3. Daftardar-Gejji Method
The Daftardar-Gejji method was developed from the Adomian decomposition and is more efficient because mathematicians are not required to formulate Adomian polynomials beforehand. According to Ref. [
27], the Daftardar-Gejji method was applied to solve the multispecies Lotka–Volterra Equation.
When
is written as a series of
and has a linear property, then
The non-linear decomposition is defined below:
where
; the equation becomes:
Since
can be defined as an addition from the non-linear and linear operator, then:
Based on the procedure above, the following recursive formula can be attained:
The following is an example of how the Daftardar-Gejji method is used to effectively solve a fractional differential equation.
Example: solve the following fractional diffusion equation:
with initial conditions
Answer.
By integrating the Riemann–Liouvile integral, both fractional sides of Equation (1) are obtained
Using the Daftardar-Gejji method obtained,
In the same way, it is obtained that
So, the following solution is obtained:
Before attempting to solve the fractional Black–Scholes equation, it is important to first explore the existence and uniqueness of its solution. This is explained in Theorem 2.
Theorem 3. (solution existence). The fractional Black–Scholes equation ( imposed on the Caputo operator has a unique solution.
Proof. The idea of the proof concerns the method of the Picard–Lindelöf theorem. The existence of a solution to the fractional Black–Scholes equation must prove that
must satisfy the Lipschitz condition of continuous functioning. Next, the
operator given as
is bounded. Meanwhile, the uniqueness of the solution must be shown
that it is a contractive mapping. The following is a step to prove the existence and uniqueness of the solution to the fractional Black–Scholes equation. □
Given the fractional Black–Scholes equation
(
, according to the Caputo operator, the equation is defined as follows:
The equation further becomes
Based on the definition above and using Riesz–Minkowski inequality [
18], the following was formulated:
with
.
where is a Lipschitz continuous function.
The equation was further defined as follows:
Let and
When there is
with definition:
The value of
bounded needs to be proven first:
Take
contractive mapping when:
The following is an example of searching for the fractional Black–Scholes equation solution using the Daftardar-Gejji method.
The Black–Scholes equation with initial conditions for a valued European call option
is:
where
for
and
where
is the volatility of the underlying asset
,
is the exercise price,
is the expiry time and is the risk-free interest rate. By using an analogy and the chain rule
,
, another form of the Black–Scholes equation is obtained, namely:
Example 1. Given the fractional Black–Scholes equation:where the following initial conditions are: Based on the Daftardar-Gejji method, the following equation was obtained:
.
Before solving
, the following needs to determined
Based on the Daftardar-Gejji method obtained:
Subsequently, the fractional Black–Scholes equations were solved as follows:
By taking the exact solution obtained from the Black–Scholes equation, namely
Using Python 3.7 software, the simulation graph of Equation (3) is obtained for the values given in
Figure 1, where
.
Figure 1 shows the graph obtained in the form of a cylinder. If the value is fixed at [0.5] and the value runs at [0.5], the monotonic graph decreases sharply. The maximum value in
Figure 1 is
298, obtained when
and
; the minimum value
, obtained when
and
.
Figure 2 shows the graph for
, which is similar to graphics
for
. However, the graph for
tends to be more concave than the graph
with
.
Figure 2 shows that the monotonic graph decreases sharply to 0, less than or equal to t less, than or equal to 5, and the value is fixed at [0.5]. The maximum value
fixed at [0.5] is a monotone graph that drops sharply. A maximum value of
298 was obtained when
, while the minimum value
when
and
.
From
Figure 1 and
Figure 2, it can be concluded that in the initial conditions,
, and the value
is large enough; the value
also had a significant value. And vice versa, for the initial value,
, and as time goes by, the value
.
Based on
Figure 1 and
Figure 2, different values of α will determine different solutions and graphs. Then, based on the α values, a graph drawn in three dimensions and cut for certain
and
values obtained graphs in two dimensions. So, the graphs shown in
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7 and
Figure 8 could be obtained. For example, in
Figure 3, α = 0.0, 0.25, 0.5, 0.75, and 1.0,
= 0.1, x = 5.0, and 0 ≤
≤ 1.0.
Figure 3 shows that the maximum value at
. What is quite interesting here is that for
and
, the graph
for
shows that the interval [0.0.85] is above the graph
for
. When expressed in economic language, the value of the option for
is greater than that
of the interval [0.0.85]. What happens for
is the opposite. The value
for
is actually greater than the value
for
, so the value of the option
is greater compared with
on
.
The same is the case for and values on the interval [0, 0.975] and On the other hand, the value on the interval [0, 0.95] for is greater than the value for the values = 0.0, 0.25, 0.5, and 0.75, while the value is lower than the value with = 0.0, 0.25, 0.5, and 0.75.
Figure 4 shows that the t-axis is expanded to
Based on
Figure 3, the behavior of the graph
becomes more interesting. In
Figure 3, at intervals
, graph
, graph
is located at the top. Then, at interval
, the graph
is at the bottom. Meanwhile, graphics
at the interval
are located above the graph
Meanwhile, at interval
, the graph is located at the bottom. Likewise, for graph
at interval [0, 0.85], it is located above graph
Meanwhile, at interval
, it is quite the opposite. Graphics
are precisely above the graph
.
It is interesting to show the behavior
for the value of the environment
.
Figure 5 and
Figure 6 demonstrate the behavior for values of
close to 1.
Figure 5 is taken for values of
, and
Figure 6 is taken for the value
, with values
= 0.0, 0.25, 0.5, 0.75, and 1.0.
Figure 5 shows that, at interval 0
, it is clear that the maximum value
at
, in which
, is interesting enough. For
and
, the graph
for
at the interval [0, 0.85] is above the graph for
for
. When expressed in economic language, the option value for
is greater than
at the interval [0, 0.85]. At the interval [0.85, 1], the opposite happens, i.e., the value of
for
is greater than the value
for
Therefore, the option value is greater for
compared to
at the interval [0.85, 1].
For and at the interval , the value of for is greater than the value of for . However, for the value of at the interval [0.975, 1], the value of for is much larger than the value of for The same method is used for and values for the interval [0, 1.0] and On the other hand, the value of at the interval [0, 0.95] for is larger than the value of for the values = 0.0, 0.25, 0.5, and 0.75; while at the interval (0.95,1.0), the value of is lower than the value of = 0.0, 0.25, 0.5, and 0.75. For the value of , the value of .
To more clearly see the behavior of the graph
the value of t is expanded to
. It is quite interesting to note that based on
Figure 6 obtained for
, the graph
, which was originally at the interval [0, 0.975] with
, is at the top. However, at the interval [0.975, 100], the graph is the lowest and
is the lowest. Meanwhile, the graph
for
at the interval [0, 0.985] was originally at the top compared to the graph for
for
0.25 and
. At the interval [0.985, 100], the graph is below the graph of
for
and
.
Figure 6 shows clearer behavior of the graph
.
Figure 7 shows the maximum value
at
. The behavior of
is interesting enough for
and
; the graph
for
at the interval [0, 0.87] is above the graph for
for
. This means that the option value for
is greater than
at the interval [0, 0.87]. At the interval [0.87, 2], the opposite occurs, namely the value of
for
is actually greater than the value
for
Thus, the option value is greater for
compared to
at the interval [0.87, 2].
The same method was used for values and at the intervals [0,1] and [1,2] as for values of and at the intervals [0, 1.06] and [1.06, 2]. On the other hand, the value of at the interval [0, 1.06] for is larger than the value of for the values = 0.25, 0.5, and 0.75; at the interval [1.06, 2], the value of is lower than the value of = 0.0, 0.25, 0.5, and 0.75.
To further clarify the behavior of graph
on the axis
, the interval is expanded to
Based on
Figure 8, it is clear that the maximum value
at the moment
. The graph shows that the highest
is for the value of
. As for the graph
with the value of
when
, a very sharp decline was experienced. So, the graph
for the value
in the interval
is at the bottom. The values of
. For the graph
with value
, after
, it monotonically decreases towards the value of
.
Here is the graph for
by taking
with the value of
Somewhat different from the graph
when x is fixed, graph
for fixed t values
tends to coincide at the interval [0,6]; then, at the interval
, graph
tends to spread and rise monotonously. Graph
for the value of
occupies the lowest position compared to the others. The maximum value for
with value
on
is 40,000. Following the value
for the value of
, the position of the graph is slightly lower than the graph of the
The maximum value
. Furthermore, the graph
for the value of
. The height of this graph occupies the third position below the graph
for
and
The behavior of these graphs is relatively the same, i.e., they overlap first and then begin to spread out
The difference is that this graph is more convex compared to the graph
above it with the maximum value
It is followed by the graph
for the value of
with the maximum value of
= 15,000. The last is graph
for the value of
, which is the lowest compared to the graph
with the maximum value of
Figure 9 shows the same pattern of behavior for graphs
, i.e., the graph is rising and cupped.
The Daftardar-Gejji method was used to solve several problems involving the fractional Black–Scholes equation. The following examples illustrate the procedure for using the method to determine the solution.
Example 2. The Daftardar-Gejji method was used to solve the fractional Black–Scholes equation of ,with the following initial condition of Solution:
Furthermore, the solution to the fractional Black–Scholes equation is:
Using Python 3.7 software, the graphical representation of the solution in Example 2 is expressed as follows:
Figure 10 shows that for
, with the value of
fixed at [0,5], the graph is obtained to be steadily increasing. The maximum value of
4.9 is obtained when
and the minimum value
when
and
Based on
Figure 10, different values of α will determine different solutions and graphs. Then, based on the α values, a graph drawn in three dimensions and cut for certain x and t values obtained graphs in two dimensions. Thus, the graphs in
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16 can be obtained. For example, in
Figure 11, α = 0.0, 0.25, 0.5, 0.75, and 1.0, k = 0.5, x = 5.0, and 0 ≤ t ≤ 5.0.
Figure 11 shows that the maximum value
at
.
Figure 11 shows the graph of
for
and
; of note, at the interval
, the graph
for
is above the graph
for
. The graph
for
is marked with a light blue color, whereas the graph
for
is marked with red color. This means that the option value for
is greater than
at the interval [0, 1]. Meanwhile, at the interval [1,5], the opposite is true, namely the value of
for
is actually greater than the value
for
This is indicated by the blue graph, which is higher than the red graph. Therefore, the option value is greater for
compared to
at the interval [1,5].
For and at the interval [0, 0.7], the value of for is greater than the value for . However, for the value at the interval [0.7, 5], the value of for is much larger than the value of for This is indicated by the green graph, which is higher than the yellow graph. As for the values of and at the interval [0, 1], the value of for is greater than for However, at the interval [1,5], the opposite happens, namely the value of for is greater than the value of for On the other hand, the value of at the interval [0, 0.8] for is larger than the value of for the values = 0.25, 0.5, 0.75, and 1.0. Meanwhile, at the interval [0.8,5], the value of is lower than the value of = 0.25, 0.5, 0, and 1.0, as indicated by the blue graph, which is the lowest compared to other graphs.
Figure 12 shows that the graph is made by expanding the interval
. It aims to see the behavior of the graph
at interval t, which is longer. It shows that the maximum value
at
, which is marked with a light blue graph, and the location of the graph is the highest. Moreover, for
and
, the graph
for
at the interval [0, 1] is above the graph for
for
. Thus, it can be interpreted that the option value for
is greater than
at the interval [0, 1]. However, at the interval [1,25], the opposite occurs, namely the value of
for
is actually greater than the value
for
This is indicated by the red graph, which is higher than the green graph at the interval [1,25]. Therefore, the value of the option is greater for
compared to
in the interval [1,25].
By using the same method for values and values on the interval [0, 0.7] and [0.7, 25]This also applies to the values and at the intervals [0, 1] and [1,25]. On the other hand, the value of at the interval [0, 0.8] for is larger than the value of for the values = 0.0, 0.25, 0.5, and 0.75; at the interval [0.8,25], the value of is lower than the value of = 0.0, 0.25, 0.5, and 0.75.
The conclusion drawn from
Figure 11 and
Figure 12 is that for a long period of time and if the value of
is sufficiently large enough, the value of
is also large.
Then,
Figure 13 shows that the maximum value of
at
. It is quite interesting that for
and
, the graph
for
at the interval [0, 0.8] is below the graph of
for
. This means that the option value for
is greater than
at the interval [0, 0.8]. Meanwhile, at the interval [0.8, 5], the opposite occurs, namely the value of
for
is actually smaller than the value
for
Therefore, the option value is greater for
compared to
in the interval [0.8, 5].
For and at the interval [0, 0.7], the value of for is greater than the value for . However, for the value of at the interval [0.7, 5], the value of for is larger than the value of for As for the values of and at the interval [0, 1], the value of for is greater than for However, at the interval [1,5], the opposite happens, namely the value of for is greater than the value of for On the other hand, the value of at the interval [0, 1] for is greater than the values of other , namely for the values = 0.25, 0.5, 0.75, and 1.0.
Figure 14 shows that the axis
was expanded to
Furthermore, the graph behavior
becomes more interesting. If at interval
the graph
for
lies topmost, then at interval
, the precise graphic
for
will exceed the graph
with value
Meanwhile, the graphic
for
at the interval
is located above the graph
on
at interval
instead, the graph is located below. Likewise for the graph
for
at interval
, it is located above the graph
for
quite the opposite. Graph
for
is precisely above graphic
for
Figure 15 shows a graphic image
with
,
with the value of
being fixed, namely
The graph
with
is a straight line. The graph
is the highest compared to the graph
with other values. The maximum value of
= 20, with a value of
achieved when
Then, the graph
for
is a straight line. The graph is located below the graph
for
. The maximum value of
is reached when
Furthermore, the graph
for the value of
is a straight line. The location of the graph is, respectively, located below the graph
for values
and
. For the graph
for
the shape of the graph is a straight line. The maximum value of
is reached when
For the graph
for
the graph is a straight line with the maximum value
achieved when
. Then, the last graph has the lowest height compared to the height of the other graphs, namely the graph
with value
, which has a maximum value
achieved when
Figure 16 shows a graphic image
with
,
in which the value of
is made fixed, namely
The graph
with
is a straight line. The location of the graph
is the highest compared to the graph
with other values. The maximum value of
= 78, with a value of
, is achieved when
. The graph
for
is a straight line. The graph is located below the graph
for
. The maximum value of
is reached when
The graph
for the value of
is a straight line. The location of the graph is located below the graph
for values
and
. The maximum value of
is reached when the value of
The graph
for the value of
, unlike the other graphs
is a straight line. The graph
with the value of
at the interval
is a straight line. At
the graph turns slightly up, although the shape of the graph remains a straight line. The maximum value of
is reached when
. The last graph
for
has the lowest height compared to the other graphs
. This graph has similarities with the graph
for the value of
. The behavior of the graph
for the value of
at the interval
is a straight line. When
, the graph turns up in the form of a straight line. The maximum value
45 is achieved when
From the fractional Black–Scholes equation, if taken for , the solution to the Black–Scholes equation approach is obtained. The solution is an iterative solution, in the form of an infinite series of the Mittag-Lefler function. The following shows a graph of the solution to the fractional Black–Scholes equation approximation by taking and a graph of the exact solution to the Black–Scholes equation.
It can be clearly seen from
Figure 17 and
Figure 18 that the obtained solutions almost coincide. Therefore, the resulting error is very small, at approximately 0 percent. This means that the exact solution to the Black–Scholes equation can be approached as a solution to the fractional Black–Scholes equation, namely in the form of an infinite series of the Mittag-Leffler function by taking
.
Table 1 shows
values for the exact solution to the Black–Scholes equation, approximate solutions, and error values. The error formula used is
%.
Table 1 shows that the value of
is made to change, namely
and
while for the value of
the error values obtained are
and 3.507089639660298 ×
.
The value fixed is the value of , while the values of changed are and . The error values obtained are and . Meanwhile, for the value of for the value of , the same error value is 0.0%. For the value of and for the value of , the error values obtained are 1.8420923999599334 × and 1.677826322045785 × . Therefore, it can be concluded that the solution to the fractional Black–Scholes equation using the Daftardar-Geiji method is very good since the error value obtained is very small, almost 0.0%. It is interesting if the value of is taken close to 1.
The following table shows approximate solutions, exact solutions, and error values by taking the value of , which is close enough to 1: .
From
Table 2, with
values that change, namely
, and
values kept constant, namely
, it is found that error values tend to decrease, namely three in a row: 74%, 2.36%, and 1.68%. Likewise, if the value of
is fixed, namely
and the value of
is varied, namely
the error values obtained are 3.74% 3.35%, and 2.69%, respectively. The error obtained is still very good since it is less than 5%.
From
Table 3, with
values that change, namely
and
and with the
value being fixed, namely
, it is found that the error value tends to decrease, namely 6.35%, 4.02%, and 2.86%. Likewise, if the value of
is fixed, namely
, and the value of
is varied, namely
error values of 6.35%, 5.64%, and 4.49%, respectively, are obtained. The error values obtained tend to decrease, but it can be said that this is still good since it is less than 10%.
From
Table 4, for the
values that change, namely
and 0.3, and with the
values made constant, namely
it is found that the error values tend to decrease, namely 13.35%, 8.45%, and 6.01%, respectively. Even though the error tends to decrease, an error value of 13.35% is not good for estimating the value of
because it is more than 10%. Likewise, if the value of
is fixed, namely
and the value of
is varied, namely
, the error values of 13.35%, 11.51%, 8.99%, and 6.72%, respectively, are obtained. For an error value of 13.35%, 11.51% is not good enough to estimate the value of
since it is above 10%.
From
Table 1,
Table 2,
Table 3 and
Table 4, it can be concluded that the closer the value of
is to 1, the closer the graph obtained is to the graph of the solution to the Black–Scholes equation. This is reinforced by the graph below:
Figure 19 shows that the graph
for the value of
, which is closer to 1, is becoming closer to the exact solution. This can be shown through the red graph that is close to the Black–Scholes equation solution graph. The red graph is the graph
for the value of
. Meanwhile, the graph
is shown in purple. It is clear that the graph with the value
shown in blue is still far from approaching the exact solution, namely the solution to the Black–Scholes equation. Then, the graph with the value of
, which is yellow, is located below the blue graph. Meanwhile, the graph with the value
, which is green, moves closer to the graph of the exact solution to the fractional Black–Scholes equation. Finally, the red graph has the value
The error comparison produced using the analysis Shehu transform method [
28] is as follows:
According to
Table 5, both the Daftardar-Geijji method approach and the homotopy analysis Shehu transform method produce the same error, which is relatively minimal. The two strategies were shown to be successful in solving the fractional Black–Scholes equation.
The value of European call options can be expressed as
with
being the exercise price. As a result, the produced graph is shown in
Figure 20 and
Figure 21.
The following is a theorem about the convergence of fractional Black–Scholes equation solutions and the upper limit for the absolute maximum error.
Theorem 4. Let Banach spaces and , . The approximation solution in the form of a series convergences to the solution to the Black–Scholes equation if for , i.e., for every there is such that for every .
Proof. Suppose that the partial sum terms .
Because and , take . It is proved that . This implies that . □
Theorem 5. Suppose that is finite and the exact solution is . If for , then the maximum absolute error is .
Proof . Based on the exact and the fractional Black–Scholes approach solutions, the difference norm is taken to find the upper bound of the absolute error value.
Based on Theorems 4 and 5, it can be concluded that the approximate solution to the fractional Black–Scholes equation converges to the exact solution and has an upper bound for the absolute error. □