξ
−rW = 0, (3.7)

W |t=T = eξ −K.

where

W = W (ξ,t)

For convenience, we are assuming that W (ξ,t) = u (ξ,t), we get

∂u

∂t
+
σ2

2

(
T − t
T

)2
∂2u

∂ξ2
+

(
r −q −

σ2

2

)(
T − t
T

)
∂u

∂ξ
−ru = 0, (3.8)

u |t=T = eξ −K.

We are taking the change of variable τ = T − t, we have

∂u

∂τ
=
σ2

2

(τ
T

)2 ∂2u
∂ξ2

+

(
r −q −

σ2

2

)(τ
T

) ∂u
∂ξ
−ru, (3.9)

u |τ=0= eξ −K.

where

u = u (ξ,τ)

By applying the RDTM, we construct the following iteration formula to Eq. (3.9)

(m + 1) Um+1 (ξ) =
σ2

2T2
∂2

∂ξ2
Um−2 (ξ) +

(
r −q − σ

2

2

T

)
∂

∂ξ
Um−1 (ξ) −rUm (ξ)

Here we used the property if v (x,t) = xmtnu (x,t) then Vk (x) = x
mUk−n (x) when k ≥ n,else 0, on first

and second term.

By considering the initial condition, we can write

U0 (ξ) = e
ξ −K

For m = 0

U1 (ξ) = −r
(
eξ −K

)
For m = 1

U2 (ξ) =
1

2!

{(
r −q − σ

2

2

T

)
eξ + r2

(
eξ −K

)}
For m = 2

3U3 (ξ) =
σ2

2T2
∂2

∂ξ2
U0 (ξ) +

(
r −q − σ

2

2

T

)
∂

∂ξ
U1 (ξ) −rU2 (ξ)



Int. J. Anal. Appl. 18 (3) (2020) 524

we get,

U3 (ξ) =
1

3!

{
σ2

T2
eξ − 3reξ

(
r −q − σ

2

2

T

)
−r3

(
eξ −K

)}
For m = 3

4U4 (ξ) =
σ2

2T2
∂2

∂ξ2
U1 (ξ) +

(
r −q − σ

2

2

T

)
∂

∂ξ
U2 (ξ) −rU3 (ξ)

Substitute U1 (ξ) ,U2 (ξ) and U3 (ξ) in above equation, we get

U4 (ξ) =
1

4!


−4 σ

2

T2
reξ + 3

(
r −q − σ

2

2

T

)2
eξ + 6

(
r −q − σ

2

2

T

)
r2eξ + r4

(
eξ −K

)
and so on.

For the solution differential inverse transform of Um (ξ) is defined as below:

u (ξ,τ) =

∞∑
m=0

Um (ξ) τ
m

u (ξ,τ) = eξe


σ2τ3

6T2
+

(
r−q− σ

2

2

)
τ2

2T
−rτ



−Ke−rτ

∵ τ = T − t

and from

ξ =
t ln J + (T − t) ln S

T

we can write

eξ = J
t
T S

(T−t)
T

So finally, we have the exact solution, in closed form

W (S,J,t) =


J

t
T S

(T−t)
T e

(T−t)


σ2

6 (
T−t
T )

2
+

(
r−q− σ

2

2

)
2 (

T−t
T )



−K


e−r(T−t)

which is the same result as obtained in [17]. Consider a six-month call option on stock. If S0 = K =

$145,r = 6,q = 3 and σ = 29.5 then graphs of W (S,J,t), for average stock price J ranging from $140 to

$150, can be found in Figure 3 on next page.

3.4. Call-Put Parity for Geometric Average Asian Option with Floating Strike Price.

Let

W (S,J,t) = C (S,J,t) −P (S,J,t)

Thus in {0 ≤ S < ∞, 0 ≤ J < ∞, 0 ≤ t ≤ T} ,W satisfies

∂W

∂t
+ J

ln S − ln J
t

∂W

∂J
+
σ2

2
S2
∂2W

∂S2
+ (r −q) S

∂W

∂S
−rW = 0, (3.10)

W |t=T = (S −J)
+ − (J −S)+ = S −J.



Int. J. Anal. Appl. 18 (3) (2020) 525

Figure 3. Plot of w(S,J,t) when S0 = K = $145,r = 6, q = 3 and σ = 29.5

under the suitable transformations,the function u satisfies the Cauchy problem [17] in the domain{
(x,y) ∈ R2, 0 ≤ t ≤ T

}
:

∂u

∂t
= rW −

(
r −q −

σ2

2

)[
T − t
T

∂u

∂y
+
∂u

∂x

]
−
σ2

2

∂2u

∂x2

−
σ2

2

(
T − t
T

)2
∂2u

∂y2
−σ2

(
T − t
T

)
∂2u

∂x∂y
(3.11)

u |t=T = ex −ey.

Taking the change of variable τ = T − t, we have

∂u

∂τ
=

σ2

2

(τ
T

)2 ∂2u
∂y2

+ σ2
(τ
T

) ∂2u
∂x∂y

+
σ2

2

∂2u

∂x2

+

(
r −q −

σ2

2

)[
τ

T

∂u

∂y
+
∂u

∂x

]
−ru (3.12)

u |τ=0= ex −ey.

where

u = u (x,y,τ)



Int. J. Anal. Appl. 18 (3) (2020) 526

Using the RDTM to Eq. (3.12), we get the following iterative formula

(m + 1) Um+1 (x,y) =
σ2

2T2
∂2

∂y2
Um−2 (x,y) +

σ2

T

∂2

∂x∂y
Um−1 (x,y) +

σ2

2

∂2

∂x2
Um (x,y) +(

r −q −
σ2

2

)[
1

T

∂

∂y
Um−1 (x,y) +

∂

∂x
Um (x,y)

]
−rUm (x,y)

with the initial condition

u (x,y, 0) = U0 (x,y) = e
x −ey

For m = 0

U1 (x,y) = 0 + 0 +
σ2

2

∂2

∂x2
U0 (x,y) +

(
r −q −

σ2

2

)[
0 +

∂

∂x
U0 (x,y)

]
−rU0 (x,y)

Thus

U1 (x,y) = −qex + rey

For m = 1

2U2 (x,y) = 0 +
σ2

T

∂2

∂x∂y
U0 (x,y) +

σ2

2

∂2

∂x2
U1 (x,y) +(

r −q −
σ2

2

)[
1

T

∂

∂y
U0 (x,y) +

∂

∂x
U1 (x,y)

]
−rU1 (x,y)

So

U2 (x,y) =
1

2!

{
q2ex −ey

(
r −q − σ

2

2

T
+ r2

)}
For m = 2

3U3 (x,y) =
σ2

2T2
∂2

∂y2
U0 (x,y) +

σ2

T

∂2

∂x∂y
U1 (x,y) +

σ2

2

∂2

∂x2
U2 (x,y) +(

r −q −
σ2

2

)[
1

T

∂

∂y
U1 (x,y) +

∂

∂x
U2 (x,y)

]
−rU2 (x,y)

So

U3 (x,y) =
1

3!

{
−q3ex −ey

(
σ2

T2
− 3

r −q − σ
2

2

T
r −r3

)}
For m = 3

4U4 (x,y) =
σ2

2T2
∂2

∂y2
U1 (x,y) +

σ2

T

∂2

∂x∂y
U2 (x,y) +

σ2

2

∂2

∂x2
U3 (x,y) +(

r −q −
σ2

2

)[
1

T

∂

∂y
U2 (x,y) +

∂

∂x
U3 (x,y)

]
−rU3 (x,y)

we have

U4 (x,y) =
1

4!


q4ex + 4 σ

2

T2
rey − 6

(
r −q − σ

2

2

T

)
r2ey − 3

(
r −q − σ

2

2

T

)2
ey −r4ey






Int. J. Anal. Appl. 18 (3) (2020) 527

For the solution, inverse differential transform of Um (x,y) is defined as below:

u (x,y,τ) =

∞∑
m=0

Um (x,y) τ
m

u (x,y,τ) = exe−qτ −eye


σ2τ3

6T2
+

(
r−q− σ

2

2

)
τ2

2T
−rτ




∵ x = ln S ⇒ ex = S

and

y =
t ln J + (T − t) ln S

T
⇒ ey = J

t
T S

T−t
T

also

τ = T − t

u (x,y,t) = Se−q(T−t) −J
t
T S

T−t
T e


σ2(T−t)3

6T2
+

(
r−q− σ

2

2

)
(T−t)2

2T
−r(T−t)




Thus we reached to the exact solution, in closed form

C (S,J,t) −P (S,J,t) = Se−q(T−t) −J
t
T S

T−t
T e

σ2(T−t)3

6T2
+

(
r−q− σ

2

2

)
(T−t)2

2T
−r(T−t)

which is the same solution as obtained in [17]. Consider a six-month call option on stock. If S0 = K =

$145,r = 6,q = 3 and σ = 29.5 then graphs of W (S,J,t), for average stock price J ranging from $140 to

$150, can be found in Figure 4 on next page.



Int. J. Anal. Appl. 18 (3) (2020) 528

Figure 4. Plot of w(S,J,t) when S0 = K = $145,r = 6, q = 3 and σ = 29.5

4. Conclusion

The Reduced Differential Transform Method (RDTM) for put-call parities PDEs of Asian options has

been presented. The solutions obtained by the method are an infinite power series for appropriate initial

condition, which can in turn in a closed form, the exact solution. The efficiency of the presented method is

validated through all four versions of PDEs of Asian options put-call parities and found exact solutions same

as Lishang solutions.We notice that the RDTM technique is highly accurate, rapidly convergent. RDTM is a

very easily implementable mathematical tool for the PDEs in Mathematical Finance subject to appropriate

initial condition.

Availability of Data and Materials: This paper contains no any studies with human participants or

animals performed by any of the authors. All the data related to the current study were provided along with

this paper.

Authors’ Contributions: Both authors contributed equally to the manuscript and typed, read, and

approved the final manuscript.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 18 (3) (2020) 529

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	1. Introduction
	2. Reduced Differential Transform Method
	3. Pricing of Four Versions of Call-Put Parities of Asian Options
	3.1. Call-Put Parity for Arithmetic Average Asian Option with Fixed Strike Price
	3.2. Call-Put Parity for Arithmetic Average Asian Option with Floating Strike Price
	3.3. Call-Put Parity for Geometric Average Asian Option with Fixed Strike Price
	3.4.  Call-Put Parity for Geometric Average Asian Option with Floating Strike Price

	4. Conclusion
	References