A modified Black-Scholes (B-S) model with time fractional derivative is studied when the price change of the underlying is considered as a fractal transmission system. It is very practical in application to study the numerical computation of this time fractional B-S model (TFBSM) governing European options. This paper constructs mixed alternative segment Crank-Nicolson (MASC-N) parallel difference scheme, which accuracy is spatially 2 order and temporally 2-α order. In addition, we have provided a theoretical proof for the stability and convergence of the MASC-N scheme, and demonstrated the accuracy and effectiveness of the proposed method through numerical experiments. The numerical results show that the MASC-N scheme is easy to implement when applied to time fractional B-S model.
Option is one of the most widely used financial derivatives, and it has shown very important significance in both theory and practice to study the option pricing. The Black-Scholes (B-S) model, which was proposed by Black F., Scholes M. and Merton R. (1973) [
Through observation and research on the stock market, it is found that the most fundamental characteristics and basic state of the capital market are random fluctuations, and some complex dynamic systems are usually fractional, such as fractional Brownian motion, continuous time walk, etc. Based on the fractional stochastic differential equation driven by fractional Brownian motion, many scholars have made some progress in the research of fractional B-S equation in recent years. W. Wyss (2000) [
The analytical solution of the fractional B-S models is usually via integral transform methods [
For solving the time fractional B-S equation, a few achievements on the numerical methods are as follow. Song L.N. and Wang W.G. (2013) [
With the rapid development of multi-core and cluster technology, it is of great theoretical significance and practical value to construct the parallel difference scheme with good stability for solving time fractional B-S model (TFBSM). In combination with the call option, we construct the mixed alternative segment C-N (MASC-N) scheme for TFBSM. The stability and convergence of the MASC-N scheme are analyzed. Finally, numerical examples demonstrate the effectiveness and accuracy of the MASC-N scheme for solving the TFBSM.
In the present work, the option price P ( S , t ) is suggested to be subject to the time fractional Black-Scholes equation with the following form [
∂ α P ( S , t ) ∂ t α + 1 2 σ 2 S 2 ∂ 2 P ( S , t ) ∂ S 2 + r S ∂ P ( S , t ) ∂ S − r P ( S , t ) = 0. (1)
where t > 0 , 0 < α ≤ 1 , P is the price of European call option; S and t are asset price and time; r is risk free interest rate; σ represents volatility of underlying asset. The fractional derivative P t ( α ) ( S , t ) is Riemann-Liouville time fractional derivative with respect to t; P S and P S S refer to first derivative and second derivative with respect to S.
The value of European call option is taken as a solution of (1) with the following terminal and boundary conditions [
1) P ( S , T ) = max { S − K , 0 } . The profit and loss when the option expires is its price. Here K is the exercise price;
2) S → ∞ , P ( S , t ) ~ S − K e − r ( T − t ) . When S is sufficiently large, the option price is close to S − K e − r ( T − t ) , T is the due date of the options;
3) P ( 0 , t ) = 0 , which implies that the option price is approaching to zero when S is zero.
The European call option pricing model is to solve the follow time fractional B-S equation:
{ ∂ α P ( S , t ) ∂ t α + 1 2 σ 2 S 2 ∂ 2 P ( S , t ) ∂ S 2 + r S ∂ P ( S , t ) ∂ S − r P ( S , t ) = 0 , P ( S , T ) = max { S − K , 0 } . (2)
The boundary conditions P ( 0 , t ) = 0 , lim S → ∞ P ( S , t ) = S − K e − r ( T − t ) , and the solution region is Σ = { 0 ≤ S ≤ ∞ , 0 ≤ t ≤ T } .
Equation (2) is an anti-variable coefficient parabolic equation. With the change of variables [
Equation (2) can be transformed into the parabolic equations as follows:
{ V τ ( α ) ( x , τ ) + ( r − 1 2 σ 2 ) V x ( x , τ ) − 1 2 σ 2 V x x ( x , τ ) − r V ( x , τ ) = 0 , V ( x , 0 ) = max { e x − K , 0 } . (3)
The solution region converses into: Σ 0 = { − ∞ ≤ x ≤ + ∞ , 0 ≤ τ ≤ T } .
To solve the TFBSM numerically it is necessary to truncate the original unbounded domain into a finite interval. Here we truncate the range of variable x in problem (1) to a finite interval [ M − , M + ] . Then the solution region we consider is of the following finite domain: Σ 1 = { M − ≤ x ≤ M + , 0 ≤ τ ≤ T } .
Meanwhile, boundary conditions are transformed into the form:
V ( M + , τ ) = e M + − K e − r τ , V ( M − , τ ) = 0 .
Let h = ( M + − M − ) / M and k = T / N , ( M , N ∈ Z + ) be space and time steps. The computation domain Σ 1 is discretized by uniform grid ( x i , τ n ) :
{ x i = M − + ( i − 1 ) h , i = 1 , 2 , ⋯ , M + 1 , τ n = ( n − 1 ) k , n = 1 , 2 , ⋯ , N + 1.
V i n = V ( x i , τ n ) denotes an approximate solution (exact solution of MASC-N scheme) of (3) in x i and τ n . The corresponding initial and boundary conditions are:
V i 1 = f ( x i ) = max { e x i − K , 0 } , V M + 1 n = f 2 ( τ n ) = e M + − K e − r ( n − 1 ) k .
The discrete format of time fractional derivative P t ( α ) ( S , t ) in the point ( x i , τ n + 1 ) using L1 interpolation approximation of the modified Riemann-Liouville fractional derivative is as follows [
∂ α V ( x i , τ n + 1 ) ∂ τ α = k − α Γ ( 2 − α ) ∑ j = 1 n [ V ( x i , τ n + 2 − j ) − V ( x i , τ n + 1 − j ) ] [ j 1 − α − ( j − 1 ) 1 − α ] + O ( k 2 − α ) .
Letting l j = j 1 − α − ( j − 1 ) 1 − α , we can define the operator L τ α V ( x i , τ n + 1 ) as:
L τ α V ( x i , τ n + 1 ) = k − α Γ ( 2 − α ) ∑ j = 1 n l j [ V ( x i , τ n + 2 − j ) − V ( x i , τ n + 1 − j ) ] . (4)
Moreover, the discrete formats of the space derivatives are:
∂ V ( x i , τ n ) ∂ x = V i + 1 n − V i − 1 n 2 h + O ( h 2 ) , ∂ 2 V ( x i , τ n ) ∂ x 2 = V i + 1 n − 2 V i n + V i − 1 n h 2 + O ( h 2 ) . (5)
Substituting (4) and (5) into (3), we can derive the classical explicit scheme, implicit scheme and C-N scheme of time fractional B-S Equation (3):
The classical explicit scheme of time fractional B-S Equation (3) is
L τ α V ( x i , τ n + 1 ) = 1 2 σ 2 V i + 1 n − 2 V i n + V i − 1 n h 2 + ( r − 1 2 σ 2 ) V i + 1 n − V i − 1 n 2 h − r V i n + O ( k 2 − α + h 2 ) .
After combining similar terms and omitting the truncation errors, it can be rewritten as
V i n + 1 = a V i − 1 n − 2 b V i n + c V i + 1 n + ∑ j = 1 n − 1 w j V i n + 1 − j + l n V i 1 . (6)
The classical implicit scheme of time fractional B-S Equation (3) gives
L τ α V ( x i , τ n + 1 ) = 1 2 σ 2 V i + 1 n + 1 − 2 V i n + 1 + V i − 1 n + 1 h 2 + ( r − 1 2 σ 2 ) V i + 1 n + 1 − V i − 1 n + 1 2 h − r V i n + 1 + O ( k 2 − α + h 2 ) .
It can be rewritten as
− a V i − 1 n + 1 + ( 1 + 2 b ) V i n + 1 − c V i + 1 n + 1 = ∑ j = 1 n − 1 w j V i n + 1 − j + l n V i 1 . (7)
The classical C-N scheme of time fractional B-S Equation (3) gives
− 1 2 a V i − 1 n + 1 + ( 1 + b ) V i n + 1 − 1 2 c V i + 1 n + 1 = 1 2 a V i − 1 n − b V i n + 1 2 c V i + 1 n + ∑ j = 1 n − 1 w j V i n + 1 − j + l n V i 1 . (8)
where
a = m σ 2 2 h 2 − m ( r − 0.5 σ 2 ) 2 h , b = m σ 2 2 h 2 + m r 2 , c = m σ 2 2 h 2 + m ( r − 0.5 σ 2 ) 2 h , m = k α Γ ( 2 − α ) , w j = 2 j 1 − α − ( j + 1 ) 1 − α − ( j − 1 ) 1 − α .
Simulating the method of alternating explicit and implicit scheme [
On the odd level, we apply explicit scheme (6) to calculate points x i ( i = i ″ 1 , i ′ 2 , i ″ 3 , i ′ 4 , ⋯ , i ″ 2 s − 1 , i ′ 2 s ) , at points x i ( i = i ″ 2 , i ′ 3 , i ″ 4 , i ′ 5 , ⋯ , i ″ 2 s , i ′ 2 s + 1 ) , we apply explicit scheme (7) to calculate. At the remaining points, we apply C-N scheme (8). When it turns to the even level, we apply explicit scheme (6) to calculate points x i ( i = i ″ 2 , i ′ 3 , i ″ 4 , i ′ 5 , ⋯ , i ″ 2 s , i ′ 2 s + 1 ) , at points x i ( i = i ″ 1 , i ′ 2 , i ″ 3 , i ′ 4 , ⋯ , i ″ 2 s − 1 , i ′ 2 s ) , we apply explicit scheme (7) to calculate. At the remaining points, we apply C-N scheme (8). The schematic design of the MASC-N scheme is shown in
Then the approximate discrete (MASC-N) scheme of Equation (1) can be expressed in the matrix form:
{ ( I + A 1 G ) V n + 1 = − A 2 G V n + w 1 V n + ⋯ + w n − 1 V 2 + l n V 1 + b n , ( I + A 2 G ) V n + 2 = − A 1 G V n + 1 + w 1 V n + 1 + ⋯ + w n V 2 + l n + 1 V 1 + b n + 1 . (9)
where n = 1 , 3 , ⋯ , V n = ( V 2 n V 3 n ⋯ V M − 1 n V M n ) T , b n = ( a 2 ( V 1 n + V 1 n + 1 ) 0 ⋯ 0 c 2 ( V M + 1 n + V M + 1 n + 1 ) ) ′ .
A 1 and A 2 are (M-1)-order diagonal matrices and they meet A 1 + A 2 = I , A 1 = d i a g ( θ 1 , θ 2 , ⋯ , θ M − 2 , θ M − 1 ) .
θ i = { 0 , i = i ″ 2 l − 1 , i ′ 2 l , 0 < l ≤ s 1 , i = i ″ 2 l , i ′ 2 l + 1 , 0 < l ≤ s 1 2 , elsewhere , G = ( 2 b − c − a 2 b − c ⋱ ⋱ ⋱ − a 2 b − c − a 2 b ) M − 1 .
Here I is an ( M − 1 ) order unit matrix.
Lemma 1. The matrices defined by the MASC-N scheme (9) are the nonnegative real matrices.
Proof.
G + G T = ( 4 b − ( a + c ) − ( a + c ) 4 b − ( a + c ) ⋱ ⋱ ⋱ − ( a + c ) 4 b − ( a + c ) − ( a + c ) 4 b ) ,
where 2 b = m σ 2 h 2 + m r , − ( a + c ) = − m σ 2 h 2 , m = k α Γ ( 2 − α ) , we have G + G ′ as a strictly diagonally dominant three diagonal matrix, since the main diagonal elements are positive real numbers, G + G ′ is a positive definite matrix, G + G ′ is nonnegative definite matrix, G is nonnegative definite matrix.
Combined with lemma 1, ( I + A 1 G ) − 1 and ( I + A 2 G ) − 1 exist, and then we have following theorem:
Theorem 1. Solution of the MASC-N scheme (9) of time fractional Black-Scholes Equation (1) is existing and unique.
Lemma 2 (Kellogg [
Lemma 3 ( [
1 > w 1 > w 2 > ⋯ > w n > 0 , ∑ j = 1 n − 1 w j = 1 − l n , l 1 = 1 .
Let V ˜ i n be the approximate solution of Equation (1) at mesh point ( x i , t n ) . Defining ε i n = V ˜ i n − V i n , E n = ( ε 2 n , ε 3 n , ⋯ , ε M n ) , 1 ≤ n ≤ N + 1 , and substituting V i n = V ˜ i n − ε i n into MASC-N scheme, we have E n satisfying:
{ ( I + A 1 G ) E n + 1 = − A 2 G E n + w 1 E n + ⋯ + w n − 1 E 2 + l n E 1 , ( I + A 2 G ) E n + 2 = − A 1 G E n + 1 + w 1 E n + 1 + ⋯ + w n E 2 + l n + 1 E 1 , n = 1 , 3 , ⋯ (10)
When n ≥ 3 , we have
E n + 2 = ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) E n + ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 2 E n − 1 + ⋯ + w n − 1 E 2 + l n E 1 ) + ( I + A 2 G ) − 1 ( w 2 E n + ⋯ + w n E 2 + l n + 1 E 1 ) . (11)
Take the norm of both sides of Equation (11)
‖ E n + 2 ‖ = ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ‖ ‖ E n ‖ + ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ ( w 2 E n − 1 + ⋯ + w n − 1 E 2 + l n E 1 ) ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ ( w 2 E n + ⋯ + w n E 2 + l n + 1 E 1 ) ‖ .
Due to the growth matrix T = ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) and we define T ˜ = ( I + A 2 G ) T ( I + A 2 G ) − 1 , assume that the eigenvalue of G is r, then
‖ T ‖ = ‖ T ˜ ‖ = ‖ ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ( I + A 2 G ) − 1 ‖ = max { | ( w 1 − θ r 1 + θ r ) 2 | } ≤ 1.
Then we use mathematical induction to prove ‖ E n ‖ ≤ ‖ E 1 ‖ .
When n = 1 , ( I + A 1 G ) E 2 = ( I − A 2 G ) E 1
‖ E 2 ‖ = ‖ ( I + A 1 G ) − 1 ( I − A 2 G ) E 1 ‖ ≤ ‖ E 1 ‖ .
When max { w 1 , θ r } = w 1 is established,
‖ E 3 ‖ ≤ ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( I − A 2 G ) ‖ ‖ E 1 ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ l 2 E 1 ‖ ≤ max { | w 1 − θ r 1 + θ r | } ‖ E 1 ‖ + max { | l 2 1 + θ r | } ‖ E 1 ‖ ≤ max { | 1 − θ r 1 + θ r | } ‖ E 1 ‖ ≤ ‖ E 1 ‖ .
When max { w 1 , θ r } = θ r is established,
‖ E 3 ‖ ≤ ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( I − A 2 G ) ‖ ‖ E 1 ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ l 2 E 1 ‖ ≤ max { | w 1 − θ r 1 + θ r | } ‖ E 1 ‖ + max { | l 2 1 + θ r | } ‖ E 1 ‖ ≤ max { θ r + l 2 − w 1 1 + θ r } ‖ E 1 ‖ ≤ ‖ E 1 ‖ .
Assuming that n ≤ k + 1 , we have ‖ E n ‖ ≤ ‖ E 1 ‖ . When n = k + 2 , we prove it in two cases:
Case 1: max { w 1 − θ r } ≤ max { w 1 , θ r } ≤ w 1
‖ E n + 2 ‖ = ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ‖ ‖ E n ‖ + ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ ( w 2 E n − 1 + ⋯ + w n − 1 E 2 + l n E 1 ) ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ ( w 2 E n + ⋯ + w n E 2 + l n + 1 E 1 ) ‖ ≤ ( w 1 1 + θ r ) 2 ‖ E 1 ‖ + w 1 ( 1 − w 1 ) ( 1 + θ r ) 2 ‖ E 1 ‖ + 1 − w 1 1 + θ r ‖ E 1 ‖ ≤ w 1 ‖ E 1 ‖ + ( 1 − w 1 ) ‖ E 1 ‖ ≤ ‖ E 1 ‖ .
Case 2: max { w 1 − θ r } ≤ max { w 1 , θ r } ≤ θ r
‖ E n + 2 ‖ = ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ‖ ‖ E n ‖ + ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ ( w 2 E n − 1 + ⋯ + w n − 1 E 2 + l n E 1 ) ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ ( w 2 E n + ⋯ + w n E 2 + l n + 1 E 1 ) ‖
≤ ( θ r 1 + θ r ) 2 ‖ E 1 ‖ + θ r ( 1 − w 1 ) ( 1 + θ r ) 2 ‖ E 1 ‖ + 1 − w 1 1 + θ r ‖ E 1 ‖ ≤ θ r 1 + θ r ( θ r 1 + θ r + 1 − w 1 1 + θ r ) ‖ E 1 ‖ + 1 − w 1 1 + θ r ‖ E 1 ‖ ≤ θ r 1 + θ r ‖ E 1 ‖ + 1 − w 1 1 + θ r ‖ E 1 ‖ ≤ ‖ E 1 ‖ .
To sum up, we obtain the following theorem.
Theorem 2. The MASC-N scheme (9) of time-fractional Black-Scholes Equation (1) is unconditional stability.
Let v i n = v ( x i , t n ) ( i = 1 , 2 , ⋯ , M + 1 ; n = 1 , 2 , ⋯ , N + 1 ) be the exact solution of Equation (1) at mesh point ( x i , t n ) . Defining e i n = v i n − V i n and e n = ( e 2 n , e 3 n , ⋯ , e M − 1 n , e M n ) T , and substituting e i n = v i n − V i n into MASC-N scheme, we have e n satisfying:
{ ( I + A 1 G ) e n + 1 = − A 2 G e n + w 1 e n + ⋯ + w n − 1 e 2 + l n e 1 + R n , ( I + A 2 G ) e n + 2 = − A 1 G e n + 1 + w 1 e n + 1 + ⋯ + w n e 2 + l n + 1 e 1 + R n + 1 , n = 1 , 3 , ⋯ (12)
and e 1 = 0 , R n = k α O ( k 2 − α + h 2 ) ,which means there exists a positive constant C such that ‖ R n ‖ ≤ C k α ( k 2 − α + h 2 ) .
Imitate the proof of stability, when n ≥ 3 , we have
e n + 2 = ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) e n + ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 2 e n − 1 + ⋯ + w n − 1 e 2 + R n ) + ( I + A 2 G ) − 1 ( w 2 e n + ⋯ + w n e 2 + R n + 1 ) .
When n = 1 , it is straightforward to see that:
‖ e 2 ‖ ≤ ‖ ( I + A 1 G ) − 1 ‖ ‖ R 1 ‖ ≤ l 1 − 1 k α C ( k 2 − α + h 2 ) .
When max { w 1 , θ r } = w 1 is established,
‖ e 3 ‖ ≤ ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ R 1 ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ R 2 ‖ ≤ | w 1 − θ r ( 1 + θ r ) 2 | C k α ( k 2 − α + h 2 ) + 1 1 + θ r C k α ( k 2 − α + h 2 ) ≤ w 1 + 1 ( 1 + θ r ) 2 C k α ( k 2 − α + h 2 ) ≤ l 2 − 1 C k α ( k 2 − α + h 2 ) .
When max { w 1 , θ r } = θ r is established,
‖ e 3 ‖ ≤ ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ R 1 ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ R 2 ‖ ≤ | θ r 1 + θ r | C k α ( k 2 − α + h 2 ) + | 1 1 + θ r | C k α ( k 2 − α + h 2 ) ≤ C k α ( k 2 − α + h 2 ) ≤ l 2 − 1 C k α ( k 2 − α + h 2 ) .
Assuming that n ≤ k + 1 , we have ‖ e n ‖ ≤ l n − 1 − 1 C k α ( k 2 − α + h 2 ) . When n = k + 2 , we prove it in two cases:
Case 1: max { w 1 − θ r } ≤ max { w 1 , θ r } ≤ w 1
‖ e n + 2 ‖ ≤ ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ‖ ‖ e n ‖ + ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ ( w 2 e n − 1 + ⋯ + w n − 1 e 2 ) + R n ‖
+ ‖ ( I + A 2 G ) − 1 ‖ ‖ ( w 2 e n + ⋯ + w n e 2 ) + R n + 1 ‖ ≤ ( w 1 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + ( w 1 − θ r ( 1 + θ r ) 2 + 1 1 + θ r ) × ( ∑ i = 2 n l n + 1 − 1 w i + 1 ) C k α ( k 2 − α + h 2 )
≤ ( w 1 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + w 1 + 1 ( 1 + θ r ) 2 ( ∑ i = 2 n w i + l n + 1 ) l n + 1 − 1 C k α ( k 2 − α + h 2 ) ≤ w 1 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + ( 1 + w 1 ) ( 1 − w 1 ) l n + 1 − 1 C k α ( k 2 − α + h 2 ) ≤ l n + 1 − 1 C k α ( k 2 − α + h 2 ) .
Case 2: max { w 1 − θ r } ≤ max { w 1 , θ r } ≤ θ r
‖ e n + 2 ‖ = ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ( w 1 I − A 2 G ) ‖ ‖ e n ‖ + ‖ ( I + A 2 G ) − 1 ( w 1 I − A 1 G ) ( I + A 1 G ) − 1 ‖ ‖ ( w 2 e n − 1 + ⋯ + w n − 1 e 2 ) + R n ‖ + ‖ ( I + A 2 G ) − 1 ‖ ‖ ( w 2 e n + ⋯ + w n e 2 ) + R n + 1 ‖ ≤ ( θ r 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + ( θ r − w 1 ( 1 + θ r ) 2 + 1 1 + θ r ) × ( ∑ i = 2 n l n + 1 − 1 w i + 1 ) C k α ( k 2 − α + h 2 )
≤ ( θ r 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + 2 θ r + 1 ( 1 + θ r ) 2 ( ∑ i = 2 n l n + 1 − 1 w i + 1 ) C k α ( k 2 − α + h 2 ) ≤ ( θ r 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + 2 θ r + 1 ( 1 + θ r ) 2 ( ∑ i = 2 n w i + l n + 1 ) l n + 1 − 1 C k α ( k 2 − α + h 2 ) ≤ ( θ r 1 + θ r ) 2 l n − 1 − 1 C k α ( k 2 − α + h 2 ) + 2 θ r + 1 ( 1 + θ r ) 2 l n + 1 − 1 C k α ( k 2 − α + h 2 ) ≤ l n + 1 − 1 C k α ( k 2 − α + h 2 ) .
Due to
lim n → ∞ l n − 1 n α = lim n → ∞ n − α n 1 − α − ( n − 1 ) 1 − α = lim n → ∞ n − 1 1 − ( 1 − 1 n ) 1 − α = 1 1 − α .
There is a positive constant C1, such that
‖ e n ‖ ≤ n α C 1 ( k 2 + k α h 2 ) = ( n k ) α C 1 ( k 2 − α + h 2 ) , n = 1 , 2 , ⋯ , N .
Here C 1 = C 1 − α , it is clear that n k < T , then we can know that | v ( x i , τ n ) − V i n | ≤ C 2 ( k 2 − α + h 2 ) and C 2 = T α C 1 .
To sum up, we obtain the following theorem.
Theorem 3. The MASC-N scheme (9) of time-fractional Black-Scholes Equation (1) is convergent, and | v ( x i , τ n ) − V i n | ≤ C ( k 2 − α + h 2 ) , whereC is a positive constant.
Numerical experiments will be done in Matlab 2013b, based on the Intel Core i3-2330 CPU@2.20GHz.The analytic solution of TFBSM is very complex to be obtained [
Example 1 [
{ D 0 τ α U ( x , τ ) = a ∂ 2 U ( x , τ ) ∂ x 2 + b ∂ U ( x , τ ) ∂ x − c U ( x , τ ) + g ( x , τ ) , U ( 0 , τ ) = 0 , U ( 1 , τ ) = 0 , U ( x , 0 ) = x 2 ( 1 − x ) .
where the source term
g = ( 2 τ 2 − α Γ ( 3 − α ) + 2 τ 1 − α Γ ( 2 − α ) ) x 2 ( 1 − x ) − ( t + 1 ) 2 [ a ( 2 − 6 x ) + b ( 2 x − 3 x 2 ) − c x 2 ( 1 − x ) ]
is chosen so that the exact solution of example 1 is U = ( t + 1 ) 2 x 2 ( 1 − x ) . Here we take the parameters values as r = 0.05 , σ = 0.25 , α = 0.7 , a = 1 2 σ 2 , b = r − a , c = r , T = 1 .
Next, we verify the calculation precision and convergence order of MASC-N parallel difference schemes for solving time fractional B-S equation. Convergence order of spatial layer (COS) and convergence order of time horizon (COT) are defined respectively [
E 2 , Δ t m = ‖ V j m − V ¯ j m ‖ 2 = { ∑ j = 1 n ( V j m − V ¯ j m ) 2 Δ t } 1 2 ,
E ∞ , Δ t m = ‖ V j m − V ¯ j m ‖ ∞ = max 1 ≤ j ≤ n | V j m − V ¯ j m | ,
COT ≈ log 2 ( E Δ x , Δ t m / E Δ x , Δ t / 2 m ) , l = 2 , ∞ ,
E 2 , Δ x n = ‖ V i n − V ¯ i n ‖ 2 = { ∑ i = 1 m ( V i n − V ¯ i n ) 2 Δ x } 1 2 ,
E ∞ , Δ x n = ‖ V i n − V ¯ i n ‖ ∞ = max 1 ≤ i ≤ m | V i n − V ¯ i n | ,
COS ≈ log 2 ( E Δ x , Δ t n / E Δ x / 2 , Δ t / 4 n ) , l = 2 , ∞ .
Example 2 [
First, let M = 101 , N = 100 , l = 200 , q = 5 . The parameters are S = 97 , K = 50 , T = 12 , r = 0.05 , σ = 0.25 , M − = ln 0.1 , M + = ln 100 .
We give the plots of the option price curves P ( S , T ) under the INM, MASC-N scheme.
The parameter values are selected according to paper [
Second, in order to verify the stability and the accuracy of MASC-N scheme, we regard the solution V i n of INM as the control solution, and the solution V ¯ i n of MASC-N is used as the perturbation solution, then define the sum of relative error for every time level (SRET) and difference total energy (DTE):
SRET ( n ) = ∑ i = 1 M | V i n − V ¯ i n | V i n , DTE ( i ) = 1 2 ∑ n = 1 N ( V i n − V ¯ i n ) 2 .
| Δt | ‖ . ‖ 2 | COT | ‖ . ‖ ∞ | COT |
|---|---|---|---|---|
| 1/40 | 0.00549407537 | 0.00642653898 | ||
| 1/80 | 0.00163256675 | 1.25073472300 | 0.00262767491 | 1.29025517900 |
| 1/160 | 0.00048949099 | 1.23778776807 | 0.00108954783 | 1.27005725963 |
| 1/320 | 0.00015143065 | 1.19262497528 | 0.000468875620 | 1.21645237116 |
| Δt | Δx | ‖ . ‖ 2 | COS | ‖ . ‖ ∞ | COS |
|---|---|---|---|---|---|
| 1/10 | 1/15 | 0.0455637460106 | 0.0206037934344 | ||
| 1/40 | 1/30 | 0.0136296111988 | 1.74114195458324 | 0.0043408515398 | 2.246859993971 |
| 1/160 | 1/60 | 0.0042681481212 | 1.67506225886374 | 0.0009528489608 | 2.187658631184 |
| 1/640 | 1/120 | 0.0013767764693 | 1.63231589868382 | 0.0002160284100 | 2.141026491177 |
Let M = 101 , N = 100 , other parameters are the same as above. The results of numerical experiments are as follows:
From
From the viewpoint of computational efficiency, INM is the serial format and needs to use the catch-up method to solve three diagonal equations. The advantage of the MASC-N scheme of this paper is that they can divide the large numerical problems A X = b into five small problems A i X = b , i = 1 , 2 , ⋯ , 5 . Using “parfor” to implement parallel computing means that the cycle is divided into independent parts and the various parts can carry out parallel. It leads to the improvement of the efficiency of program execution and the effective reduction of computation time [
Finally, in order to better reflect the superiority of parallel difference schemes, we choose INM, MASC-N scheme (9) to perform numerical tests. The number of time layers is 1000 (N = 1000), and the spatial grid points gradually increase from 201 to 1001. Next, the number of space grid points is 1001 (M = 1001), and the time layers gradually increase from 200 to 1000. And results are shown in
Obviously, MASC-N parallel scheme has superiority in saving the computing time with the increase of time layer as shown in
| Space grid (N) | 200 | 400 | 600 | 800 | 1000 |
|---|---|---|---|---|---|
| INM [ | 25.7785534 | 52.3843632 | 80.9568081 | 109.7452905 | 140.4224646 |
| MASC-N | 1.312375357 | 3.716396959 | 7.247458624 | 12.07522917 | 18.10542618 |
Example 3. Similarly, we consider the case that the underlying asset is a call option. Here we take the parameters values as S = 97 , K = 50 , r = 0.05 , σ = 0.25 and T = 6 . Using INM, MASC-N difference scheme to compute the price of European call options with different values of α and the relative error when α = 0.7, the results are calculated as follows:
In general, option price obtained by the classical B-S equation (α = 1) is lower than real financial market price whose deadline is 6 months [
| α | 0.5 | 0.7 | 0.9 | 1.0 | Time/(s) | Relativeerror |
|---|---|---|---|---|---|---|
| INM [ | 48.39938155 | 48.38462401 | 48.37244188 | 48.36813716 | 138.02153018 | —— |
| MASC-N | 48.40010322 | 48.38539992 | 48.37338163 | 48.36918903 | 17.133438792 | 1.60362453 e-05 |
For the time fractional B-S equation with boundary conditions satisfied by standard European call options, this paper constructs the MASC-N parallel difference scheme. This scheme has been shown to be uniquely solved, unconditionally stable and convergent. And convergence rates are temporally 2-α order and spatially 2-order.
Moreover, compared with the implicit difference scheme, the MASC-N difference scheme is easy to realize parallel computing. So this scheme can greatly improve the computational efficiency, and it has obvious advantages in solving the multi-asset option pricing problems. All the results imply that MASC-N difference scheme is feasible to solve the time fractional B-S equation.
The work is supported by the Fundamental Research Funds for the Central Universities (No. 2021MS045).
The authors declare no conflicts of interest regarding the publication of this paper.
Yang, X.B., Wu, L.F. and Zhang, Y. (2022) A New Parallel Difference Method for Solving Time Fractional Black-Scholes Model. Journal of Mathematical Finance, 12, 683-701. https://doi.org/10.4236/jmf.2022.124036