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

Transcription

1 Xiaozhog et al Advaces i Differece Equatios (016) 016:71 DOI /s R E S E A R C H Ope Access A uiversal differece method for time-space fractioal Black-Scholes equatio Yag Xiaozhog, Wu Lifei *, Su Shuzhe ad Zhag Xue * Correspodece: wulf@cepueduc School of Mathematics ad Physics, North Chia Electric Power Uiversity, Beijig, 1006, PR Chia Abstract The fractioal Black-Scholes (B-S) equatio is a importat mathematical model i fiace egieerig, ad the study of its umerical methods has very sigificat practical applicatios This paper costructs a ew kid of uiversal differece method to solve the time-space fractioal B-S equatio The uiversal differece method is aalyzed to be stable, coverget, ad uiquely solvable Furthermore, it is proved that with umerical experimets the uiversal differece method is valid ad efficiet for solvig the time-space fractioal B-S equatio At the same time, umerical experimets idicate that the time-space fractioal B-S equatio is more cosistet with the actual fiacial market Keywords: time-space fractioal Black-Scholes equatio; uiversal differece method; stability; covergece; umerical experimets 1 Itroductio The Black-Scholes (B-S) equatio is a importat mathematical model i optio pricig theory of fiace egieerig I the fiacial market, the extesive applicatio of B-S optio pricig model has bee drive by the rapid developmet of the fiacial derivatives market [1, ] However, as we kow the classical B-S model was established uder some strict assumptios Accordig to the research o the stock market, the hypothesis of the traditioal B-S equatio is so idealistic that it is ot completely cosistet with the actual stock movemet Some extesios of the B-S model are obtaied by weakeig these assumptios, such as the fractioal B-S model [3 8], the B-S model with trasactios costs [9], the jump-diffusio model [10] etc Durig the past few decades, may importat pheomea i electromagetics, acoustics, viscoelasticity, ad material sciece could be well described by fractioal differetial equatios [11 13] This is due to the fact that a realistic model of a physical pheomeo has a depedece ot oly o the time istat, but also the previous time history ca be successfully described by usig fractioal calculus [14 18] I recet years, progress has bee made i the study of the fractioal B-S equatio Wyss first deduced the fractioal B-S equatio with a time fractioal derivative to price Europea call optio [3] Later, Jumarie applied the fractioal Taylor formula to derive the fractioal B-S equatio based o the classical B-S equatio [4] Jumariepromotedtheprevious work ad gave a optimal fractioal Merto s portfolio, which has wider applicatios i the actual fiacial market [5] Cartea ad del-castillo-negrete obtaied several frac- 016 Xiaozhog et al This article is distributed uder the terms of the Creative Commos Attributio 40 Iteratioal Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author(s) ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made

2 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page of 14 tioal diffusio models of optio prices i markets with jumps ad priced barrier optio usig the fractioal partial differetial equatio [6] The time-space fractioal optio pricig model - the time-space fractioal B-S equatio [3, 5] takes the form ( P (α) t (S, t)= r Ɣ( α) P rsα P (α) S ) t 1 α Ɣ3 (1 + α) Ɣ(1 + α) Ɣ ( α)σ S α P (α) S (1) Here t >0,0<α 1adP(S, t) is the Europea call optio price at asset price S ad time t, r is the risk free iterest rate σ represets the volatility of uderlyig asset, ad α deotes the fractioal order The fractioal derivatives P (α) t (S, t), P (α) S,adP(α) S are Riema- Liouville time fractioal derivatives There exists o perfect aalytic solutio of the time-space fractioal B-S equatio, so it is importat to study its umerical solutios At preset, there are a few achievemets o the umerical methods for solvig the fractioal B-S equatio [19 4] Kumar et al provided aalytic solutio of the fractioal B-S optio pricig equatio by homotopy perturbatio method with couplig of the Laplace trasform [19]I 014,they also preseted a umerical algorithm for the time fractioal B-S equatio with boudary coditio by homotopy perturbatio method ad homotopy aalysis method [0] Sog ad Wag employed implicit fiite differece method to solve the time fractioal B-S equatio together with the coditios satisfied by the stadard put optios [] Yag et al proposed a Implicit- Explicit ad Explicit-Implicit differece scheme for the time fractioal B-S equatio [3] However, up to ow there is o research o the umerical methods of the time-space fractioal B-S equatio Based o the existig problems, this paper maily studies the umerical methods of the time-space fractioal B-S optio pricig model i the actual fiacial market We combie the call optios to costruct the uiversal differece scheme for solvig the time-space fractioal B-S equatio The existece ad uiqueess of a umerical solutio, computatioal stability, ad covergece of the uiversal differece scheme are aalyzed Fially, umerical experimets demostrate the effectiveess of the uiversal differece scheme for solvig the time-space fractioal B-S equatio Uiversal differece scheme of time-space fractioal B-S equatio 1 Time-space fractioal B-S equatio I order to get the value of a Europea call optio, equatio (1) mustbeitegratedwith boudary coditios for umerical solutios There are three boudary coditios: (1) P(S, T) =max{s K, 0} This coditio is quite clear, the profit ad loss whe the optio expires is its price Here, K is the exercise price () S, P(S, t) S Ke r(t t) This coditio meas whe S is sufficietly great, the optio price is close to S Ke r(t t) T is the due date of the optios (3) P(0, t)=0 This coditio meas whe S is zero, the optio price is approximat to zero Therefore, the Europea call optio pricig is to solve the followig equatio: { P (α) t =( r Ɣ( α) P rsα P (α) S P(S, T)=max(S K,0) )t1 α Ɣ3 (1+α) Ɣ(1+α) Ɣ ( α)σ S α P (α) S, Equatio () is a ati-variable coefficiet parabolic equatio ()

3 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 3 of 14 Boudary coditios: P(0, t)=0, lim S + P(S, t)=s Ke r(t t) Solutio regio: = {0 S,0 t T} I order to costruct differece scheme, we make the followig coordiate trasform [5]: S = e x, t = T τ, P(S, t)=e rτ V(x, τ) Equatio () coverses ito the followig parabolic equatio: V τ (α) (x, τ) [γ (α) Ɣ(1 α) Ɣ(1 α) σ + r(t τ) 1 α ]τ 1 α (T τ) α 1 V x (x, τ) γ (α)σ τ 1 α (T τ) α 1 V xx (x, τ)=0, (3) V(x,0)=max(e x K,0) Here, γ (α)= Ɣ3 (1+α)Ɣ ( α) Ɣ(1+α) The solutio regio coverses ito 0 = { x <+,0 τ T} I the theory, the price of the uderlyig asset will ot always appear to be zero or ifiity Therefore, we provide a small eough umber M as the lower boudary ad a large eough umber M + as the upper boudary i actual computatio Therefore, the solutio regio coverses ito a fiite domai: 1 = { M x < M +,0 τ T } At the same time, the boudary coditios coverse ito V ( M +, τ ) = e M+ +rτ K, V ( M, τ ) =0 Uiversal differece scheme h, k are defied, respectively, as a spatial step ad a time step, here h = M+ M M, k = T N M ad N are positive itegers { x i = M +(i 1)h, i =1,,,M +1, τ =( 1)k, =1,,,N +1 The approximate value of equatio (3)ithepoit(x i, τ )isdefiedasv i I order to costruct the uiversal differece scheme (θ-differece scheme), we shall itroduce the classic explicit scheme ad implicit scheme of equatio (3) Theclassicexplicitschemeofequatio(3): α V(x i, τ +1 ) τ α = [ ab + r(t k + k) 1 α] (k k) 1 α (T k + k) α 1 V i+1 V i 1 h + a(k k) 1 α (T k + k) α 1 V i+1 V i + Vi 1 (4) h

4 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 4 of 14 The classic implicit scheme of equatio (3): α V(x i, τ +1 ) = [ ab + r(t k) 1 α] (k) 1 α +1 α 1 Vi+1 (T k) V i 1 +1 τ α h + a(k) 1 α (T k) +1 α 1 Vi+1 +1 Vi + Vi 1 +1 (5) h Here, a = γ (α)σ, b = Ɣ(1 α), =1,,,N, i =1,,,M Ɣ(1 α) The we assume a parameter θ (0 θ 1), ad let (1 θ)multiplyequatio(4), θ multiply equatio (5), the add up their results, α V(x i, τ +1 ) τ α [ (ab =(1 θ) + r(t k + k) 1 α ) (k k) 1 α (T k + k) α 1 V i+1 V h + a(k k) 1 α (T k + k) α 1 V i+1 V i + Vi 1 ] [ (ab + θ + r(t k) 1 α ) (k) 1 α +1 α 1 Vi+1 (T k) V i 1 +1 h + a(k) 1 α α 1 Vi+1 Vi + V +1 ] i 1 (T k) The discrete scheme of time fractioal derivative is as follows: α V(x i, τ +1 ) k α = τ α Ɣ( α) Igorig the errors, we ca get equatio (6): k α Ɣ( α) h h [ V(xi, τ + j ) V(x i, τ +1 j ) ][ j 1 α (j 1) 1 α] [ V(xi, τ + j ) V(x i, τ +1 j ) ][ j 1 α (j 1) 1 α] [ (ab =(1 θ) + r(t k + k) 1 α ) (k k) 1 α (T k + k) α 1 V i+1 V h + a(k k) 1 α (T k + k) α 1 V i+1 V i + Vi 1 ] h [ (ab + θ + r(t k) 1 α ) (k) 1 α +1 α 1 Vi+1 (T k) V i 1 +1 h + a(k) 1 α α 1 Vi+1 Vi + V +1 ] i 1 (T k) (6) h Equatio (6) is the uiversal differece scheme for equatio (3), it ca be writte as follows: Vi +1 Vi + [ V(xi, τ + j ) V(x i, τ +1 j ) ] l j j= = m 1 [ (1 θ)(abg + rq ) ( V i+1 V i 1 + m [ (1 θ)g ( V i+1 Vi + V i 1 i 1 i 1 ) + θ(abg+1 + rq +1 ) ( Vi+1 +1 V i 1 +1 )] ) ( + θg+1 V i+1 Vi + Vi 1 +1 )]

5 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 5 of 14 Sortig out the last equatio, the we ca get the followig equatio: [ ] m1 (abg +1 + rq +1 ) m g +1 θv +1 i+1 +(1+θm g +1 )Vi +1 + [ m 1 (abg +1 + rq +1 ) m g +1 ] θv +1 i 1 =(1 θ) [ ] m 1 (abg + rq )+m g V i+1 + [ 1 m g (1 θ) ] Vi +(1 θ) [ m 1 (abg + rq )+m g ] V i 1 ( 1 α 1 ) V 1 i + =(1 θ) [ ] m 1 (abg + rq )+m g V i+1 + [ m g (1 θ) ] Vi +(1 θ) [ m 1 (abg + rq )+m g ] V 1 i 1 + j= d j V +1 j i + l Vi 1 d j V +1 j i + l V 1 i (7) The matrix form of uiversal differece scheme is as follows: G 1 V +1 =(G + Id 1 )V + B, B = 1 j= d jv +1 j + l V 1 + C, C =(a V 1 a +1V1 +1,0,,0,c V M+1 c +1V +1 M+1 ), =1,,3,,N (8) The matrix (8) cabe writte asfollows: b +1 c +1 V +1 a +1 b +1 c +1 V +1 3 a +1 b +1 c +1 VM 1 +1 a +1 b +1 VM +1 b c V a b c V 3 = 1 + a b c 1 d j + l V 1 V 1 3 VM 1 1 VM 1 V +1 j V +1 j 3 V +1 j M 1 V +1 j M a b VM 1 VM d j a V 1 a +1V , 0 c V M+1 c +1VM+1 +1 V V 1 V 3 V 1 3 = d 1 + d + d 3 V 1 VM 1 VM M 1 VM 1 V +1 j V +1 j 3 V +1 j M 1 V +1 j M V V 3 VM 1 VM + + d 1 V V 3 VM 1 VM

6 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 6 of 14 Here, m 1 = Ɣ( α)k α /h, m = γ (α)σ Ɣ( α)k α /h, l j = j 1 α (j 1) 1 α, d j =j 1 α (j +1) 1 α (j 1) 1 α, j =1,,,, g =(k k) 1 α (T k + k) α 1, q =(k k) 1 α, a +1 = θ [ m 1 (abg +1 + rq +1 ) m g +1 ], b +1 =1+θm g +1, c +1 = θ [ m 1 (abg +1 + rq +1 ) m g +1 ], a =(1 θ)[ m 1 (abg + rq )+m g ], b = m g (1 θ), c =(1 θ)[ m 1 (abg + rq )+m g ] 3 The theoretical aalysis of uiversal differece scheme for the time-space fractioal B-S equatio 31 Existece ad uiqueess of the uiversal differece scheme solutio For G 1,therearea +1 <0,c +1 <0,b +1 >0adb +1 a +1 + c +1 =1,sothematrixG 1 is a diagoally domiat matrix I other words, the coefficiet matrix G 1 is a ivertible matrix For G,therearea >0,c >0,b <0ad b a + c =0,sothematrixG + Id 1 is a diagoally domiat matrix I other words, the coefficiet matrix G + Id 1 is a ivertible matrix Therefore, the uiversal differece scheme (6) hasauiquesolutio Theorem 1 The uiversaldifferece scheme (6) for the time-space fractioal B-S equatio is uiquely solvable 3 Stability ad covergece of uiversal differece scheme Lemma 1 The followig equatios hold [14]: 0<d < < d < d 1 <1, d j = l j l j+1, 1 d j =1 l, l 1 =1 Lemma Assume that Ṽi is the approximate solutio of uiversal differece scheme (6), ad εi = Ṽi Vi, E =(ε1, ε,,ε m 1 ), the whe 1 θ 1 for ay 1 N +1,oe will set E E 1 ; whe 0 θ < 1 aɣ( α)kα ad N α 1 <1,oe will set E h E 1 Proof Applyig mathematical iductio Whe =1,oewillset c ε i+1 + b ε i + a ε i 1 = ε1 i Whe >1,oewillset 1 c +1 εi b +1εi +1 + a +1 εi 1 +1 = c ε i+1 + b ε i + a ε i 1 + d j ε j i + l εi 1 Defie εl = max i M εi,theoewillset ε l c ε l+1 + b ε l + a εl 1 c εl+1 + b εl + a εl 1 = ε 1 i = E 1 Assumig s,wewillhave E E 1

7 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 7 of 14 Whe = s + 1, assume εl s+1 = max i M εi s+1,theoewillset ε s+1 l cs+1 ε s+1 l+1 + bs+1 ε s+1 l + as+1 ε s+1 l 1 cs+1 εl+1 s+1 + b s+1εl s+1 + a s+1 εl 1 s+1 s 1 = c s εs i+1 + b s εs i + a s εs i 1 + d j ε s+1 j i + l s εi 1 c s E s + b s E s + a s E s + d 1 E s + d E s + + d s 1 E s + l s E 1 = d 1 E s + d E s + + d s 1 E s + l s E 1 (d 1 + d + + d s 1 + l s ) E 1 = E 1 Obviously, we have the coclusio E s+1 E 1 Therefore, we ca obtai the followig theorem Theorem Whe 1 θ 1, theuiversaldifferecescheme(6) for the time-space fractioalb-sequatioisstable; whe 0 θ < 1 aɣ( α)kα ad the iequality N α 1 <1holds, h the uiversal differece scheme (6) for the time-space fractioal B-S equatio is stable Lemma 3 Assumig V(x i, τ ) is the exact solutio of the differetial equatio o the mesh poit (x i, τ ), ad defiig e i = V(x i, τ ) Vi, e1 =0,e =(e 1, e,,e m 1 ), e = max 1 i m 1 e i, here, =1,,,N, the whe 1 θ 1, oe will set e l 1 1 H(τ 1 α + τ α h ); whe 0 θ < 1 l 1 1 H(τ 1 α + τ α h ) Here, Hisacostat ad the iequality aɣ( α)kα h = V(x i, τ ) e i ito the differ- Proof We will apply mathematical iductio Substitute Vi ece scheme Whe =1,oewillset c e i+1 + b e i + a e i 1 = R1 i N α 1 <1holds, oe will set e Whe >1,oewillset 1 c +1 e +1 i+1 + b +1e +1 i + a +1 e +1 i 1 = c e i+1 + b e i + a e i 1 + d j e j i + R +1 i Here, R +1 i H(τ 1+α + τ α h ), H is a costat, =1,,,N Whe =1,assumig e l = max i M e i the relatio is as follows: e = e l c e l+1 + b e l + a e l 1 c e l+1 + b e l + a e l 1 = ( R 1 l H τ 1+α + τ α h ) = l 1 1 H( τ 1+α + τ α h )

8 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 8 of 14 Assumig k s, wewillhave e k+1 l k 1H(τ 1+α + τ α h ) We already have l j 1 l k 1, j = 1,,,k The,whek = s +1adassumig e s+1 l = max i M e s+1 i, weobtaithe followig result: e s+1 = e s+1 l cs+1 e s+1 l+1 + bs+1 e s+1 l + as+1 e s+1 l 1 cs+1 e s+1 l+1 + b s+1e s+1 l + a s+1 e s+1 l 1 s 1 = c s es l+1 + b s es l + a s es l 1 + d j e s+1 j i + R s+1 l c s e s + b s e s + a s e s + d 1 e s + d e s d s 1 e + H ( τ 1+α + τ α h ) = d 1 e s + d e s d s 1 e + H ( τ 1+α + τ α h ) ( d 1 l s d l s d s 1l ) H ( τ 1+α + τ α h ) ) Because ( s 1 = l 1 s ( s 1 l 1 lim = lim α l 1 s d j +1 H ( τ 1+α + τ α h ) ) d j + l s H ( τ 1+α + τ α h ) = l 1 s H ( τ 1+α + τ α h ) α = lim 1 α ( 1) 1 α there is a costat c >0,bywhichwecaobtai 1 1 (1 1 )1 α = 1 1 α, e α c ( τ 1+α + τ α h ) =(τ) α c ( τ + h ), =1,,,N We kow τ T, which is a limit umber, the we ca get the coclusio as follows: V(x i, τ ) Vi c ( τ + h ), i =,3,,M, =1,,,N Here, c = T α c Theorem 3 Whe 1 θ 1, the uiversal differece scheme (6) for the time-space fractioal B-S equatio is coverget; whe 0 θ < 1 aɣ( α)kα ad the iequality N α 1 <1 h holds, the uiversal differece scheme (6) for the time-space fractioal B-S equatio is coverget, ad the degree of covergece is first-order time ad secod-order space 4 Numerical examples Usig a Petium (R) Dual Core CPU 300 GHz, we will experimet by utilizig the uiversal differece scheme i the Matlab 70 eviromet I order to compare with the iteger order B-S equatio, we use the uiversal differece scheme to calculate the price of a Europea call optio

9 Xiaozhogetal Advaces i Differece Equatios (016) 016:71 Page 9 of 14 Table 1 The price of a Europea call optio (α =5/7,M = 00, N =40) Time (T/moth) The stability θ = stable θ = / stable θ = 1/ stable θ = 1/ ustable (a) θ =1 (b)θ =/3 (c) θ =1/ (d)θ =1/3 Figure 1 The price of a Europea call optio (α =5/7,M =00,N =40) Example Supposig a Europea call optio, whose maturity is 3, 6, 1 moths, the curret price of the stock is 97$, the strike price is 50$, the risk free omial iterest rate is 1%, ad the stock s volatility is 0% Solutio The parameters are S =97, K =50, T =1, r =001, σ =0, M + = l 300, M =1 The we take differet spatial steps (see case I ad case II) ad temporal steps to compute the umerical solutios Case I: α = 5/7, θ = 1, /3, 1/, 1/3, M =00, N =40, k =005, h =0035, aɣ( α)k α h N α 1 =113>1 We get the results i Table 1 ad Figure 1

10 XiaozhogetalAdvaces i Differece Equatios (016) 016:71 Page 10 of 14 Table The price of a Europea call optio (α =5/7,M = 00, N = 10) Time (T/moth) The stability θ = stable θ = / stable θ = 1/ stable θ = 1/ stable (a) θ =1 (b)θ =/3 (c) θ =1/ (d)θ =1/3 Figure The price of a Europea call optio (α =5/7,M =00,N =10) From the umerical solutio i Table 1 ad Figure 1, we ca see that whe θ = 1/3, the uiversal differece scheme does ot satisfy the stability coditio, so the calculatio is ustable ad the umerical solutios have few refereces; whe selectig θ = 1/, /3, 1, the uiversal differece scheme satisfies the stability coditio, so the calculatio is stable Case II: α = 5/7, θ = 1, /3, 1/, 1/3, M =00, N =10, k =00083, h =0035, aɣ( α)k α h N α 1 =037<1 We get the results i Table ad Figure Accordig to the give date i Table ad Figure ad the theoretical aalysis, we ca see that whe selectig θ = 1, /3, 1/, 1/3, the uiversal differece scheme satisfies the stability coditio, so the calculatio is stable The results above idicate that the uiversal differece scheme (6) for the time-space fractioal B-S equatio is a efficiet ad practical differece scheme

11 XiaozhogetalAdvaces i Differece Equatios (016) 016:71 Page 11 of 14 Table 3 The price of a Europea call optio (θ =1,α =1,/3,1/,1/3) Time (T/moth) CPU time (s) α = α = / α = 1/ α = 1/ (a) α =1 (b)α =/3 (c) α =1/ (d)α =1/3 Figure 3 The price of a Europea call optio (θ =1,α =1,/3,1/,1/3) I order to further examie the effectiveess of the time-space fractioal B-S optio pricig modelig ad the feasibility of the uiversal differece scheme method for solvig the time-space fractioal B-S equatio, we will compute the umerical solutios with the coditio of case III ad IV Specific pla is as follows: select a ucoditioal stability implicit scheme (θ = 1) ad the Crak-Nicolso scheme (θ = 1/), study of the effect of α value for optio price, the value of α selected /3, 1/, 1/3, respectively Case III: θ =1, α = 1, /3, 1/, 1/3, M =00, N =10, k =00083, h =0035 We get the results i Table 3 ad Figure 3 Case IV: θ = 1/, α = 1, /3, 1/, 1/3, M =00, N =10, k =00083, h =0035 We get the results i Table 4 ad Figure 4

12 XiaozhogetalAdvaces i Differece Equatios (016) 016:71 Page 1 of 14 Table 4 The price of a Europea call optio (θ =1/,α =1,/3,1/,1/3) Time (T/moth) CPU time (s) α = α = / α = 1/ α = 1/ (a) α =1 (b)α =/3 (c) α =1/ (d)α =1/3 Figure 4 The price of a Europea call optio (θ =1/,α =1,/3,1/,1/3) From Figures 3 ad 4, we ca see that the visible shapes ad the tred of the time-space fractioal B-S equatio are similar to the classical Europea call optio pricig model based o the stadard B-S equatio (α = 1), which illustrates the essetial characteristics of the Europea call optios Therefore, they illustrate that the time-space fractioal B-S optio pricig equatio is effective ad the uiversal differece scheme (6)isfeasiblefor solvig the time-space fractioal B-S equatio Accordig to the umerical results i Tables 3, 4 ad Figures 3 ad 4, whe 1 α <1 for the time-space fractioal B-S equatio, the results of the time-space fractioal B-S equatio are better tha the stadard B-S equatio The optio price of the stadard B-S equatio (α = 1) is small, lower tha for the actual fiacial market (see [4]) It affirms that the time-space fractioal B-S equatio is more cosistet with the actual fiacial market Whe 0 < α < 1, the ifluece of the optio price is larger for the time-space fractioal B-S equatio, that is to say, the optio price of 1 moths is higher tha the stadard B-S

13 XiaozhogetalAdvaces i Differece Equatios (016) 016:71 Page 13 of 14 equatio I order to meet the actual fiacial market, the parameter of the time-space B-S equatio should be selected properly accordig to the actual data 5 Coclusios I this work, the uiversal differece method is employed to solve the time-space fractioal B-S equatio with the boudary coditios satisfied by stadard Europea call optios Theoretical aalysis demostrates that the uiversal differece method satisfies coditioal stability ad covergece Numerical experimets are well i agreemet with theoretical aalysis All the results illustrate that the time-space fractioal B-S equatio is effective ad the uiversal differece scheme is feasible to solve the time-space fractioal B-S equatio Competig iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Authors cotributios All authors cotributed equally ad sigificatly i writig this article All authors read ad approved the fial mauscript Ackowledgemets This work is sposored by the project Natioal Sciece Foudatio of Chia (No ), the Fudametal Research Fuds for the Cetral Uiversities (No 13QN30) Received: 4 November 015 Accepted: 4 February 016 Refereces 1 Kwok, Y: Mathematical Models of Fiacial Derivatives, d ed Spriger, Berli (008) Jiag, LS, Xu, CL, et al: Mathematical Model ad Case Aalysis of the Pricig of Fiacial Derivatives Higher Educatio Press, Beijig (008) (i Chiese) 3 Wyss, W: The fractioal Black-Scholes equatios Fract Calc Appl Aal 3(1),51-61 (000) 4 Jumarie, G: Stock exchage fractioal dyamics defied as fractioal expoetial growth drive by Gaussia white oise Applicatio to fractioal Black-Scholes equatios Isur Math Eco 4(1),71-87 (008) 5 Jumarie, G: Derivatio ad solutios of some fractioal Black-Scholes equatios i coarse-graied space ad time Applicatio to Merto s optimal portfolio Comput Math Appl 59(3), (010) 6 Cartea, A, del-castillo-negrete, D: Fractioal diffusio models of optio prices i markets with jumps Phys A, Stat Mech Appl 374(), (007) 7 Zeg, CB, Che, YQ, Yag, QG: Almost sure ad momet stability properties of fractioal order Black-Scholes model Fract Calc Appl Aal 16(), (013) 8 Meg, L, Wag, M: Compariso of Black-Scholes formula with fractioal Black-Scholes formula i the foreig exchage optio market with chagig volatility Asia-Pac Fiac Mark 17(), (010) 9 Barles, G, Soer, HM: Optio pricig with trasactio costs ad a oliear Black-Scholes equatio Fiace Stoch (4), (1998) 10 Kou, SG: A jump-diffusio model for optio pricig Maag Sci 48(8), (00) 11 Diethelm, K: The Aalysis of Fractio Differetial Equatios Spriger, Berli (010) 1 Guo, BL, Pu, XK, Huag, FH: Fractioal Partial Differetial Equatios ad Their Numerical Solutios Sciece Press, Beijig (011) (i Chiese) 13 Su, ZZ, Gao, GH: Fiite Differece Method for Fractioal Differetial Equatios Sciece Press, Beijig (015) (i Chiese) 14 Ta, PY, Zhag, XD: A umerical method for the space-time fractioal covectio-diffusio equatio Math Numer Si 30(3), (008) (i Chiese) 15 Laglads, TAM, Hery, BI: The accuracy ad stability of a implicit solutio method for the fractioal diffusio equatio J Comput Phys 05, (005) 16 Tadjera, C, Meerschaert, MM, Scheffler, HP: A secod-order accurate umerical approximatio for the fractio diffusio equatio J Comput Phys 13(1), (006) 17 Liu, F, Zhuag, P, Ah, V, Turer, I, Burrage, K: Stability ad covergece of the differece methods for the space-time fractioal advectio-diffusio equatio Appl Math Comput 191(1), 1-0 (007) 18 Che, CM, Liu, FW, Kevi, B: Fiite differece methods ad a Fourier aalysis for the fractioal reactio-subdiffusio equatio Appl Math Comput 198(), (008) 19 Kumar, S, Yildirim, A, et al: Aalytical solutio of fractioal Black-Scholes Europea optio pricig equatio by usig Laplace trasform J Fract Calc Appl (8), 1-9 (01) 0 Kumar, S, Kumar, D, Sigh, J: Numerical computatio of fractioal Black-Scholes equatio arisig i fiacial market Egypt J Basic Appl Sci 1(3-4), (014) 1 Ghadehari, MAM, Rajbar, M: Europea optio pricig of fractioal Black-Scholes model with ew Lagrage multipliers Comput Methods Differ Equ (1), 1-10 (014) Sog, LN, Wag, WG: Solutio of the fractioal Black-Scholes optio pricig model by fiite differece method Abstr Appl Aal 013, Article ID (013)

14 XiaozhogetalAdvaces i Differece Equatios (016) 016:71 Page 14 of 14 3 Yag, XZ, Zhag, X, Wu, LF: A kid of efficiet differece method for time-fractioal optio pricig model Appl Math J Chi Uiv Ser A 30(), (015) (i Chiese) 4 Carr, P, Wu, LR: Time-chaged Levy processes ad optio pricig J Fiac Eco 71(1), (004)