A universal difference method for time-space fractional Black-Scholes equation

Abstract

The fractional Black-Scholes (B-S) equation is an important mathematical model in finance engineering, and the study of its numerical methods has very significant practical applications. This paper constructs a new kind of universal difference method to solve the time-space fractional B-S equation. The universal difference method is analyzed to be stable, convergent, and uniquely solvable. Furthermore, it is proved that with numerical experiments the universal difference method is valid and efficient for solving the time-space fractional B-S equation. At the same time, numerical experiments indicate that the time-space fractional B-S equation is more consistent with the actual financial market.

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 DOI 10.1186/s13662-016-0792-8

*Correspondence: mailto:[email protected]

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

R E S E A R C H Open Access

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Web End = A universal difference method for time-space fractional Black-Scholes equation

Yang Xiaozhong, Wu Lifei*, Sun Shuzhen and Zhang Xue

Abstract

The fractional Black-Scholes (B-S) equation is an important mathematical model in nance engineering, and the study of its numerical methods has very signicant practical applications. This paper constructs a new kind of universal dierence method to solve the time-space fractional B-S equation. The universal dierence method is analyzed to be stable, convergent, and uniquely solvable. Furthermore, it is proved that with numerical experiments the universal dierence method is valid and ecient for solving the time-space fractional B-S equation. At the same time, numerical experiments indicate that the time-space fractional B-S equation is more consistent with the actual nancial market.

Keywords: time-space fractional Black-Scholes equation; universal dierence method; stability; convergence; numerical experiments

1 Introduction

The Black-Scholes (B-S) equation is an important mathematical model in option pricing theory of nance engineering. In the nancial market, the extensive application of B-S option pricing model has been driven by the rapid development of the nancial derivatives market [, ]. However, as we know the classical B-S model was established under some strict assumptions. According to the research on the stock market, the hypothesis of the traditional B-S equation is so idealistic that it is not completely consistent with the actual stock movement. Some extensions of the B-S model are obtained by weakening these assumptions, such as the fractional B-S model [], the B-S model with transactions costs [], the jump-diusion model [] etc.

During the past few decades, many important phenomena in electromagnetics, acoustics, viscoelasticity, and material science could be well described by fractional dierential equations []. This is due to the fact that a realistic model of a physical phenomenon has a dependence not only on the time instant, but also the previous time history can be successfully described by using fractional calculus [].

In recent years, progress has been made in the study of the fractional B-S equation. Wyss rst deduced the fractional B-S equation with a time fractional derivative to price European call option []. Later, Jumarie applied the fractional Taylor formula to derive the fractional B-S equation based on the classical B-S equation []. Jumarie promoted the previous work and gave an optimal fractional Mertons portfolio, which has wider applications in the actual nancial market []. Cartea and del-Castillo-Negrete obtained several frac-

2016 Xiaozhong 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.

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 2 of 14

tional diusion models of option prices in markets with jumps and priced barrier option

using the fractional partial dierential equation [].

The time-space fractional option pricing model - the time-space fractional B-S equation [, ] takes the form

P()t(S, t) =

r ( )P rSP()S t

( )SP()S. ()

Here t > , < and P(S, t) is the European call option price at asset price S and time t, r is the risk free interest rate. represents the volatility of underlying asset, and denotes the fractional order. The fractional derivatives P()t(S, t), P()S, and P()S are Riemann-

Liouville time fractional derivatives.

There exists no perfect analytic solution of the time-space fractional B-S equation, so it is important to study its numerical solutions. At present, there are a few achievements on the numerical methods for solving the fractional B-S equation []. Kumar et al. provided analytic solution of the fractional B-S option pricing equation by homotopy perturbation method with coupling of the Laplace transform []. In , they also presented a numerical algorithm for the time fractional B-S equation with boundary condition by homotopy perturbation method and homotopy analysis method []. Song and Wang employed implicit nite dierence method to solve the time fractional B-S equation together with the conditions satised by the standard put options []. Yang et al. proposed an Implicit-Explicit and Explicit-Implicit dierence scheme for the time fractional B-S equation []. However, up to now there is no research on the numerical methods of the time-space fractional B-S equation.

Based on the existing problems, this paper mainly studies the numerical methods of the time-space fractional B-S option pricing model in the actual nancial market. We combine the call options to construct the universal dierence scheme for solving the time-space fractional B-S equation. The existence and uniqueness of a numerical solution, computational stability, and convergence of the universal dierence scheme are analyzed. Finally, numerical experiments demonstrate the eectiveness of the universal dierence scheme for solving the time-space fractional B-S equation.

2 Universal difference scheme of time-space fractional B-S equation2.1 Time-space fractional B-S equation

In order to get the value of a European call option, equation () must be integrated with boundary conditions for numerical solutions. There are three boundary conditions:

() P(S, T) = max{S K, }. This condition is quite clear, the prot and loss when the option expires is its price. Here, K is the exercise price.

() S , P(S, t) S Ker(Tt). This condition means when S is suciently great, the option price is close to S Ker(Tt). T is the due date of the options.

() P(, t) = . This condition means when S is zero, the option price is approximant to zero.

Therefore, the European call option pricing is to solve the following equation:

P()t = ( r () P rSP()S)t (+) (+) ( )SP()S,

P(S, T) = max(S K, ). ()

Equation () is an anti-variable coecient parabolic equation.

( + ) ( + )

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 3 of 14

Boundary conditions:

P(, t) = , lim

Solution region:

= { S , t T}.

In order to construct dierence scheme, we make the following coordinate transform []: S = ex, t = T , P(S, t) = er V(x, ). Equation () converses into the following parabolic equation:

Here, () =

2.2 Universal difference scheme

h, k are dened, respectively, as a spatial step and a time step, here h = M+MM , k = TN .

M and N are positive integers.

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

n = (n )k, n = , , . . . , N + .

The approximate value of equation () in the point (xi, n) is dened as Vni.

In order to construct the universal dierence scheme (-dierence scheme), we shall introduce the classic explicit scheme and implicit scheme of equation ().

The classic explicit scheme of equation ():

V(xi, n+)

= ab

(nk k)(T nk + k) Vni+ Vni h

+ a(nk k)(T nk + k) Vni+ Vni + Vnih . ()

S+ P(S, t) = S Ker(Tt).

V()(x, ) [ () () () + r(T )](T )Vx(x, ) ()(T )Vxx(x, ) = ,

V(x, ) = max(ex K, ).

(+) ()

(+) .

The solution region converses into

= { x < +, T}.

In the theory, the price of the underlying asset will not always appear to be zero or innity. Therefore, we provide a small enough number M as the lower boundary and a large enough number M+ as the upper boundary in actual computation.

Therefore, the solution region converses into a nite domain:

=

M x < M+, T .

At the same time, the boundary conditions converse into

V

M+,

= eM++r K, V

()

M, = .

+ r(T nk + k)

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The classic implicit scheme of equation ():

V(xi, n+)

= ab

+ r(T nk)

(nk)(T nk) Vn+i+ Vn+i h

+ a(nk)(T nk) Vn+i+ Vn+i + Vn+ih . ()

Here, a = (), b = () (), n = , , . . . , N, i = , , . . . , M.

Then we assume a parameter ( ), and let ( ) multiply equation (), multiply equation (), then add up their results,

V(xi, n+)

= ( )

ab + r(T nk + k)

(nk k)(T nk + k) Vni+ Vni h

+ a(nk k)(T nk + k) Vni+ Vni + Vni h

(nk)(T nk) Vn+i+ Vn+i h

+ a(nk)(T nk) Vn+i+ Vn+i + Vn+i h

.

+

ab + r(T nk)

The discrete scheme of time fractional derivative is as follows:

V(xi, n+)

=

k

( )

n

j=

V(xi, n+j) V(xi, n+j)

j

(j )

.

Ignoring the errors, we can get equation ():

k

( )

n

j=

V(xi, n+j) V(xi, n+j)

j

(j )

(nk k)(T nk + k) Vni+ Vni h

+ a(nk k)(T nk + k) Vni+ Vni + Vni h

= ( )

ab + r(T nk + k)

(nk)(T nk) Vn+i+ Vn+i h

+ a(nk)(T nk) Vn+i+ Vn+i + Vn+i h

+

ab + r(T nk)

. ()

Equation () is the universal dierence scheme for equation (), it can be written as follows:

Vn+i Vni +

n

j=

V(xi, n+j) V(xi, n+j)

lj

= m

( )(abgn + rqn)

Vni+ Vni

+ (abgn+ + rqn+)

Vn+i+ Vn+i

+ m

( )gn

Vni+ Vni + Vni + gn+

Vn+i+ Vn+i + Vn+i .

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Sorting out the last equation, then we can get the following equation:

m(abgn+ + rqn+) mgn+ Vn+i+ + ( + mgn+)Vn+i

+ m(abgn+ + rqn+) mgn+

Vn+i

= ( ) m(abgn + rqn) + mgn

Vni+ +

mgn( )

Vni

+ ( )

m(abgn + rqn) + mgn

Vni

Vni +

n

j=

djVn+ji + lnVi

= ( )

m(abgn + rqn) + mgn

Vni+ +

mgn( )

Vni

+ ( )

m(abgn + rqn) + mgn

Vni +

n

j=

djVn+ji + lnVi. ()

The matrix form of universal dierence scheme is as follows:

GVn+ = (G + Id)Vn + B, B =

nj= djVn+j + lnV + C,

C = (a nVn an+Vn+, , . . . , , c nVnM+ cn+Vn+M+) ,

n = , , , . . . , N. ()

The matrix () can be written as follows:

bn+ cn+

an+ bn+ cn+

... ... ...

an+ bn+ cn+

an+ bn+

Vn+

Vn+

...

Vn+M

Vn+M

=

... ... ...

Vn

Vn

...

VnM

VnM

Vn+j

Vn+j

...

Vn+jM

Vn+jM

b n c n

a n b n c n

+

n

j=

dj

a n b n c n

a n b n

+ ln

V

V

...

VM

VM

+

a nVn an+Vn+

...

c nVnM+ cn+Vn+M+

,

n

j=

dj

Vn+j

Vn+j

...

Vn+jM

Vn+jM

= d

Vn

Vn

...

VnM

VnM

+ d

Vn

Vn

...

VnM

VnM

+ d

Vn

Vn

...

VnM

VnM

+ + dn

V

V

...

VM

VM

.

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 6 of 14

Here, m = ( )k/h, m = () ( )k/h,

lj = j (j ), dj = j (j + ) (j ), j = , , . . . , n,

gn = (nk k)(T nk + k), qn = (nk k),

an+ =

m(abgn+ + rqn+) mgn+

,

bn+ = + mgn+, cn+ =

m(abgn+ + rqn+) mgn+

,

a n = ( )

m(abgn + rqn) + mgn

,

b n = mgn( ), c n = ( )

m(abgn + rqn) + mgn

.

3 The theoretical analysis of universal difference scheme for the time-space fractional B-S equation

3.1 Existence and uniqueness of the universal difference scheme solution

For G, there are an+ < , cn+ < , bn+ > and bn+ |an+ + cn+| = , so the matrix G is a diagonally dominant matrix. In other words, the coecient matrix G is an invertible matrix.

For G, there are a n > , c n > , b n < and |b n| |a n + c n| = , so the matrix G + Id is a diagonally dominant matrix. In other words, the coecient matrix G +Id is an invertible matrix. Therefore, the universal dierence scheme () has an unique solution.

Theorem The universal dierence scheme () for the time-space fractional B-S equation is uniquely solvable.

3.2 Stability and convergence of universal difference scheme Lemma The following equations hold []:

< dn < < d < d < , dj = lj lj+,

n

j=

dj = ln, l = .

Lemma Assume that Vni is the approximate solution of universal dierence scheme (), and ni = Vni Vni, En = (n, n, . . . , nm), then when for any n N +, one will set En E ; when < and a ()k

h N < , one will set En E .

Proof Applying mathematical induction.When n = , one will set

ci+ + bi + ai = i.

When n > , one will set

cn+n+i+ + bn+n+i + an+n+i = c nni+ + b nni + a nni +

n

j=

djnji + lni.

Dene |l| = maxiM |i|, then one will set

l

c

l+

+ b

l + a|l| c

l+ + bl + al

=

i

=

E

.

Assuming n s, we will have En E .

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When n = s + , assume |s+l| = maxiM |s+i|, then one will set

s+

l cs+ s+

l+

+ bs+

s+

l

+ as+

s+

l

c

s+s+l+ + bs+s+l + as+s+l

= c ssi+ + b ssi + a ssi +

s

j=

djs+ji + lsi

c s Es

+ b s Es

+ a s Es

+ d Es

+ d Es

+

+ ds

Es

+ ls E

= d

Es

+ d Es

+ + ds Es

+ ls E

(d + d + + ds + ls) E

= E

.

Obviously, we have the conclusion Es+ E .

Therefore, we can obtain the following theorem.

Theorem When , the universal dierence scheme () for the time-space fractional B-S equation is stable; when < and the inequality a ()k

h N < holds, the universal dierence scheme () for the time-space fractional B-S equation is stable.

Lemma Assuming V(xi, n) is the exact solution of the dierential equation on the mesh point (xi, n), and dening eni = V(xi, n) Vni, e = , en = (en, en, . . . , enm), en =

maxim |eni|, here, n = , , . . . , N, then when , one will set en lnH( +

h); when < and the inequality a ()k

h N < holds, one will set en lnH( + h). Here, H is a constant.

Proof We will apply mathematical induction. Substitute Vni = V(xi, n) eni into the dierence scheme.

When n = , one will set

cei+ + bei + aei = Ri.

When n > , one will set

cn+en+i+ + bn+en+i + an+en+i = c neni+ + b neni + a neni +

n

j=

djenji + Rn+i.

Here, Rn+i H(+ + h), H is a constant, n = , , . . . , N.When n = , assuming |el| = maxiM |ei| the relation is as follows:

e

e

c

e

l+

+ b

e

l

+ a

e

l

c

= l el+ + bel + ael

=

R

l

H

+ + h

= lH

+ + h

.

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Assuming k s, we will have ek+ lkH(+ + h). We already have lj lk, j = , , . . . , k. Then, when k = s + and assuming |es+l| = maxiM |es+i|, we obtain the following result:

es+

= es+

l cs+ es+

l+

+ bs+

es+

l

+ as+

es+

l

c

s+es+l+ + bs+es+l + as+es+l

= c sesl+ + b sesl + a sesl +

s

j=

djes+ji + Rs+l

c s es

+ b s es

+ a s es

+ d es

+ d es

+

+ ds

e

+ H

+ + h

= d

es

+ d es

+ + ds e

+ H

+ + h

d

ls + dls + + dsl + H

+ + h

s

j=

lsdj + H

+ + h

= ls

s

j=

dj + ls

H

+ + h

= lsH

+ + h

.

Because

lim

n

ln

n = lim n

n

n (n ) =

lim

n

n

( n)

=

,

there is a constant c > , by which we can obtain

en

nc

, n = , , . . . , N.

We know n T, which is a limit number, then we can get the conclusion as follows:

V

(xi, n) Vni

+ + h

= (n)c

+ h

c

, i = , , . . . , M, n = , , . . . , N.

Here, c = Tc.

Theorem When , the universal dierence scheme () for the time-space fractional B-S equation is convergent; when < and the inequality a ()k

+ h

h N <

holds, the universal dierence scheme () for the time-space fractional B-S equation is convergent, and the degree of convergence is rst-order time and second-order space.

4 Numerical examples

Using a Pentium (R) Dual Core CPU . GHz, we will experiment by utilizing the universal dierence scheme in the Matlab . environment. In order to compare with the integer order B-S equation, we use the universal dierence scheme to calculate the price of a European call option.

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 9 of 14

Table 1 The price of a European call option ( = 5/7, M = 200, N = 40)

Time (T/month) 3 6 9 12 The stability

= 1 45.0580 45.3705 45.7238 46.1680 stable = 2/3 45.2422 45.8177 46.5242 47.7146 stable = 1/2 45.3345 46.0423 46.9278 48.4993 stable = 1/3 45.4269 46.2676 47.3336 49.2917 unstable

(a) = (b) = /

(c) = / (d) = /

Figure 1 The price of a European call option ( = 5/7, M = 200, N = 40).

Example Supposing a European call option, whose maturity is , , months, the current price of the stock is $, the strike price is $, the risk free nominal interest rate is %, and the stocks volatility is %.

Solution The parameters are

S = , K = , T = , r = ., = ., M+ = ln , M = .

Then we take dierent spatial steps (see case I and case II) and temporal steps to compute the numerical solutions.

Case I:

= /, = , /, /, /, M = , N = , k = ., h = .,

a ( )kh N = . > .

We get the results in Table and Figure .

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Table 2 The price of a European call option ( = 5/7, M = 200, N = 120)

Time (T/month) 3 6 9 12 The stability

= 1 45.0578 45.3698 45.7219 46.2439 stable = 2/3 45.2641 45.8436 46.5591 47.9096 stable = 1/2 45.3675 46.0816 46.9812 48.7626 stable = 1/3 45.4709 46.3204 47.4058 49.6250 stable

(a) = (b) = /

(c) = / (d) = /

Figure 2 The price of a European call option ( = 5/7, M = 200, N = 120).

From the numerical solution in Table and Figure , we can see that when = /, the universal dierence scheme does not satisfy the stability condition, so the calculation is unstable and the numerical solutions have few references; when selecting = /, /, , the universal dierence scheme satises the stability condition, so the calculation is stable.

Case II:

= /, = , /, /, /, M = , N = , k = ., h = .,

a ( )kh N = . < .

We get the results in Table and Figure .

According to the given date in Table and Figure and the theoretical analysis, we can see that when selecting = , /, /, /, the universal dierence scheme satises the stability condition, so the calculation is stable. The results above indicate that the universal dierence scheme () for the time-space fractional B-S equation is an ecient and practical dierence scheme.

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Table 3 The price of a European call option ( = 1, = 1, 2/3, 1/2, 1/3)

Time (T/month) 3 6 9 12 CPU time (s)

= 1 44.9980 45.2358 45.4745 45.7157 0.45 = 2/3 45.1751 45.6286 46.1718 47.1348 0.43 = 1/2 45.4855 46.3806 47.6890 51.5890 0.45 = 1/3 45.6278 46.8333 49.0078 59.5237 0.46

(a) = (b) = /

(c) = / (d) = /

Figure 3 The price of a European call option ( = 1, = 1, 2/3, 1/2, 1/3).

In order to further examine the eectiveness of the time-space fractional B-S option pricing modeling and the feasibility of the universal dierence scheme method for solving the time-space fractional B-S equation, we will compute the numerical solutions with the condition of case III and IV. Specic plan is as follows: select an unconditional stability implicit scheme ( = ) and the Crank-Nicolson scheme ( = /), study of the eect of value for option price, the value of selected /, /, /, respectively.

Case III:

= , = , /, /, /, M = , N = , k = ., h = ..

We get the results in Table and Figure . Case IV:

= /, = , /, /, /, M = , N = , k = ., h = ..

We get the results in Table and Figure .

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 12 of 14

Table 4 The price of a European call option ( = 1/2, = 1, 2/3, 1/2, 1/3)

Time (T/month) 3 6 9 12 CPU time (s)

= 1 44.9940 45.2318 45.4705 45.7114 0.33 = 2/3 45.3411 46.0406 46.9641 49.0869 0.42 = 1/2 45.2673 45.9415 47.0145 51.3765 0.40 = 1/3 45.1913 45.8369 47.1096 55.8129 0.37

(a) = (b) = /

(c) = / (d) = /

Figure 4 The price of a European call option ( = 1/2, = 1, 2/3, 1/2, 1/3).

From Figures and , we can see that the visible shapes and the trend of the time-space fractional B-S equation are similar to the classical European call option pricing model based on the standard B-S equation ( = ), which illustrates the essential characteristics of the European call options. Therefore, they illustrate that the time-space fractional B-S option pricing equation is eective and the universal dierence scheme () is feasible for solving the time-space fractional B-S equation.

According to the numerical results in Tables , and Figures and , when <

for the time-space fractional B-S equation, the results of the time-space fractional B-S equation are better than the standard B-S equation. The option price of the standard B-S equation ( = ) is small, lower than for the actual nancial market (see []). It arms that the time-space fractional B-S equation is more consistent with the actual nancial market.

When < < , the inuence of the option price is larger for the time-space fractional B-S equation, that is to say, the option price of months is higher than the standard B-S

Xiaozhong et al. Advances in Dierence Equations (2016) 2016:71 Page 13 of 14

equation. In order to meet the actual nancial market, the parameter of the time-space B-S equation should be selected properly according to the actual data.

5 Conclusions

In this work, the universal dierence method is employed to solve the time-space fractional B-S equation with the boundary conditions satised by standard European call options. Theoretical analysis demonstrates that the universal dierence method satises conditional stability and convergence. Numerical experiments are well in agreement with theoretical analysis. All the results illustrate that the time-space fractional B-S equation is eective and the universal dierence scheme is feasible to solve the time-space fractional B-S equation.

Competing interests

The authors declare that there is no conict of interests regarding the publication of this paper.

Authors contributions

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

Acknowledgements

This work is sponsored by the project National Science Foundation of China (No. 11371135), the Fundamental Research Funds for the Central Universities (No. 13QN30).

Received: 4 November 2015 Accepted: 24 February 2016

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