ORIGINAL_ARTICLE
European option pricing of fractional Black-Scholes model with new Lagrange multipliers
In this paper, a new identification of the Lagrange multipliers by means of the Sumudu transform, is employed to btain a quick and accurate solution to the fractional Black-Scholes equation with the initial condition for a European option pricing problem. Undoubtedly this model is the most well known model for pricing financial derivatives. The fractional derivatives is described in Caputo sense. This method finds the analytical solution without any discretization or additive assumption. The analytical method has been applied in the form of convergent power series with easily computable components. Some illustrative examples are presented to explain the efficiency and simplicity of the proposed method.
https://cmde.tabrizu.ac.ir/article_1118_016b1d6fb802cae6f2eb541551438d26.pdf
2014-07-01
1
10
Sumudu transforms
Fractional Black- Scholes equation
European option pricing problem
Mohammad Ali Mohebbi
Ghandehari
mohammadalimohebbi@yahoo.com
1
Azarbijan Shahid Madani University
AUTHOR
Mojtaba
Ranjbar
ranjbar633@gmail.com
2
Azarbijan Shahid Madani University
LEAD_AUTHOR
[1] M.A. Asiru, Further properties of the Sumudu transform and its applications, International
1
Journal of Mathematical Education in Science and Technology 33 (2002), 441-449.
2
[2] G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes
3
equation, Finance and Stochastics 2 (1998), 369-397.
4
[3] F.B.M. Belgacem, A.A. Karaballi, Sumudu transform fundamental properties investigations
5
and applications, Journal of Applied Mathematics and Stochastic Analysis, (2006), 1-23.
6
[4] T. Bjork and H. Hult, A note on Wick products and the fractional Black-Scholes model,
7
Finance and Stochastics 9 (2005), 197-209.
8
[5] C. Castelli. The theory of options in stocks and shares. Fc Mathieson, London, 1877.
9
[6] Z. Cen and A. Le, A robust and accurate finite difference method for a generalized BlackScholes
10
equation. J. Comput. Appl. Math, 235 (2011), 3728-3733.
11
[7] V.B.L. Chaurasia and J. Singh, Application of Sumudu transform in Schodinger equation
12
occurring in quantum mechanics, Applied Mathematical Sciences 4 (2010), 2843- 2850.
13
[8] M.H.A. Davis, V.G. Panas, and T. Zariphopoulou, European option pricing with transaction
14
costs, SIAMJournal on Control and Optimization 31 (1993), 470-493.
15
[9] K. Diethelm and N.J. Ford, Multi-order fractional differential equations and their numerical
16
solution, Applied Mathematics and Computation 154 (2004), 621-640.
17
[10] J.S. Duan, R. Rach, D. Buleanu, and A.M. Wazwaz, A review of the Adomian decomposition
18
method and its applicaitons to fractional differential equations, Communications in Fractional
19
Calculus 3 (2012), 73-99.
20
[11] M.A.M. Ghandehari and M. Ranjbar, A numerical method for solving a fractional partial
21
differential equation through converting it into an NLP problem, Computers and Mathematics
22
with Applications. 65 (2013), 975-982.
23
[12] V. Gulkac, The homotopy perturbation method for the Black-Scholes equation, J. Stat. Comput.
24
Simul. 80 (2010), 1349-1354.
25
[13] J.H. He, Approximate analytical solution for seepage flow with fractional derivative in porous
26
media.Comput. Methods Appl. Mech. Eng. 167 (1998), 57-68.
27
[14] J.H. He, Variational iteration methoda kind of non-linear analytical technique: some examples,
28
International Journal of Non-Linear Mechanics 34 (1999), 699-708.
29
[15] J.C. Hull, Options future and other derivatives. Sixth edition, Pearson prentice Hall, Toronto,
30
[16] J.C. Hull and A.D. White, Thepricingof optionsonassetswith stochastic volatilities, Journal
31
of Finance 42 (1987), 281-300.
32
[17] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical
33
physics. In: Nemat-Nassed S, editor. Variational method in the mechanics of solids,
34
pp. 156-162, Pergamon press, 1978.
35
[18] X.Y. Jiang and H.T. Qi, Thermal wave model of bioheat transfer with modified RiemannLiouville
36
fractional derivative, Journal of Physics 45 (2012).
37
[19] S. Kumar, A. Yildirim, Y. Khan, H. Jafari, K. Sayevand and L. Wey, Analytical solution of
38
fractional Black-Scholes European option pricing equation by using laplace transform. Journal
39
of Fractional Calculus and Applications, 2 (2012), 1-9.
40
[20] J.R. Liang, J. Wang, W.J. Zhang, W.Y. Qiu, and F.Y. Ren, Option pricing of a bi-fractional
41
Black-Merton-Scholes model with the Hurst exponent H in [1/2,1], Applied Mathematics
42
Letters 23 (2010), 859-863.
43
[21] F.W. Liu, V. Anh, and I. Turner, Numerical solutionof the space fractional Fokker-Planck
44
equation, Journal of Computational and Applied Mathematics 166 (2004), 209-219.
45
[22] J.H. Ma and Y.Q. Liu, Exact solutions for a generalized nonlinear fractional Fokker-Planck
46
equation, Nonlinear Analysis. RealWorld Applications 11 (2010), 515-521.
47
[23] R.C. Merton, On the pricing of corporate debt: the risk structure of interest rates, Journal
48
of Finance 29 (1974), 449- 470.
49
[24] R.C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of
50
Financial Economics, 3 (1976), 125-144.
51
[25] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in: S.
52
Rionero, T. Ruggeeri, Waves and stability in continuous media, World scientific, Singapore ,
53
[26] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,
54
Academic Press, San Diego, Calif, USA, 1999.
55
[27] X.T. Wang, Scaling and long-range dependence in option pricing I: Pricing European option
56
with transaction costs under the fractional Black-Scholes model, Physica A 389 (2010), 438-
57
[28] G.C. Wu, D. Baleanu, Variational iteration method for the Burgers flow with fractional
58
derivatives-New Lagrange multipliers, Applied Mathematical Modelling 37 (2013), 6183-6190.
59
ORIGINAL_ARTICLE
Exact travelling wave solutions for some complex nonlinear partial
differential equations
This paper reflects the implementation of a reliable technique which is called $left(frac{G'}{G}right)$-expansion ethod for constructing exact travelling wave solutions of nonlinear partial differential equations. The proposed algorithm has been successfully tested on two two selected equations, the balance numbers of which are not positive integers namely Kundu-Eckhaus equation and Derivative nonlinear Schr"{o}dinger’s equation. This method is a powerful tool for searching exact travelling solutions in closed form.
https://cmde.tabrizu.ac.ir/article_1199_fb3739857771654c2517f5bfd6a6baeb.pdf
2014-07-01
11
18
$frac{G'}{G}$-expansion method
Kundu-Eckhaus
equation
Derivative nonlinear Schr"{o}dinger’s equation
N.
Taghizadeh
1
University of Guilan
AUTHOR
Mohammad
Mirzazadeh
mirzazadehs2@guilan.ac.ir
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
LEAD_AUTHOR
M.
Eslami
3
University of Mazandaran
AUTHOR
M.
Moradi
4
University of Guilan
AUTHOR
[1] H. Naher, F. A. Abdullah, The improved ( G′/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), (2014) 220-235.
1
[2] H. Naher, F. A. Abdullah, New generalized and improved ( G′/G)-expansion method for nonlinear evolution equations in mathematical physics. Journal of the Egyptian Mathematical Society, http://dx.doi.org/10.1016/j.joems.2013.11.008.
2
[3] H. Naher, F. A. Abdullah, New approach of ( G′/G)-expansion method and new approach of generalized ( G′/G)-expansion method for nonlinear evolution equation, AIP Advances, 3(3) (2013) 032116.
3
[4] M.L. Wang, X.Z. Li, J.L. Zhang, The ( G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A, 372 (2008) 417-423.
4
[5] W.M. Taha, M.S.M. Noorani, I. Hashim, New exact solutions of sixth-order thin-film equation.
5
Journal of King Saud University- Science, 26 (2014) 75-78.
6
[6] G. Ebadi, A. Biswas, The ( G′/G) method and topological soliton solution of the K(m, n) equation. Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2377-2382.
7
[7] E. Zayed, K.A. Gepreel, Some applications of the ( G′/G)-expansionmethod to non-linear partial differential equations. Appl. Math. Comput. 212(1) (2009) 113.
8
[8] M. Mirzazadeh, M. Eslami, A. Biswas, Soliton solutions of the generalized Klein-Gordon equation by using ( G′/G)-expansion method, Comp. Appl. Math. DOI 10.1007/s40314-013-0098-3.
9
[9] M. Eslami, M. Mirzazadeh, A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger’s equation in optical fibers with time-dependent coefficients by simplest equation approach.Journal of Modern Optics, 60(19) (2013) 1627-1636.
10
[10] N. Taghizadeh, M. Mirzazadeh, The simplest equation method to study perturbed nonlinear Schrodingers equation with Kerr law nonlinearity. Commun. Nonlinear Sci. Numer Simulat. 17 (2012) 1493-1499.
11
[11] A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei, A. Biswas, New exact travelling wave solutions for DS-I and DS-II equations. Nonlinear Anal.: Modell. Control, 17 (3) (2012) 369-378.
12
[12] A. Biswas, Optical Solitons with Time-Dependent Dispersion, Nonlinearity and Attenuation in a Kerr-Law Media. Int. J. Theor. Phys. 48 (2009) 256-260.
13
[13] A. Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution. Phys. Lett. A. 372(25) (2008) 4601-460.
14
[14] A. Biswas, 1-Soliton solution of the K(m,n) equation with generalized evolution and timedependent damping and dispersion. Comput. Math. Appl. 59(8) (2010) 2538-2542.
15
[15] M. Eslami, M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrodinger equation with dual-power law nonlinearity in nonlinear optical fibers. Eur. Phys. J. Plus, (2013) 128-140.
16
[16] N. Taghizadeh, M. Mirzazadeh, A. Samiei Paghaleh, Exact solutions of some nonlinear evolution equations via the first integral method. Ain Shams Engineering Journal, 4 (2013) 493-499.
17
[17] F. Tascan, A. Bekir, M. Koparan, Travelling wave solutions of nonlinear evolutions by using the first integral method. Commun. Nonlinear Sci. Numer Simul. 14 (2009) 1810-1815.
18
[18] F. Tascan, A. Bekir, Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl. Math. Comput. 207 (2009) 279-282.
19
[19] A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana J. Phys. 81 (2013) 225-236.
20
[20] M. Mirzazadeh, M. Eslami, Exact solutions for nonlinear variants of Kadomtsev-Petviashvili (n,n) equation using functional variable method.Pramana J. Phys. 81 (2013) 225-236.
21
[21] A.C. Cevikel, A. Bekir, M. Akar and S. San, A procedure to construct exact solutions of nonlinear evolution equations. Pramana J. Phys. 79(3) (2012) 337-344.
22
[22] Dmitry Levko, Alexander Volkov, Modeling of Kundu-Eckhaus equation, (2006), ArXiv:nlin.PS/0702050.
23
[23] A. Biswas, K. Porsezian, Soliton perturbation theory for the modified nonlinear Schrodingers equation. Commun. Nonlinear Sci. Numer. Simulat. 12 (2007) 886-903.
24
ORIGINAL_ARTICLE
Asymptotic distributions of Neumann problem for Sturm-Liouville equation
In this paper we apply the Homotopy perturbation method to derive the higher-order asymptotic distribution of the eigenvalues and eigenfunctions associated with the linear real second order equation of Sturm-liouville type on $[0,pi]$ with Neumann conditions $(y'(0)=y'(pi)=0)$ where $q$ is a real-valued Sign-indefinite number of $C^{1}[0,pi]$ and $lambda$ is a real parameter.
https://cmde.tabrizu.ac.ir/article_1322_90f31a367ef89be733f0c5ba5934a118.pdf
2014-07-01
19
25
Sturm-Liouville
Nondefinite problem
Homotopy perturbation method
Asymptotic distribution
Hamidreza
Marasi
hamidreza.marasi@gmail.com
1
University of Bonab, Bonab, Iran
LEAD_AUTHOR
Esmail
Khezri
ekhezri@yahoo.com
2
University of Bonab, Bonab, Iran
AUTHOR
[1] F. V. Atkinson, A. B. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted Sturm-Liouville problems, J. Reine angev. Math. 375 (1987), 380-393.
1
[2] M. Duman, Asymptotics for the Sturm-Liouville problem by homotopy perturbation method, Applied Mathematics and Computation, 216 (2010), 492-496.
2
[3] J. H. He, Homotopy perturbation technique, computer Methods in Applied mathematics and Engineering, 178 (1999), 257-262.
3
[4] J. H. He, Homotopy perturbation method: a new nonlinear technique, Applied Mathematics and Computation, 135 (2003), 73-79.
4
[5] H. Hochstsdt, Differential equations, Dover, New york, 1957.
5
[6] E. L. Ince, Ordinary differential equations, Dover, New york,1956.
6
[7] B. M. Levitan, G. Gasymov, Determination of a differential equation by its spectra, Russian Math. Surveys. 19 (1964), 1-63.
7
[8] B. M. Levitan, G. Gasymov, Introduction to spectral theory, Translation of Math. Amer. Math. Society (1975).
8
[9] A. B. Mingarelli, A survey of the regular weighted Sturm-Liouville problem, The non-definite case, P. Fuquan, X. Shutie(Eds.), Applied Differential Equations, World scientific, Singapoure and Philadelphia(1986), 109-137.
9
[10] Z. M. Odibat, A new modification of the homotopy perturbation method for linear and nonlinear operators, Applied Mathematics and Computation, 189 (2007), 746-753.
10
[11] J. I. Ramos, Piecewise homotopy methods for nonlinear ordinary differential equations, Appled Mathematics and Computation, 198 (2008), 92-116.
11
[12] F. G. Tricomi, Differential equations, Hofner, New york, 1961.
12
[13] E. yusufoglu, A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modeling, 47 (2008), 1099-1107.
13
ORIGINAL_ARTICLE
Exact solutions of distinct physical structures to the fractional potential Kadomtsev-Petviashvili equation
In this paper, Exp-function and (G′/G)expansion methods are presented to derive traveling wave solutions for a class of nonlinear space-time fractional differential equations. As a results, some new exact traveling wave solutions are obtained.
https://cmde.tabrizu.ac.ir/article_1334_c581afeccef02be0b79b53ac365d021a.pdf
2014-07-01
26
36
Exact solution
Fractional differential equations
modified Riemann--Liouville derivative
space-time fractional Potential Kadomtsev-Petviashvili equation
solitons
Ahmet
Bekir
bekirahmet@gmail.com
1
Eskisehir Osmangazi University, Art-Science Faculty,
Department of Mathematics-Computer
LEAD_AUTHOR
Ozkan
Guner
ozkanguner@karatekin.edu.tr
2
Dumlupınar University
AUTHOR
ORIGINAL_ARTICLE
Solving The Stefan Problem with Kinetics
We introduce and discuss the Homotopy perturbation method, the Adomian decomposition method and the variational iteration method for solving the stefan problem with kinetics. Then, we give an example of the stefan problem with kinetics and solve it by these methods.
https://cmde.tabrizu.ac.ir/article_1569_7ca4dc8443ac085037c248f5d280329e.pdf
2014-07-01
37
49
stefan problem
kinetics
Homotopy perturbation method
Adomian Decomposition Method
variational iteration method
Ali
Beiranvand
alibeiranvand36@gmail.com
1
Faculty of mathematical sciences, university of tabriz, tabriz, Iran.
AUTHOR
Karim
Ivaz
ivaz@tabrizu.ac.ir
2
University of Tabriz, Iran
LEAD_AUTHOR
ORIGINAL_ARTICLE
Application of the Kudryashov method and the functional variable method for the complex KdV equation
In this present work, the Kudryashov method and the functional variable method are used to construct exact solutions of the complex KdV equation. The Kudryashov method and the functional variable method are powerful methods for obtaining exact solutions of nonlinear evolution equations.
https://cmde.tabrizu.ac.ir/article_1585_379a8017132c4021d5f76ff4a353ed26.pdf
2014-07-01
50
55
Kudryashov method
functional variable method
complex KdV equation
Mojgan
Akbari
m_akbari@guilan.ac.ir
1
P.h.D
LEAD_AUTHOR
ORIGINAL_ARTICLE
Inverse Laplace transform method for multiple solutions of the fractional Sturm-Liouville problems
In this paper, inverse Laplace transform method is applied to analytical solution of the fractional Sturm-Liouville problems. The method introduces a powerful tool for solving the eigenvalues of the fractional Sturm-Liouville problems. The results how that the simplicity and efficiency of this method.
https://cmde.tabrizu.ac.ir/article_2498_1309b9e8503adfd2a6e6e1bb6afc7769.pdf
2014-01-01
56
61
Laplace transform
Fractional Sturm-Liouville problem
Caputo's fractional derivative
eigenvalue
Farhad
Dastmalchi Saei
farhadsaei@gmail.com
1
Tabriz Azad University
LEAD_AUTHOR
Sadegh
Abbasi
s.abbasi2000@yahoo.com
2
Tabriz Azad University
AUTHOR
Zhila
Mirzayi
mirzayi93@yahoo.com
3
Tabriz Azad University
AUTHOR
[1] S. Abbasbandy, A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional
1
Sturm-Liouville problems, Numer Algor 54 (2010), 521-532.
2
[2] Q.M. Al-Mdallal, An efficient method for solving fractional Sturm-Liouville problems, Chaos
3
Soluitions Fractals 40 (2009), 138-189.
4
[3] A.M. Cohen, Numerical method for Laplace transform inversion, springer, New York, 2007.
5
[4] G.M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), Comptes Rendusdel Academie des Sciences
6
SerieII Paris 137 (1903), 554-558.
7
[5] P. Humbert and R.P. Agarwal, Surlafonction de Mittag-Leffler et quelques-unes de ses generalisations,
8
Bulletin des Sciences Mathematiques SeriesII 77 (1953), 180-185.
9
[6] K.B. Oldham and j. Spanier, The Fractional Calculus, Academic, New York, 1974.
10
[7] K.S. Miller and B Ross, An introduction to the fractional calculus and fractional differential
11
equations, John Wiley and Sons, New York, 1993.
12
[8] I. Podlubny, Fractional differential equations, Academic, New York, 1999.
13
[9] SG. Samko, AA. Kilbas, and OI. Marichev, Fractional integrals and derivatives, Berlin: Gorden
14
and Breach, 1993.
15
[10] H. Sheng, Y. Li, and Y. Chen, Application of numerical inverse Laplace transform algorithms
16
in fractional calculus, Journal of the Franklin Institute 348 (2011), 315-330.
17