A space-time fractional derivative model for European option pricing with transaction costs in fractal market

Chaos, Solitons and Fractals 103 (2017) 123–130

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A space-time fractional derivative model for European option pricing with transaction costs in fractal marketR Lina Song School of Mathematics, Dongbei University of Finance and Economics, Dalian 116025, China

a r t i c l e

i n f o

Article history: Received 5 March 2017 Revised 29 April 2017 Accepted 30 May 2017

Keywords: Fractional differential equation Option pricing Approximate solution Transaction cost

a b s t r a c t From the point of view of fractional calculus and fractional differential equation, the work handles European option pricing problems with transaction costs in fractal market. Under the definition of the modified Riemman-Liouville fractional derivative, the pricing model based on a space-time fractional patrial differential equation is presented by the replicating portfolio, containing the Hurst exponent taken as the order of fractional derivative. And then, European call and put options are constructed and calculated by the enhanced technique of Adomian decomposition method under the finite difference frame. The fractional derivative model is finally tested by the data from the option market.

1. Introduction Market friction exists in the real financial world. The existence of transaction costs relates the number of hedging and the price of options. The pricing models with transaction costs are important improvements for the classical Black-Scholes model. As early as 1985, Leland [1] gave a technique to replicate option returns in the presence of transaction costs. In 1992, Boyle and Vorst [2] took transaction costs into account and extended the CoxRoss-Rubinstein binomial option pricing model. The model can be expressed by the Black-Scholes model with a modified volatility. Davis et al. [3] priced European options with proportional transaction charges based on a model similar to Black-Scholes one. With transaction costs, Barles and Soner [4] derived a nonlinear Black-Scholes equation with an adjusted volatility. Considering transaction costs and the risk from a volatile portfolio, Kratka derived a mathematical pricing model [5]. Afterwards, Jandacˇ ka and Ševcˇ ovicˇ [6] extended the classical Black-Scholes equation and Lelands equation to a new model for pricing derivative securities under both transaction costs and the risk from the unprotected portfolio. A large amount of researches have found that time series have long-range dependence and the market returns display scaling properties. That the financial market has fractal character is an important discovery and provides a new perspective for the theoretical researches of financial derivatives. In a fractal market, the R The work is partially supported by National Natural Science Foundation of China (No. 71501031, No. 71171035, No. 71273044, No. 71571033, No. 11601068). E-mail address: [email protected]

http://dx.doi.org/10.1016/j.chaos.2017.05.043 0960-0779/© 2017 Published by Elsevier Ltd.

© 2017 Published by Elsevier Ltd.

fractional Black-Scholes models [7–9] are deduced by replacing the standard Brownian motion involved in the classical model with fractional Brownian motion. Further, Wang et al. [10,11] obtained a option pricing model with transaction costs in the fractional version of the Merton model. Gu et al. [12] deal with the option pricing with transaction costs by a fractional sub-diffusive BlackScholes model. Liu et al. [13] proposed a pricing formula for the European option with transaction costs and provided an approximate solution of the nonlinear Hoggard-Whalley-Wilmott equation. Zhang et al. [14] solved the pricing problem of geometric average Asian option with transaction costs under fractional Brownian motion. Xiao et al. [15] used the sub-fractional Brownian to construct the warrants pricing model with transaction costs. The above-mentioned models made great progress and coverd the gap of the classical Black-Scholes model, but these equations are still ones with integer-order derivatives. Comparing with these equations, the fractional differential equations provide an excellent instrument for description of memory and hereditary properties of various materials and processes. The introduction of fractional differential equation into the financial theory provides a new idea and tool for the researches of pricing theory. Wyss [16] presented the fractional Black-Scholes equation with a time-fractional derivative to price European call option. Cartea et al. [17] deduced the space-fractional diffusion models of option prices under three special processes of FMLS, CGMY and KoBoL in markets with jumps and priced barrier option FMLS model. Jumarie [18,19] derived the time and space-time fractional Black-Scholes equations and gave optimal fractional Merton’s portfolio. Based on Jumarie’ ideas, Liang et al. [20,21] gained a Black-Scholes model with time and space fractional derivatives. On the basis, Marom et al. [22], Xi

124

L. Song / Chaos, Solitons and Fractals 103 (2017) 123–130

et al. [23], Song et al. [24] and Chen et al. [25] employed analytical and numerical methods to solve these fractional option pricing models. As far as we know, the study on the fractional derivative model with transaction costs is few. The aims of the work is to establish and solve the space-time fractional pricing model in the presence of transaction costs and testy the practicability of the results by the real data. The paper has been organized as follows. In Section 2, the definition and properties of the modified Riemmna-Liouville derivative are introduced and the space-time fractional derivative model of option pricing is derived. In Section 3, the semi-analytical solutions of fractional model are solved by the enhanced technique and the finite difference method. In Section 4, fractional derivative model is tested by the data. Conclusions and discussions are presented in Section 5.

In the approximation of order 2α , this series provides the equality

f (x + ξ , y + η ) ∼ = f (x, y )+ + +

2.1. Modified Riemmna–Liouville derivative

Definition 2.1 ([19] (Riemann–Liouville definition revisited)). (i) Assume that f(x) is a constant K. Then its fractional derivative of order α is

Dαx K =

K −α (1−α ) x ,

0,

if if

α ≤ 0, α > 0.

(1)

(ii) Assume that f(x) is not a constant. Then its fractional derivative of order α is

Dαx f (x )

⎧ 1 x −α −1 ( f (ξ )− f (0 ))dξ , if α < 0, ⎪ ⎨(−α ) 0 (x − ξ ) x 1 d −α = (1−α ) dx 0 (x − ξ ) ( f (ξ )− f (0 ))dξ , if 0<α <1, (2) ⎪ ⎩ (α−n) ( n) (f (x )) , if n ≤ α
Where Gamma function (z ) = 0 τ z−1 exp(−τ )dτ . Especial for a positive integer n, (n ) = (n − 1 )!. Under the modified Riemann–Liouville definition, Jumarie offered a generalized Taylor expansion on the single variable and multi-variable functions. Proposition 2.1 ([19]). Assume that the continuous function f: R → R, x → f(x) has fractional derivative of order kα , for any positive integer k and any α , 0 < α ≤ 1, then the following equality holds, which is

f (x + h ) =

∞  k=0

h (α k )

(1 + k )

f ( α k ) ( x ), 0 < α ≤ 1,

(3)

1

((1 + α ))2

( 2α ) fxy (x, y )ξ α ηα .

(5)

Corollary 2.1 ([19]). The following equalities hold, which are

Dα xγ = (γ + 1 ) −1 (γ + 1 − α )xγ −α ,

γ > 0, (u(x )v(x )) = u (x )v(x ) + u(x )v (x ), ( f [(x )] )(α ) = fu u(α ) (x ) = fuα (u )(u (x ))(α ) . (α )

(α )

(α )

(6)

Corollary 2.2 ([19]). Assume that f(x) and x(t) are two R → R functions which both have derivatives of order α , 0 < α ≤ 1, then one has the chain rule

ft(α ) (x(t )) = (2 − α )xα −1 fx(α ) (x )x(α ) (t ).

(7)

In the work, the replicating technique is adopted to establish fractional derivative model. The following assumptions are made in financial market with transaction costs. I The change of the value for the replicating portfolio t in [t , t + dt ] is subject to the fractional differential equation

dH t = X1 (t )(dSt )H + X2 (t )dH Dt .

Proposition 2.2 ([19]). Multi-variable fractional Taylor’s series

f (x + ξ , y + η ) = Eα (ξ α Dαx )Eα (ηα Dαy ) f (x, y ).

(4) ∞

zk k=0 (α k+1 ) .

(8)

Where St and Dt denote the price of underlying asset and the riskless bond, respectively. X1 (t ) = X1 (t, St ) and X2 (t) are the corresponding shares. H ∈ [0, 1] is Hurst exponent. When 1/2 < H ≤ 1, the time sequence has long range dependence or long memory. When H = 1/2, the time sequence can be described by random walks. When 0 < H < 1/2, the time sequence shows the antipermanence character. The bond D is risk-less during the time dt, then it satisfies the following equality according to Refs.[20,21],

dH Dt = rDt (dt )H .

(9)

(dt)H

Here, the form comes from Refs. [18,19], where Jumarie extended the Maruyamas notation for Brownian motion b(t, α ) and introduced db(t, α ) = σ w(t )(dt )α , and further gave the following Lemma. Lemma 2.1 ([19]). Let f(t) denote a continuous function, then the solution of dx = f (t )(dt )α , x(0 ) = x0 is defined by the equality



0

t

f (τ )(dτ )α = α



t 0

(t − τ )α−1 f (τ )dτ , 0 < α ≤ 1.

(10)

Integration with respect to (dt)α and its application can refer to Refs.[18,19]. II Transaction cost is a direct cost due to trading and it is the fixed proportion c of the trading amount for the underlying. It is expressed as

Cost = cSt |νt |,

where f(α k) (·) is the derivative of order α k of f(x).

Where Eα is a Mittag-Leffler function and Eα (z ) =

( fx(2α ) (x, y )ξ 2α + fy(2α ) (x, y )η2α )

2.2. Mathematical deduction

The modified fractional derivative is proposed by Jumarie to cover some shortages involved in the classical Riemann-Liouville derivative. Reviewing the literatures [18,19], the definition and main properties of the modified fractional derivative are described, as follows.



1

(1 + 2α )

( fx(α ) (x, y )ξ α + fy(α ) (x, y )ηα )

As a direct application of the fractional Taylors series, Jumarie gave the following corollaries.

2. Fractional derivative model In the Section, the space-time fractional Black-Scholes equation for European option pricing with transaction costs is derived under the definition of modified Riemmna-Liouville derivative.

1

(1+α )

(11)

where ν t denotes the shares of the underlying that are bought (ν t > 0) or sold (ν t < 0) at the price St . III Based on Refs.[20,21], the price St of the underlying asset follows the fractional exponential equation

(dSt )H = μStH (dt )H + σ StH dBH (t ).

(12)

L. Song / Chaos, Solitons and Fractals 103 (2017) 123–130

Where μ is the expected return rate and σ is the volatility of the asset price. BH (t) is a fractional Brownian motion with Hurst exponent H and has the following proposition. Proposition 2.3 ([26]). If BH (t) is a fractional Brownian motion with Hurst exponent H ∈ (0, 1), then for any given A > 0 we have

|BH (t + h ) − BH (t )| = 1.  h→0 0≤t≤A−h hH 2 log(h/A )−1 lim sup

(13)

The following work turns to the derivation of fractional model. It is well known that fractional derivative can reflect normal diffusion, super-diffusion and sub-diffusion. The fractional differential equation can describe complex physical and mechanical processes. The differential and integral operator of the definition of fractional derivatives depicts the temporal memory, path dependence and global correlation of time series. That is to say, the fractional model reflects that the price changes for the option, stocks and bonds are dependent on the past state. From the view of fractional calculus and fractional differential equation, one can get the following conclusion. Theorem 2.1. Based on the assumptions I − III, the price V(S, t) of option fulfills the following space-time fractional patrial differential equation under the modified Riemman–Liouville definition of fractional derivative,

∂ HV (1 + H ) 2 2H ∂ 2H V + σ S (dt )H 2H + ∂ t H (1 + 2H ) ∂S + rS





2H 2 cσ 1+H ∂ V S

∂ S 2H π (1 + H )

∂ HV − r (1 + H )V = 0. ∂ SH

(14)

Proof. Under the assumptions I and II, the change of the value for the portfolio  after the time dt is

dH  = X1 (t )(dS )H + X2 (t )rD(dt )H − cS|dH X1 (t )|.

(15)

Where dt is a finite, small and fixed timestep. The portfolio is considered to be revised every dt. European option price V = V (t, S ) is replicated by the portfolio , then

V =  = X1 (t )S + X2 (t )D.

(16)

In the light of Jumarie’s fractional Taylor’s series (4), one can obtain

dH V (S, t ) =





∂ HV ∂ HV H ( dt ) + (dS )H ∂tH ∂ SH  H  1 ∂ HV ∂ V 2H 2H + (dt ) + H (dS ) (1 + 2H ) ∂ t H ∂S 1 (1 + H )

+

dH V (S, t ) 1 = (1 + H ) +





1

(1+2H )

into

Eq.

1

(1+H )

considering



∂ V ∂ V (dt )2H + H (μ2 S2H (dt )2H +σ 2 S2H (dBH (t ))2 ∂tH ∂S H

H

1 ∂ 2H V (dt )H (μSH (dt )H (1 + H )2 ∂ t H ∂ SH



∂ V σS ∂ V ∂ V (dt )H + μSH H (dt )H + dB (t ) (1+H ) ∂ SH H ∂tH ∂S H

(18)

and

dH X1 (t ) =

σ SH ∂ H X1 (t ) dBH (t ) + O((dt )H ). (1 + H ) ∂ SH

(19)

Eqs. (16), (18) and (19) derive the following equality under the 1 ∂ H V [21], condition that X1 (t ) = (1+ H ) ∂ SH

E[dH  − dH V ]



= E (rV −

1 rS ∂ HV ∂ HV − )(dt )H (1 + H ) ∂ t H (1 + H ) ∂ SH

 cσ S1+H ∂ 2H V σ 2 S 2H ∂ 2H V 2 H ( dB ) − | || dB | + O ( ( dt ) ) H H (1 + 2H ) ∂ S2H  2 ( 1 + H ) ∂ S 2H   1 rS ∂ HV ∂ H V σ 2 S2H (dt )H ∂ 2H V = rV − − − (1 + H ) ∂ t H (1 + H ) ∂ SH (1 + 2H ) ∂ S2H

cσ S1+H

∂ 2H V

2 H ×(dt ) − 2 (dt )H + O( (dt )H ) = 0. (20)  ( 1 + H ) ∂ S 2H π −

So, the work establishes the pricing Eq. (14).



Taking the variable replacement T = t − τ , Eq. (14) is rewritten as

∂ HV (1 + H ) 2 2H ∂ 2H V (−1 )H H + σ S (dt )H 2H (1 + 2H ) ∂τ ∂S

2H H

∂ V 2 cσ ∂ V + S1+H

2H

+ rS H − r(1 + H )V = 0. (21) π (1 + H ) ∂S ∂S The values of European call and put option are suggested to be a solution of Eq. (21) with the following initial and boundary conditions, respectively:

V (S, 0 ) = max(S − K, 0 ), V ( 0, τ ) = 0, V (S, τ ) = S − K exp(−rτ ), S → +∞,

(22)

and

V (S, 0 ) = max(K − S, 0 ), V (0, τ ) = K exp(−rτ ), V (S, τ ) → 0, S → +∞.

(23)

In this section, a direct and simple algorithm is employed to solve Eq. (21). The following is to introduce an enhanced technique of Adomian decomposition method. The technique together with the finite difference method are employed to solve semianalytically solutions of the pricing model. 3.1. Description of the enhanced method

+σ SH dBH (t ))+ · · · · · · =

and

(17)

∂ HV ∂ HV (dt )H + H (μSH (dt )H + σ SH dBH (t )) H ∂t ∂S

+2μσ S2H (dt )H dBH (t ))+



(17)

σ 2 S 2H ∂ 2H V (dBH (t ))2 + O((dt )H · dBH (t )) (1 + 2H ) ∂ S2H

3. Semi-analytical solutions

1 ∂ 2H V (dt )H (dS )H + · · · · · · . 2 (1 + H ) ∂ t H ∂ SH

Substituting Eq. (12) Proposition 2.1 obtain

+

125

H

H

H

Consider a general differential equation

[L + R + N]u(x, t ) = 0,

(24)

where L is an trivially invertible linear operator. R and N are the remaining linear part and nonlinear operator, respectively. After integration, Eq. (24) can be written as

u = − L−1 [R + N]u,

(25)

where is determined by the given conditions. Actually, the choice of the operator L of Eq. (24) has the characteristic of arbitrary. It is important that the next calculation of Eq. (25) is feasible.

126

L. Song / Chaos, Solitons and Fractals 103 (2017) 123–130

rmh1−H (1 + H ) 2 ×(V m+1 − V m−1 ) − (1 + H )rV m .

The classical Adomian decomposition method[27,28] proposes that the solution u can be decomposed into the infinite series and the nonlinear term Nu is decomposed, namely

u=

∞ 

un ,

(26)

n=0

and

Nu =

∞  n=0

∞  1 An = n!



n=0

n



d N dλn

∞  i=0

, n ≥ 0.

(27)

(28)

un = −L−1 [Run−1 + An−1 ], n = 1, 2, · · · . Further, a so-called convergence-control parameter ω is introduced into Eq. (28) and these new components are defined by the expressions,

Cnk ωn−k (1 − ω )k+1 uk , Cnk =

k=0

n! , n = 0, 1, · · · . k ! ( n − k )! (29)

Such thoughts with different forms have been adopted in Refs.[29,30]. The existence of the parameter ω can adjust the convergence region and help to value the option in our algorithm. The efficient values of the parameter ω is can be identified by the analysis presented in Refs.[31–33]. In the work, a solution u derived by the improved technique is the following infinite series

u=

∞ 

 un .

(30)

n=0

For constructing semi-analytical solution framework, the difference approximation is applied in the space variable. Starting by S ∈ [0, Smax ], the step h = Smax and Sm = mh (m = 0, 1, · · · , M ), M where SM = Smax is a realistic and practical approximation to infinity. Vm (τ ) denotes a solution of Eq. (21) in the space point Sm . The fractional derivative of Eq. (21) is under Jumarie’s definition. Jumarie [18,19] gave the following relationship between fractional difference and finite difference based on the fractional Taylor expansion,

α f ∼ = (1 + α ) f, 0 < α < 1.

2

π

cσ m1+H h1−H (1 + H )sgn(m )

π

cσ m sgn( ) 2

m

rm m+1 (V − V m−1 ) − rV m . 2 (35)

V 0 ( τ ) = 0, V M (τ ) = Mh − K exp(−rτ ).

(36)

For European put option, one obtains

V m (0 ) = max(K − mh, 0 ), m = 0, · · · , M, V 0 (τ ) = K exp(−rτ ),

V0m (τ ) = max(mh − K, 0 ), m = 0 · · · M, Vk0 (τ ) = 0, k = 1, 2, · · · ,



Vkm

(τ ) = (−1 )

+ +

(1+H ) −1

L

2

π

 3 ( 1 + H ) 2 2H σ m (dt )H (1 + 2H )

cσ m1+H h1−H (1 + H )sgn(m k−1 )



rmh1−H (1 + H ) +1 Vkm−1 − 2





2 3 ( 1 + H ) 2 2 H σ m (dt )H (1 + 2H )

2

m cσ m1+H h1−H (1+H )sgn(m k−1 )+r (1+H ))Vk−1 π  3  ( 1 + H ) 2 2H + σ m (dt )H (1 + 2H )

+2

+

(33)



(37)

In accordance with the Adomian decomposition method, the recursive relationships of the call option are defined as

2

π

cσ m1+H h1−H (1 + H )sgn(m k−1 )



rmh1−H (1 + H ) m−1 − )Vk−1 , m = 1 · · · M − 1, k = 1, · · · , 2

Substituting Eqs. (32) and (33) to Eq. (21), one can get

+



2

V m (0 ) = max(mh − K, 0 ), m = 0, · · · , M,

and

 3 ( 1 + H ) 2 2H σ m (dt )H (1 + 2H )



Where the numerical approximations (32) and (33) in space are the standard difference form. If time difference is made, the discrete frame is just the classical finite difference models [35,36]. Actually, the simple forms have great improvements and there exists some numerical methods with the higher precision. It is worthy to be mentioned that Ballestra and Cecere [37] presented an optimization algorithm using a finite difference scheme enhanced by a space-time Richardson extrapolation procedure. In the work, the semi-discrete approximation (34) is introduced and established based on the integer difference. The work focuses on the cases that 1 2 < H < 1, because the actual market shows long memory properties. From the initial and boundary conditions of European call option, one can get

(32)

∂ 2H V  2 (1 + H ) m+1 ≈ (V − 2V m + V m−1 ). ∂ S 2H h2H

(34)

H L = ∂∂t H and it is chosen as a in-

×(V m+1 − 2V m + V m−1 ) +

(31)

∂ H V (1 + H ) m+1 ≈ (V − V m−1 ) ∂ SH 2 hH



1 2 2 σ m dt + 2

L(V ) =

Using Eq. (31) and the central differences of the integer-order derivatives [34], one can get the difference approximations to spatial-fractional derivatives,



+ V m−1 .

V M ( τ ) = 0.

3.2. Derivation of the solutions

L(V m ) = (−1 )(1+H )

− 2V m

Where = vertible and linear operator in Adomian decomposition method. If let H = 1, Eq. (34) is reduced to the following form

λ=0

u0 = ,

n 

V m+1

m

After substituting the decomposition series (26) and (27) into both sides of Eq. (24), Adomian decomposition method defines the components un by the following recursive relationship

 un =

m



 λi ui

×(V m+1 − 2V m + V m−1 ) +

VkM (τ ) = −K

(−rτ )k k!

, k = 1, 2, · · · .

For the put option, we have

V0m

(τ ) = max(K − mh, 0 ), m = 0 · · · M,

(38)

L. Song / Chaos, Solitons and Fractals 103 (2017) 123–130

(−rτ )k

Vk0 (τ ) = K Vkm

k!

+

2

π

ten thousand of trading amount, so c = 0.0 0 03. The risk-free rate r equals to the annual interest rate of the 3-year savings bonds2 , then r = 0.038. Strike price of the option 10 0 0 0645 is 2, namely K = 2. the parameters are chosen, Smax = 3, M = 30 0 0, h = 0.0 01, successively. The following is to determine the value of dt. For that, the work considers a special case that H = 1, hence Eq. (14) is reduced to

, k = 1, 2, · · · ,



(τ ) = (−1 )(1+H ) L−1

 3 ( 1 + H ) 2 2H σ m (dt )H (1 + 2H )

cσ m1+H h1−H (1 + H )



∂ V 1 2 2 ∂ 2V + σ S dt 2 + ∂t 2 ∂S

rmh1−H (1 + H ) +1 ×sgn(m Vkm−1 k−1 ) + 2



− +2

π

 +





m cσ m1+H h1−H (1+H )sgn(m k−1 )+r (1+H ) Vk−1

 3 ( 1 + H ) 2 2H σ m (dt )H (1 + 2H ) 2

+ − VkM

2

π



rmh1−H (1 + H ) −1 Vkm−1 , m=1 · · · M−1, k=1, · · · , 2

( τ ) = 0, k = 1, 2, · · · .

(39)

m are defined by the following expresThe new components V n sions

0 (τ ) = V 0 (τ ), n = 0, 1, 2, · · · , V n n m (τ ) = V n

n 

Cnk ωn−k (1 − ω )k+1Vkm (τ ), m = 1 · · · M − 1,

k=0

n = 0, 1, · · · , M (τ ) = V M (τ ), n = 0, 1, 2, · · · . V n n

(40)

The semi-analytical approximations to the fractional model (21) together with Eq. (22) or (23) are finally expressed as

V¯ m (τ ) =

∞ 

m (τ ). V n

(41)

n=0

4. Empirical analysis With the benefit of the mathematical calculation software, the results of Section 3 are tested in China’s option market. It is known 2 that ∂∂ SV2 is positive for European call and put options in the absence of transaction costs, most literatures on option pricing model with transaction costs postulate the same sign. So, the following 2H analysis takes ∂∂ S2HV > 0 into account. Example 4.1. The data1 of SSE 50ETF option 10 0 0 0645 are taken as the set of samples to illustrate that the fractional call option pricing model is effective. The date from 06/23/2016 to 08/04/2016 are selected to estimate the parameters. The first step is to employ the R/S analysis S [38] to calculate the Hurst exponent. The formula ln( iS+1 ) comi

putes the logarithmic return rate of 50ETF, where Si is the daily closing price. By taking these values as time series, R/S analysis tells us that H = 0.6055 and gives the order of fractional derivative. The traditional historical volatility gives that σ = 0.1096, where the number of trading days is considered to be 242 in a year. The transaction cost for 50ETF option is considered to be three per

1

The data came from the trading software of Essence Securities.

σ˜ = σ 2

2

1 dt + 2



∂ 2V ∂V

cσ S 2

+ rS − rV = 0. π ∂S ∂S 2

2

(42)





∂ 2V sign πσ ∂ S2 2 c

 .

(43)

2 Based on the consideration of Ref.[10], ∂∂ SV2 > 0 is adopted and one can obtain

σ˜ 2 1 = dt + σ2 2

cσ m1+H h1−H (1 + H )sgn(m k−1 )





The modified volatility is given by

2 3 ( 1 + H ) 2 2 H σ m (dt )H (1 + 2H )



127



2 c

πσ

≥

 8  14  cdt  12 π

σ

.

According to mathematical inequality, when 12 dt = equality sign of Eq. (44) holds. So the work takes



dt = 2

2 c

πσ

(44)



.

2 c π σ , the

(45)

Based on the above parameters estimation and Eq. (45), dt takes 0.0044. The next is to calculate option value on 08/05/2016. V¯3 (τ ) n (τ ). One needs is the third approximation, namely V¯3 (τ ) = 3n=0 V 2186 to investigate the solutions V¯3 (14/242 ) and V¯32195 (13/242 ). Substituting the option price V = 0.1849 on 08/04/2016 into the expression V¯32186 (14/242 ), one can get a set of values for ω, as follows

ω1 = 0.0191144414 − 0.274584764i, ω2 = −0.272433809 − 0.0199096639i, ω3 = 0.27350915 + 0.0170174424i, ω4 = −0.0180418409 + 0.271695002i.

(46)

In Fig. 1, the curves depict the relationship between the real part (Re) and the image part (Im) of the solution V¯32195 (13/242 ) and the real value ω. Although there is a distance between the figures and the values (41), it can be thought that the values ω1 − ω4 are proper by the aid of experiences and analysis. Then, solutions (46) replace ω of V¯32195 (13/242 ) and the corresponding option prices are listed blow,

V¯32195 (1 ) (13/242 ) = 0.193838847 + 0.0 0 0 027141959i, V¯32195 (2 ) (13/242 ) = 0.193836076 + 0.0 0 0 0284206166i, V¯32195 (3 ) (13/242 ) = 0.19383826 + 0.0 0 0 0243695747i, V¯32195 (4 ) (13/242 ) = 0.193834784 + 0.0 0 0 0249381969i.

(47)

The classical Black-Scholes solution [39,40] and the fractional Black-Scholes solution [41] are 0.1991. The actual value is 0.1943. Extending the detection range under the established values of H, σ and dt, the real parts of two sets of approximate prices estimated by the paper algorithm on 07/07/2016-08/15/2016 are drawn in Fig. 2, together with those of the fractional Black-Scholes model and the Black-Scholes model as well as the actual values. Example 4.2. The second example is to explain the application of the fractional model in the put option using the data3 from SSE 50ETF option 10 0 0 060 0. 2 This was the third period savings bonds and the information came from the Ministry of finance people’s republic of China. 3 The data came from the trading software of Essence Securities.

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Fig. 1. ω-curves of the call option, Solid line: V¯32195 (13/242 ); Dashed line: (V¯32195 (13/242 ))ω .

Fig. 2. Numerical comparisons for the call option, Solid line: paper algorithm; Dashed line: fractional Black-Scholes model; Diamond: Black-Scholes model; Star: actual value.

Repeat the above procedure, the following parameters are determined by the date from 03/22/2016 to 07/18/2016,

K = 2.3, H = 0.6832,

σ = 0.1346, r = 0.038,

c = 0.0 0 03, Smax = 3, M = 30 0 0, h = 0.0 01, dt = 0.0 036. n (τ ). Substituting the option price V = Let V¯4 (τ ) = 4n=0 V 0.1176 on 07/18/2016 into the expression V¯42228 (50/242 ) to evaluate the option price on 07/19/2016, one can get the following values for ω,

ω1 = −0.221451681 − 0.89768176i, ω2 = 0.773584635 − 0.556229974i, ω3 = −0.863735005 − 0.0537410909i, ω4 = −0.264686397 + 0.81080036i, ω5 = 0.746260872 + 0.500677682i.

(48)

The parameter ω of V¯42216 (49/242 ) is replaced by this set of values and the corresponding option prices are listed blow,

V¯42216 (1 ) (49/242 ) = 0.137212328 + 0.00254059628i, V¯42216 (2 ) (49/242 ) = 0.13881043 + 0.0 0 0204126907i, V¯42216 (3 ) (49/242 ) = 0.135386816 + 0.00161158414i, V¯42216 (4 ) (49/242 ) = 0.135246485 + 0.0 0 0 0434357828i, V¯42216 (5 ) (49/242 ) = 0.136589761 − 0.0011606563i.

(49)

The actual value of the option is 0.1263 on 07/19/2016. The fourth approximation (49) does not get good results. But anyway, the work wants to take the “bad” results to explain the fractional model. See the results (49) made by the fractional Black-Scholes model. Referring to the ω-curves of V¯42216 (49/242 ) are drawn in Fig. 3, the values ω1 and ω4 are acceptable and the corresponding solutions are reasonable. The classical Black-Scholes solution and the fractional Black-Scholes solution are 0.0938 and 0.0941, respectively. After computational error, it can be known that the values, not only the ones associated to ω1 and ω4 but also the others produced by ω2 , ω3 and ω5 , are reliable than the estimations from the classical and fractional Black-Scholes solution. Further, numerical comparisons from 06/02/2016 to 08/12/2016 are shown in Fig. 4. It’s evident that the estimates are close to the actual price and they may serve as reliable estimates to guide practice. 5. Conclusions and discussions In the work, European option pricing with transaction cost is handled based on a fractional partial differential equation. By the two examples in the Section 4, one can find that the application of the fractional derivative model in the actual market is feasible. In the completing process of the manuscript, the existing problems and some gains are worthy to be presented and discussed. ∗ Eq. (14) is a space-time fractional partial differential equation and derived using replicating portfolio under the assumptions I − III, which is different from the known fractional Black-Scholes

L. Song / Chaos, Solitons and Fractals 103 (2017) 123–130

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Fig. 3. ω-curves of the put option, Solid line: V¯42216 (49/242 ); Dashed line: (V¯42216 (49/242 ))ω .

Fig. 4. Numerical comparisons for the put option, Solid line: paper algorithm; Dashed line: fractional Black-Scholes model; Diamond: Black-Scholes model; Star: actual value.

in Refs.[16–21]. The model employs the replicating technique and cares about the market friction. The direct participation of Hurst exponent H made itself connected with the fractional derivative and fixes the order of fractional derivative. Comparing with the known literatures considering transaction cost, Eq. (14) introduces the fractional calculus and fractional differential equation, and it contributes to describe the fractal feature of the market. ∗ Taking the variable substitution T = t − τ , Jumarie provided three equalities for the chain rule of the compound functions under modified Riemann-Liouville derivatives. Eq. (7) hampers the next calculation and the first equal mark in the last of Eq. (5) lacks of the fractional derivative for variable V, so the second mark is adopted in this work. This also directly leads to the appearance of complex results for some values of H. The sample analysis in Section 4 tells us that the form (21) does not influence the effectiveness of the fractional model (14). ∗ Empirical analysis of the fractional derivative models is few. The estimation of the parameters have always been a thorny issue. Besides some generic parameters, the parameter ω of the work need to take into account in addition. The parameter ω appeared in the semi-analytic expressions (40) is an adjustable parameter. For a approximate solution of a general differential equation, the existence of the parameter ω can adjust the convergence region and be beneficial to improve the accuracy of the truncated series. But for the practical application in the pricing problems, it shortens the distance between the history data and the value on the

day. In our work, the values of ω determined by the data from these previous days are used to calculate option price on that day. Such a practice complies with the history dependence feature of the market development. But the best problem of ω is still unsettled. The optimal determinations of the parameters are the key to the empirical analysis. ∗ The algorithm exists the computational difficulties, which causes the limitations in the actual applications. But, undeniable is, the work is effective from the numerical analysis of Examples 4.1 and 4.2, and makes every effort to testy the feasibility of the fractional derivative model in the financial market. Acknowledgments The author would like to express gratitude to the reviewers for the careful reading of the manuscript and for their constructive comments. And the author is grateful to editors for their contributions to the publication of the manuscript. References [1] Leland HE. Option pricing and replication with transactions costs. J Finance 1985;40:1283–301. [2] Boyle P, Vorst T. Option replication in discrete time with transaction costs. J Finance 1992;47(1):271–93. [3] Davis MHA, Panas VG, Zariphopoulou T. European option pricing with transaction costs. SIAM J Control Optim 1993;31(2):470–93. [4] Barles G, Soner HM. Option pricing with transaction costs and a nonlinear black-scholes equation. Finance Stochast 1998;2(4):369–97.

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