Alternating segment explicit-implicit and implicit-explicit parallel difference method for the nonlinear Leland equation

Abstract

The nonlinear Leland equation is a Black-Scholes option pricing model with transaction costs and the research of its numerical methods has theoretical significance and practical application value. This paper constructs a kind of difference scheme with intrinsic parallelism-alternating segment explicit-implicit (ASE-I) scheme and alternating segment implicit-explicit (ASI-E) scheme based on the improved Saul'yev asymmetric scheme, explicit-implicit (E-I) scheme, and implicit-explicit (I-E) scheme. Theoretical analysis demonstrates that this kind of scheme is unconditional stable parallel difference scheme. Numerical experiments show that the computational accuracy of this kind of scheme is very close to the classical Crank-Nicolson (C-N) scheme and the alternating segment Crank-Nicolson (ASC-N) scheme. But the computational time of this kind of scheme can save nearly 81% for the classical C-N scheme and save nearly 40% for the ASC-N scheme. Numerical experiments confirm the theoretical analysis, showing the higher efficiency of this kind of scheme given by this paper for solving a nonlinear Leland equation.

Zhao et al. Advances in Dierence Equations (2016) 2016:103 DOI 10.1186/s13662-016-0823-5

R E S E A R C H Open Access

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Web End = Alternating segment explicit-implicit and implicit-explicit parallel difference method for the nonlinear Leland equation

Weijuan Zhao, Xiaozhong Yang* and Lifei Wu

*Correspondence: mailto:[email protected]

Web End [email protected] School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

Abstract

The nonlinear Leland equation is a Black-Scholes option pricing model with transaction costs and the research of its numerical methods has theoretical signicance and practical application value. This paper constructs a kind of dierence scheme with intrinsic parallelism-alternating segment explicit-implicit (ASE-I) scheme and alternating segment implicit-explicit (ASI-E) scheme based on the improved Saulyev asymmetric scheme, explicit-implicit (E-I) scheme, and implicit-explicit (I-E) scheme. Theoretical analysis demonstrates that this kind of scheme is unconditional stable parallel dierence scheme. Numerical experiments show that the computational accuracy of this kind of scheme is very close to the classical Crank-Nicolson (C-N) scheme and the alternating segment Crank-Nicolson (ASC-N) scheme. But the computational time of this kind of scheme can save nearly 81% for the classical C-N scheme and save nearly 40% for the ASC-N scheme. Numerical experiments conrm the theoretical analysis, showing the higher eciency of this kind of scheme given by this paper for solving a nonlinear Leland equation.

MSC: 65M06; 65Y05

Keywords: nonlinear Leland equation; alternating segment explicit-implicit (ASE-I) scheme; alternating segment implicit-explicit (ASI-E) scheme; parallel computing; numerical experiments

1 Introduction

The Black-Scholes (B-S) option pricing model can be accepted by practice elds and theory elds, not only because it has abundant nancial implications, but also it is linear and is a simple model. The B-S model can be transformed into a heat conduction equation with a more mature theory in mathematics and can get the analytical solution of the European call option and put option pricing. However, there exist certain dierences between the assumptions of the B-S model and the real nancial market, such as there being no transaction costs and the xed volatility hypothesis. In order to meet the needs of the actual nancial market, we need to broaden the idealized assumptions and improve the standard B-S model. That has been the focus of academic research; see []. In the real nancial market, because of transaction costs which one needs to pay in securities trading, using the continuous trading strategy is not realistic. So studying the option pricing model with transaction costs (nonlinear Leland model) has great nancial practical signicance.

2016 Zhao et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 2 of 18

The nonlinear Leland equation is one of the nonlinear B-S option pricing models which need to consider transaction costs and received extensive attention of economists and applied mathematicians in the past years []. Because one is unable to export the accurate analytical expression of the European option and American option pricing in the case of considering the transaction costs, many researches focus on the study of numerical solutions. In the numerical solution, in order to make the numerical scheme have good computing stability and precision, we often design implicit or half implicit dierence scheme. In recent years, Ankudinova and Ehrhardt proposed the Crank-Nicolson (C-N) scheme for solving a nonlinear B-S equation (the Leland model, the Barles-Soner model, and the risk adjustment pricing model) []. Wu and Yang put forward the explicit-implicit (E-I) and implicit-explicit (I-E) dierence schemes for solving the payment of dividend B-S equation []. However, most of the schemes are calculated in a serial way and the eciency is low.

In order to make full use of the computer advantages of multi-core processors, a parallel algorithm and a parallel program design have become a necessary means to improve the computing eciency []. The implicit scheme generally has good stability, but it is unfavorable for parallel computing. Inspired by the grouping explicit method [], Zhang et al. put forward the thought that using the Saulyev asymmetric scheme to construct a segment implicit scheme, and one properly used the alternating technology to establish a variety of explicit-implicit and pure implicit alternating parallel schemes (such as an alternating segment explicit-implicit (ASE-I) scheme, an alternating segment Crank-Nicolson (ASC-N) scheme), then one got some numerical results which contained stability and parallelism []. Yang et al. constructed a new kind of parallel dierence scheme-the alternating band Crank-Nicolson (ABdC-N) scheme for solving the quanto option pricing model and proved that it is close to second-order accuracy and unconditionally stable []. Yuan et al. had put forward a parallel dierence scheme with second-order accuracy and unconditional stability for a nonlinear parabolic equation []. Wang also gave a kind of alternating segment dierence scheme with intrinsic parallelism for the KdV equation and proved that the scheme is linearly absolute stable []. Zhang showed the alternating segment explicit-implicit parallel dierence scheme for a class of nonlinear evolution equations and got the result that the method has unconditional stability and parallelism [].

For the research of the parallel dierence method for solving the nonlinear Leland equation, Wu et al. presented a dierence method with intrinsic parallelism-the ASC-N parallel dierence scheme []. Because of the high timeliness of the option, constructing a dierence scheme with good stability and intrinsic parallelism has important practical application value. We apply the E-I and I-E schemes at segment interior points, and the improved asymmetric dierence scheme at interior boundary points, and we get a kind of dierence numerical dierence scheme with intrinsic parallelism-the alternating segment explicit-implicit (ASE-I) scheme and the alternating segment implicit-explicit (ASI-E) scheme.

The plan of this paper is as follows. In Section , we construct the ASE-I dierence scheme for the nonlinear Leland equation. In Section , by using three lemmas, the unique solvability of the dierence solution is discussed. We analyze the stability of the ASE-I scheme in Section and the accuracy in Section . In Section , the ASI-E scheme is put forward by simulating the ASE-I scheme and a theorem is given. Numerical examples are

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 3 of 18

provided to show the eectiveness of the ASE-I and ASI-E schemes in Section . Some concluding remarks are included nally.

2 ASE-I parallel difference method2.1 Nonlinear Leland equation

Assuming that the underlying asset is the transaction cost-paying stock, by the -hedging principle, we can get the following nonlinear Leland equation [], which we will consider for the European options:

V

t +

S

V

S + rS

V

S rV = , ()

here V is the price of a European call option (dollar), S is the price of the underlying asset, r is risk-free interest rate,

is the revised volatility,

= (+Le sign(VSS)). In the revised

volatility, Le = k

t is the Leland number, is the volatility, k is a volume of transaction cost, t is the time dierence between the two transactions, t is the time.

The Leland equation is a denite solution problem of nonlinear partial dierential equations. When k > t/, equation () will become a terminal value problem of a positive parabolic equation which is an ill-posed problem [, ]. In order to transform the problem () into a well-posed problem, we can assume that k < t/, and that the transaction cost should be smaller or the process of hedging risk cannot be too often.

In order to solve the equation of the European call option pricing with transaction costs by using numerical methods, equation () is to be satised on the following boundary conditions [, ]:

() The value of the option is the pay-o function i.e. V(S, T) = (S K)+.

() limS V(S,t)S = . When S is suciently great, the option price is approximately

S K.() If S(t) = , then V(S, t) = for t > t.

Hence, for the European call option, we need to solve the following equation on the domain = { S < , t T}:

Vt + S VS + rS VS rV = ,

V(S, T) = max{S K, }.

()

In order to be able to solve equation (), we can substitute its variables as follows []:

S = Kex; = (T t); V(S, t) = KexU(x, ).

Then equation () will be transformed into the initial-boundary value problem of a partial dierential equation with constant coecients:

U DUx (D + L)Ux = ,

U(x, ) = max{ ex, }, x R,

()

here D = , L = r , x R, T = T .

Meanwhile, the initial and boundary conditions will be translated into

U(x, ) = max ex, , x R,

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 4 of 18

x+ U(x,

lim

) = ex, lim

x U(x,

) = .

In the specic calculation, we can select a large enough M+ and a small enough M making the solving area and the boundary conditions

= M x M+, T ,

U M+, = eM+, U M, = .

2.2 Construction of the ASE-I scheme

Let us make a mesh partition on the area and consider the function U(x, ) at the discrete set of points

xi = M + (i )h, i = , , . . . , m, m + ; h = M+ M

m ;

j = (j )p, j = , , . . . , n, n + ; p = T

n .

Here h is the space step, p is the time step, and m, n are the number of grid points in the x direction and direction, respectively. We use Uji to denote the solution of () at a nite dierence point (xi, j). In order to construct the ASE-I scheme, we give some dierence schemes of equation (). Let a = p(D+L)h, b = pDh .

First, the classical explicit scheme is

Uj+i Uji

p = D

Uji+ Uji + Ujih + (D + L)

Uji+ Uji

h .

The above scheme can be written as

Uj+i = (b a)Uji + ( b)Uji + (a + b)Uji+. ()

Second, the classical implicit scheme is

Uj+i Uji

p = D

Uj+i+ Uj+i + Uj+ih + (D + L)

Uj+i+ Uj+i

h .

The above scheme can be written as

(b a)Uj+i + ( + b)Uj+i (a + b)Uj+i+ = Uji. ()

At last, we present the two improved Saulyev asymmetric schemes,

Uj+i Uji

p = D

Uj+i+ Uj+i Uji + Uji

h + (D + L)

Uj+i+ Uji

h ,

Uj+i Uji

p = D

Uji+ Uji Uj+i + Uj+i

h + (D + L)

Uji+ Uj+i

h .

The above schemes can be written as

( + b)Uj+i (a + b)Uj+i+ = (b a)Uji + ( b)Uji, ()

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Figure 1 The diagram of schemes (6) and (7).

(a b)Uj+i + ( + b)Uj+i = ( b)Uji + (a + b)Uji+. ()

Among the schemes mentioned above, the classic explicit scheme () has the property of parallelism and is very suitable for parallel computing, but it is conditionally stable. The classic implicit scheme () is unconditionally stable, but it needs to solve an algebraic equation which cannot be implemented on a parallel computer []. The improved Saulyev asymmetric schemes (), () are convenient to parallel computing, but they are conditionally stable (see Figure ).

The ASE-I scheme which we constructed is combined with the advantages of the above schemes and the design is as follows:

Let m = Nl, here N is a positive odd number, l is a positive integer (N, l ) and we divide the points on each time level into N sections. And on the odd level, we arrange the computation according to the rule of the explicit segment - the implicit segment -the explicit segment. When it turns to the even level, the rule changes into the implicit segment - the explicit segment - the implicit segment thus making the implicit segment and the explicit segment doing alternatively at dierent time levels.

For realizing the parallel computing of the ASE-I scheme, for i , we consider the calculation of the implicit segment point (i + i, j + ), i = , , . . . , l. The left endpoint (i + , j + ) of the implicit segment is calculated with the improved Saulyev scheme (), the right endpoint (i + l, j + ) is calculated with the improved Saulyev scheme (), and the interior points (i + i, j + ), i = , , . . . , l , are calculated with the classical implicit scheme (), leading to the following implicit segment (see Figure ).

+ b (a + b)a b + b (a + b)

... ... ...

a b + b (a + b) a b + b

Uj+i+

Uj+i+...Uj+i+l

Uj+i+l

=

( b)Uji+ + (b a)Uji Uji+

...

Uji+l

( b)Uji+l + (a + b)Uji+l+

, ()

here, i = l, l, . . . , (N )l.

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Figure 2 Schematic of implicit segment.

In order to improve the calculation accuracy, the implicit segment will be translated into

+ b (a + b)a b + b (a + b)

... ... ...a b + b (a + b)

a b + b

Uj+

Uj+ ...Uj+l

Uj+l

=

Uj + (b a)Uj+

Uj

...

Ujl

( b)Ujl + (a + b)Ujl+

()

when i = and

+ b (a + b)a b + b (a + b)

... ... ...

a b + b (a + b) a b + b

Uj+i+

Uj+i+...Uj+i+l

Uj+i+l

=

( b)Uji+ + (b a)Uji Uji+

...

Uji+l

Uji+l + (a + b)Uj+i+l+

()

when i = (N )l.The explicit segment is

Uj+i+

Uj+i+...Uj+i+l

Uj+i+l

=

b (a + b)b a b (a + b)

... ... ...

b a b (a + b) b a b

Uji+

Uji+...Uji+l

Uji+l

+

(b a)Uji

...(b + a)Uji+l+

. ()

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Figure 3 Schematic with segment nodes of the ASE-I scheme.

We use to denote the classical explicit scheme, to denote the classical implicit

scheme, to denote the two improved Saulyev asymmetric schemes. Let m = , l = ,

N = and let the schematic of the ASE-I scheme be as given (see Figure ).

A complete calculation step of the ASE-I scheme is as follows. For odd level:

() for i = : N

() if mod(i, ) ==

() if i ==

() Solve equation () to get Uj+;

() else if i == N

() Solve equation () to get Uj+N;

() else

() Solve equation () to get Uj+i;

() end

() end

() else

() Solve equation () to get Uj+i;

() end

() end for

For even level, we just switch the segment implicit scheme and the segment explicit

scheme of the odd level to calculate Uj+i.

The ASE-I scheme can also be expressed as

(I + G)Uj+ = (I G)Uj + bj,

(I + G)Uj+ = (I G)Uj+ + bj+, j = , , , . . . , ()

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where Uj = (Uj, Uj, . . . , Ujm, Ujm)T, bj = ((b a)Uj, , . . . , , (b + a)Ujm+)T, j = , , , . . . , n + ,

G =

G()l

Ql

G()l

...

Ql

,

Ql

G(

N

) l

Ql

G()l+

Ql

G()l+

Ql

...

G =

,

G(

N

) l+

Ql

G(

N+

) l+

in which

G()l+ =

b (a + b)a b b (a + b)

... ... ...

a b b (a + b) a b b

(l+)(l+)

,

G(

N+

) l+ =

b (a + b)a b b (a + b)

... ... ...

a b b (a + b) a b b

(l+)(l+)

,

G(i)(l ) =

b (a + b)a b b (a + b)

... ... ...

a b b (a + b) a b b

l l

,

l = l or l + , i = , , . . . , N

,

and Ql (l = l, l ) is a l l zero matrix.

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3 Existence and uniqueness of the ASE-I scheme solution

In order to discuss the existence and the uniqueness of the ASE-I scheme solution, we need to introduce the following three lemmas.

Lemma (Kellogg [, ]) If > and (C + CT) is a non-negative (or positive) denite, then (I + C) exists, and

(I

+ C)

.

Lemma (Kellogg [, ]) If (C +CT) is a non-negative (or positive) denite, for any , then

(I

C)(I + C)

.

Lemma G and G in the ASE-I scheme () for solving the nonlinear Leland equation are non-negative matrices.

Proof We only need to prove G + GT and G + GT are non-negative matrices. Because of

b b b b b

... ... ...

b b b

b b

N+

)l+ + ( G(

)l+ )T are also non-negative matrices.

Therefore, G and G are non-negative matrices.

From the initial conditions and the boundary conditions of the nonlinear Leland equation, we know the dierence solution of the rst time layer. Assuming the value Uji of the

(j)th time layer is known, the value Uj+i of the (j + )th time layer waits for calculating. From the ASE-I scheme (), the matrix equation for calculating the value of the (j + )th time layer is

(I + G)Uj+ = (I G)Uj + bj. ()

Apparently the right of equation () is known and (I + G) exists by Lemma and Lemma . Then equation () has a unique solution.

In the same way, applying the ASE-I scheme to calculate the value of the (j + )th time layer, the matrix equation is

(I + G)Uj+ = (I G)Uj+ + bj+. ()

We could also prove that the matrix equation () has a unique solution. Then we could get the following.

G(i)l/ + G(i)l T =

l l

, l = l or l + ,

we know that G(i)l + (G(i)l )T is a diagonally dominant matrix and the diagonal elements of G(i)l + (G(i)l )T are non-negative real numbers. Therefore, G(i)l + (G(i)l )T is a non-negative ma

trix. In the same way, G()l+ + ( G()l+)T and G(

N+

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Theorem The solution of the ASE-I scheme () for solving a nonlinear Leland equation exists and is unique.

4 Stability of the ASE-I scheme

By eliminating Uj+ from equation (), we obtain

Uj+ = YUj + b ,

here Y is the growth matrix of the ASE-I scheme. The growth matrix of the ASE-I scheme

is

Y = (I + G)(I G)(I + G)(I G).

From Lemmas -, we can get the following inequality easily:

(I

+ G)

,

(I

Gi)(I + Gi)

, i = , .

So

Y

n

(I

+ G)

(I

G)(I + G) n

(I

G)(I + G) n

(I

G)

(I

G)

+ b + a.

Therefore we have the following theorem.

Theorem The ASE-I scheme () for solving the nonlinear Leland equation is absolutely stable.

5 Accuracy of the ASE-I scheme

We take the inside points without interior boundary points as interior points. From the segment construction of the ASE-I scheme, we know that the ASE-I scheme uses the classic E-I scheme at an interior point of odd and even levels, and it uses the two improved Saulyev asymmetric schemes at the interior boundary points. The truncation error of the classic E-I scheme is of second order in time and space []. The ASE-I scheme just has a nite number of interior boundary points, so the overall accuracy of the ASE-I scheme is close to that of the C-N scheme.

The truncation error of the ASE-I scheme at the interior boundary points will be given in the following. We denote the truncation error as T(p, h) when we use (), we denote the truncation error as T(p, h) when we use (), and we let each point of () and () be expanded as the Taylor series at the point (xi, j), (xi+, j). Then we get

T(p, h) =

U

+ h

U

x +

h

U

x +

p

U

+

ph

U

x +

p

U

D

U

x + h

U

x +

p h

U

x +

p

U

x +

p h

U

x

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(D + L)

U

x + h

U

x +

h

U

x +

p h

U

+ p

U

x

+ hp

U

x +

p h

U

+

p

U

x +

p h

U

+ O ph ,

T(p, h) =

U

h

U

x +

h

U

x +

p

U

ph

U

x +

p

U

D

U

x h

U

x

p h

U

x +

p

U

x

p h

U

x

(D + L)

U

x h

U

x +

h

U

x

p h

U

+ p

U

x

hp

U

x

p h

U

+

p

U

x

p h

U

+ O ph ,

where + = . Because of

U

D

U

x (D + L)

U

x = ,

x

U D

U

x (D + L)

U

x = ,

U

D

U

x (D + L)

U

x = ,

we can get

T(p, h) = h

U

x +

ph

U

x +

p

U

D p

h

U

x + p

U

x +

p h

U

x

(D + L) h

U

x +

p h

U

+

p

U

x + hp

U

x +

p h

U

+ p

U

x +

p h

U

+ O ph ,

T(p, h) = h

U

x

ph

U

x +

p

U

D p

h

U

x + p

U

x

p h

U

x

(D + L) h

U

x

p h

U

+

p

U

x hp

U

x

p h

U

+ p

U

x

+ O ph .

Noticing that T(p, h) and T(p, h) contain the same form as regards the expression of

the function, respectively, but we have the reversed symbol. For these items we have the

p h

U

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following:

Dp

h

U

x

U

x

p h

U

x

j

i + D

j

i+ = pD

j

, i

i + .

This part of the interior boundary point can be oset when the ASE-I scheme alternatively uses () and () at dierent times. Ultimately, we can get the following theorem.

Theorem The truncation error of the ASE-I scheme () for solving a nonlinear Leland equation at interior points is O(p + h), and at the improved Saulyev asymmetric schemes (), () of interior boundary points it is O(p + h).

Hence the error of the points which are near the interior boundary point is bigger than that of the other interior points. The result will be proved in the following numerical experiments.

6 ASI-E parallel difference method

Imitating the method constructed in the ASE-I scheme, we give the ASI-E scheme for solving the nonlinear Leland equation.

On the odd level, we arrange the computation according to the rule of the implicit segment-the explicit segment-the implicit segment. When it turns to the even level, the rule changes into the explicit segment-the implicit segment-the explicit segment. Getting the ASI-E dierence scheme for solving the nonlinear Leland equation, we have

(I + G)Uj+ = (I G)Uj + bj+,

(I + G)Uj+ = (I G)Uj+ + bj+, ()

here j = , , , . . . ; G, G and b are as in the above denition.

Imitating the analytical and proved method of the ASE-I scheme (), we have the following theorem.

Theorem The ASI-E scheme () for solving a nonlinear Leland equation is uniquely solvable, absolutely stable, its truncation error is O(p + h) at the interior points, and it is O(p + h) at the interior boundary points.

7 Numerical experiments

Numerical experiments will be done in Matlab a, based on the Intel Core i- [email protected]. We use the ASE-I scheme () and the ASI-E scheme () of this paper, the ASC-N scheme in [] and the classic C-N scheme to calculate European call option prices with transaction costs. For the nonlinear Leland equation () it is very dicult to obtain an analytical solution [, ]. Therefore, we will let the numerical solution of the C-N scheme approximately substitute the exact solution of a European call option pricing problem with transaction costs and compare these various dierence schemes.

Example We consider a European call option on stocks with transaction cost. Assuming the initial price of the underlying stock is dollars, the strike price of an option is dollars, the risk-free interest rate is . per year, the deadline of the option is months, the volatility is . per year, the ratio of the transaction cost is ., t is /.

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Figure 4 Numerical solutions comparison of ASE-I, ASI-E, and C-N schemes.

Table 1 Comparison of several schemes numerical results

Schemes S ($) Time (s)

55 65 75 85 95

C-N 8.4734 17.5879 27.4546 37.4391 47.4317 5.2358 ASC-N [16] 8.4734 17.5879 27.4546 37.4391 47.4317 1.2225 ASE-I 8.4734 17.5879 27.4546 37.4391 47.4317 0.5195 ASI-E 8.4734 17.5879 27.4546 37.4391 47.4317 0.4995

Solution We use the following symbols:

S = , K = , T = , r = ., = ., k = ., t =

.

Let

M+ = ln ., M = ln ., m = ,, n = ,, l = , N = m

l = .

First of all, we give the numerical solutions of the ASE-I and ASI-E schemes.

From Figure and Table we can see that the numerical solutions of the ASE-I and ASI-E schemes are very close to those of the C-N and ASC-N schemes.

Second, we regard the solution Uji of the classical C-N scheme as the control solution and the solutionji of the other schemes as perturbation solutions. Let the grid ratio be r =

ph and give the absolute error (AE) under the dierent r. The denition of AE is as follows:

AE = Un

i ni .

Observing Figures , and Tables , , we see that the AE of the numerical solutions between he ASE-I, ASI-E schemes, and C-N scheme has the same magnitudes as that of the ASC-N scheme, showing that the accuracy of the ASE-I and the ASI-E schemes is close to that of the ASC-N scheme. Because grid points correspond with the stock price ($ or $) near the interior boundary and the AEs of these points are bigger than that of other points, this accords with the theory (see the details in Section and Theorem ). In addition, when r is increasing, the ASE-I and ASI-E schemes still have a good accuracy.

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Figure 5 AE of numerical solution when r1 2.066.

Figure 6 AE of numerical solution when r1 20.66.

Table 2 AE of numerical solution when r1 2.066 (m = 1,001, n = 1,000)

Schemes S ($)

55 65 75 85 95

ASC-N [16] 6.581 106 8.112 106 1.677 106 1.925 106 4.838 106

ASE-I 2.367 106 1.106 106 2.314 106 4.638 106 6.585 106

ASI-E 2.323 106 1.146 106 2.232 106 4.609 106 6.693 106

Table 3 AE of numerical solution when r1 20.66 (m = 1,001, n = 100)

Schemes S ($)

55 65 75 85 95

ASC-N [16] 6.314 104 8.119 104 1.471 104 1.922 104 4.748 104

ASE-I 2.380 104 1.140 104 2.319 104 4.649 104 6.615 104

ASI-E 2.335 104 1.180 104 2.236 104 4.620 104 6.723 104

Thirdly, we will give the proof of stability and the convergence order of the ASE-I and the ASI-E schemes. We analyze the sum of the relative error at every time level (SRET) and the convergence order in the temporal direction (COT) and the spatial direction (COS).

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 15 of 18

Figure 7 Curves of SRET at time layer.

The denitions of SRET, COT, and COS are as follows:

SRET(j) =

m

i=

|Uji ji|

Uji

,

COT =

log(L /L ) log( / ) ,

COS =

log(L x/L x) log( x/ x) .

The error of the L measurement norm is dened as follows:

L ,m = Uj

m jm

=

,

n

j=

Ujm jm p

Ln, x = Un

i ni

=

.

m

i=

Uni ni h

From Figure we can see that the SRET of the ASE-I and ASI-E schemes is larger in the beginning and decreasing along with the movement of the time step; and it is bounded. This shows that the ASE-I and ASI-E schemes have better stability.

Table and Table show that the convergence order of the ASE-I and ASI-E schemes in the temporal direction is approaching O(p) and in the spatial direction it is O(h).

Next, observing Table , the computing times of the ASE-I and ASI-E schemes (.s, .s) are less than that of the C-N and ASC-N schemes (.s, .s). In order to better compare the computing eciency of the several dierence schemes, we choose different points at the space grid and let m = , , , , , ,, n = ,. Because the calculated amount of the ASI-E scheme is the same as that of the ASI-E scheme, we just need to compare the ASE-I scheme, the ASC-N scheme, and the C-N scheme, and the results are in Figure and Table .

From Figure and Table we see that when the number of grid points we need calculated is greater than a certain range, the ASE-I and ASI-E schemes of this paper show a clear superiority in computation time. With the increase of the grid number, the computing time of the dierence schemes rises for the nonlinear Leland equation. But the

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 16 of 18

Table 4 COT of ASC-N, ASE-I, and ASI-E schemes when m = 1,001

Time grid (n) ASC-N [16] ASE-I ASI-E

L2-error COT L2-error COT L2-error COT

100 1.3959 107 1.4273 107 1.4249 107

400 1.0276 108 1.8818 1.0318 108 1.8949 1.0315 108 1.8940

700 3.3784 109 1.9123 3.3862 109 1.9226 3.3856 109 1.9218

1,000 1.6579 109 1.9252 1.6606 109 1.9342 1.6604 109 1.9335

Table 5 COS of ASC-N, ASE-I, and ASI-E schemes when n = 1,000

Space grid (m) ASC-N [16] ASE-I ASI-E

L2-error COS L2-error COS L2-error COS

301 2.1853 107 4.6645 107 4.3609 107

501 7.2702 107 2.3592 1.4714 106 2.2754 1.3071 106 2.1544

701 1.3532 106 2.1567 2.6557 106 2.3109 2.8593 106 2.2244

901 1.8025 106 1.9545 3.7376 106 2.0935 3.5504 106 1.9926

Figure 8 Comparison of the three schemes calculation time.

Table 6 Comparison of the three schemes calculation time at n = 1,000

Space grid (m) Time (s)

C-N ASC-N [16] ASE-I

101 0.0526 0.0880 0.0563 301 0.4738 0.2396 0.1371 501 1.8090 0.4338 0.2221 701 2.5048 0.5632 0.3075 901 4.2826 0.7152 0.3999 1,001 5.0936 0.8897 0.5744

increased amplitude of the computing time of the C-N scheme is greater than that of the ASE-I, ASC-N schemes. The computing time of the ASE-I scheme saves nearly % for the C-N scheme by calculating and saves nearly % for the ASC-N scheme, showing the computing eciency of the ASE-I scheme is best.

As is well known, the parallel scheme has superiority in computing time. But when the amount of calculation data is small, the impact of the data communication on the cycle can reduce the computing eciency. For programming of the ASE-I scheme in our example, we, respectively, adopt the serial for loop and the parallel parfor loop. For the serial

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 17 of 18

Figure 9 Comparison of the serial and parallel calculation time.

for loop, numerical array and the loop body are performed in the same Matlab process, so there are no data communication problems. But, for a parallel parfor loop, numerical arrays are created in the Matlab client, while parallel computing of the parfor loop body is nished under the Matlab worker, so numerical arrays need to be transmitted from the Matlab client to the Matlab worker. Because of taking up time and processor resources in data communication, we need to consider the data communication problem in parallel programming [].

Last, we give the computation time of the ASE-I scheme in the case of the single-core cpu and quad-core cpu. The result is in Figure .

Figure and the rst line of Table show that when the number of grid points is less than a certain range, the serial scheme is more eective than the parallel scheme, meaning that the data communication problems have an eect on the execution eciency of the programming in the case of small data quantity (grid points). And when we have a larger amount of data (grid points), the inuence of the loop body execution is greater than that of the data communication, meaning that using the parallel computing is more eective.

In the practical application, in order to make the numerical results more precise, we tend to a dense mesh and the number of space points becomes higher. The ASE-I parallel method has obvious localization characteristics in computing and communications and is very suitable for large-scale parallel computing in a distributed storage system on the application.

8 Conclusion

For the nonlinear Leland equation, this paper constructs the ASE-I and ASI-E parallel dierence schemes with unconditional stability and high accuracy characteristics. Theoretical analysis gets the result that the numerical solutions of the ASE-I and ASI-E schemes are very close to that of the C-N and ASC-N schemes. Under the same computing accuracy, the ASE-I and ASI-E schemes are greatly improved as regards the computing eciency. Numerical experiment demonstrates that the computing time of the ASE-I and ASI-E schemes save nearly % for the C-N scheme and saved nearly % for the ASC-N scheme, showing the practicability of this kind of parallel dierence schemes for solving a nonlinear Leland equation.

Zhao et al. Advances in Dierence Equations (2016) 2016:103 Page 18 of 18

The ASE-I and ASI-E schemes given by this paper can be extended to solve other nonlinear B-S models with transaction costs, such as the Barles-Soner model and the risk adjustment pricing model, and they can better solve the timeliness problem of the option pricing.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally and signicantly in writing this article. All authors read and approved the nal manuscript.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (No. 11371135), the Special Research Funds for the Central Universities of China (Nos. 2014ZZD10, 13QN30).

Received: 12 January 2016 Accepted: 30 March 2016

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