Page 33 - IJOCTA-15-2

M. M. Parsa, K. Sayevand, H. Jafari, I. Masti / IJOCTA, Vol.15, No.2, pp.225-244 (2025)
By placing τ = T − t and x = log(s) in Eqs. σ 2
where (x, t) ∈ (x L , x R ) × (0, T), A = ,
(3-5), we have 2
σ 2
B = r − , and C = r. On the other hand,
2
β
x
2
2
x
∂ υ(e , T − t) σ ∂ υ(e , T − t) ( u(x L , t) = λ(t),
− + + (15)
∂t β 2 ∂x 2 u(x R , t) = ζ(t),
x
σ 2 ∂υ(e , T − t) x (7)
(r − ) − rυ(e , T − t) = 0, where
2 ∂x u(x, 0) = υ(x). (16)
+
(x, t) ∈ R × (0, T),
Theorem 1. The solution of the TFB-S model
(14) with the BICs (15-16) exists and is unique.
with the following border and initial condi- Proof. See. 31
tions:
3. Method in action
(
υ(e −∞ , T − t) = λ(T − t),
(8) In this section, we outline the method used for
υ(e +∞ , T − t) = ζ(T − t), solving the Black-Scholes equation. To enhance
clarity and maintain the reader’s focus, the sec-
x
x
υ(e , T) = η(e ) := υ(x). (9) tion is divided into two subsections. In the first
subsection, we address the time discretization
process described in Eqs. (14-16). By introducing
x
Considering u(x, t) = υ(e , T −t), Eq. (7) will a series of theorems, we derive an equation for the
fractional time derivative. In the second subsec-
look like this
tion, we define the necessary spatial derivatives
β
2
2
∂ u(x, t) σ ∂ u(x, t)
= ( ) and proceed with location discretization. Finally,
∂t β 2 ∂x 2 (10) we present the Crank-Nicholson scheme, including
σ 2 ∂u(x, t)
+ (r − ) − ru(x, t), its computational molecule, to provide a clearer
2 ∂x
understanding of this method.
where (x, t) ∈ R × (0, T). With this assump- 3.1. Time discretization
tion, the boundary and initial conditions (BICs)
For time discretization, suppose that N t is the
Eqs. (8-9) also change as follows:
number of node points and k is the time step
( T
u(−∞, t) = λ(t), length defined as k = . In this case, we con-
(11)
u(+∞, t) = ζ(t), N t
sider the node points as 0 = t 0 < t 1 < t 2 < · · · <
u(x, 0) = υ(x), (12) t N t = T, where for n = 0, 1, 2, · · · , N t , we have
t n = nk.
where
β
∂ u(x, t) 1 d Z t u(x, α) − u(x, 0) Lemma 1. The Riemann-Liouville time deriva-
= dα. ∂ u(x, t)
β
∂t β Γ(1 − β) dt 0 (t − α) β tive that appears in (14) is equivalent to
(13) ∂t β
the Caputo fractional derivative of order β.
Proof. From Eq. (13), we have
As can be seen, the TFB-S model (10) has β Z t
∂ u(x, t) 1 d u(x, α) − u(x, 0)
an infinite domain R × (0, T). For the numerical β = β dα
∂t Γ(1 − β) dt 0 (t − α)
solution of the equation, we consider the finite Z t
1 du(x, α)
domain (x L , x R ) × (0, T). Therefore, the TFB-S = (t − α) −β dα
Γ(1 − β) dα
model will be as follows: 0
β
c
2
β
∂ u(x, t) ∂ u(x, t) = D u(x, t).
0
t
= A (17)
∂t β ∂x 2 (14) β
c
∂u(x, t) Here, D u(x, t) is the Caputo fractional deriva-
0
t
+ B − Cu(x, t) + f(x, t), tive (refer to 32–34 for more details).
∂x
228