Page 42 - IJOCTA-15-2

Analyzing the Black-Scholes equation with fractional coordinate derivatives using . . .
where γ = max{1, η}. We know that if i > j, where (x, t) ∈ (0, 1) × (0, 1), r = 0.05, σ =
β 2 2
b σ σ
β β i
then b > b . Hence, β ≤ 1. Therefore 0.25, A = , B = r− and C = r, u(0, t) = 0,
j
i
b 2 2 2
j u(1, t) = 0, u(x, 0) = x (1 − x) and
1
γ|r |t n−1 β β
n
X
|s | ≤ −1 + 2 b n−l−1 − b n−l
b β
n−1 l=1 2t 2−β 2t 1−β
2
f(x, t) = ( + )x (1 − x)
1 n−1
β γ|r | X β β Γ(3 − β) Γ(2 − β)
+ b ≤ −1 + 2 b − b
n−1 β n−l−1 n−l 2 2 (136)
b − (t + 1) (A(2 − 6x) + B(2x − 3x )
n−1 l=1
2
β − Cx (1 − x)).
+ b .
n−1
(129)
Consequently from Lemma 4, we have In this case, by considering β = 0.75, the
γ exact solution of the equation will be u(x, t) =
1
n
|s | ≤ β |r |, (130) (x(t + 1)) (1 − x).
2
b n−1
which completes the proof.
Theorem 5. The numerical scheme (41) for
solving the TFB-S model is convergent and the To measure the accuracy of the method, we
solution satisfies compute the errors norm (e N t ) and computa-
M x
b n
2
n
∥U − U ∥ ≤ C(∆t 2−β + h ), n = 1, 2, . . . , N t , tional orders (R N t ):
M x
(131)
b n
where U n and U are the approximate and exact
solutions of Eq. (41), respectively, and C is a e N t := ∥.∥ ∞ = max |U(x i , t j ) − u(x i , t j )|,
M x
positive constant. 0≤i≤M x
0≤j≤N t
(137)
Proof. By the Lemma 5
γ 1 n γ 1 and
n
|s | ≤ β |r |, and ∥E ∥ ≤ β ∥R ∥. (132)
b b
n−1 n−1
!
n
Furthermore, by Eq. (107), we have ∥R ∥ ≤ e N t
√ R N t M x . (138)
2
η D(∆t 2−β + h ). Therefore M x = log 2 e 2N t
γ √ 2M x
2
n
∥E ∥ ≤ η D(∆t 2−β + h ). (133)
b β
n−1 We measure the maximum numerical er-
√
γη D rors and computational orders in the computed
Thus by definition C = β , we have approximation with respect to the time vari-
b n−1 able. Table 1 shows the numerical results for
2
n
∥E ∥ ≤ C(∆t 2−β + h ), (134) β = (0.1, 0.5, 0.9) and the time variable N t =
(32, 64, 128, 256). In Table 2, the numerical er-
which completes the proof. rors and their computational orders for β = 0.7
and M x = 100 using the present method and the
discrete implicit numerical scheme presented in 36
are compared. In Table 3, the numerical results
5. Numerical feedbacks
for β = 0.7 and M x = 150 using the present
method and the method of radial basis functions
Three examples here will verify the performance 30
presented in are compared. It is clear that we
of the suggested scheme. obtain better results with the proposed algorithm.
This shows the high accuracy and efficiency of the
Example 1. Consider the following TFB-S proposed method. In addition, it can be observed
model: a13,ex11 from Tables 1-3 that the computational orders for
β
2
∂ u(x, t) ∂ u(x, t) ∂u(x, t) β = (0.1, 0.5, 0.7, 0.9) are 1.3, 1.4, 1.5, and 1.6, re-
= A + B − Cu(x, t) spectively. This shows the dependence of the com-
∂t β ∂x 2 ∂x putational orders on the order of the fractional
+ f(x, t), derivative. Figures 2-4 show the approximate so-
(135) lution for β = (0.5, 0.7, 0.9).
237