-
Previous Article
Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry
- CPAA Home
- This Issue
-
Next Article
Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space
(1+2)-dimensional Black-Scholes equations with mixed boundary conditions
| 1. | Department of Applied Mathematics & Institute of Natural Science, Kyung Hee University, Yongin, 17104, Republic of Korea |
| 2. | Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea |
In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximum-quanto lookback options and derive the pricing formula of this options.
References:
| [1] |
J. Bertrand, P. Bertrand and J. P. Ovarlez, The Mellin transform, in The Transforms and Applications Handbook (ed. A.D. Poularikas), IEEE Press, New York, (1996), 829–885. |
| [2] |
M. Dai, H. Y. Wong and Y. K. Kwok,
Quanto lookback options, Math. Finance, 14 (2004), 445-467.
doi: 10.1111/j.0960-1627.2004.00199.x. |
| [3] |
A. Dravid, M. Richardson and T. S Sun, Pricing foreign index contingent claims: An application to Nikkei index warrants, Journal of Derivatives, 1 (1993), 33-51. Google Scholar |
| [4] |
H. Eltayeb and A. Kılıçman,
A note on Mellin transform and partial differential equations, Int. J. Pure Appl. Math., 34 (2007), 457-467.
|
| [5] |
J. Jeon, H. Han, H. Kim and M. Kang,
An integral equation representation approach for valuing Russian options with a finite time horizon, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 496-516.
doi: 10.1016/j.cnsns.2015.12.019. |
| [6] |
I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York., 1972. Google Scholar |
| [7] |
J. H. Yoon and J. H. Kim,
The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl., 422 (2015), 838-857.
doi: 10.1016/j.jmaa.2014.09.015. |
show all references
References:
| [1] |
J. Bertrand, P. Bertrand and J. P. Ovarlez, The Mellin transform, in The Transforms and Applications Handbook (ed. A.D. Poularikas), IEEE Press, New York, (1996), 829–885. |
| [2] |
M. Dai, H. Y. Wong and Y. K. Kwok,
Quanto lookback options, Math. Finance, 14 (2004), 445-467.
doi: 10.1111/j.0960-1627.2004.00199.x. |
| [3] |
A. Dravid, M. Richardson and T. S Sun, Pricing foreign index contingent claims: An application to Nikkei index warrants, Journal of Derivatives, 1 (1993), 33-51. Google Scholar |
| [4] |
H. Eltayeb and A. Kılıçman,
A note on Mellin transform and partial differential equations, Int. J. Pure Appl. Math., 34 (2007), 457-467.
|
| [5] |
J. Jeon, H. Han, H. Kim and M. Kang,
An integral equation representation approach for valuing Russian options with a finite time horizon, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 496-516.
doi: 10.1016/j.cnsns.2015.12.019. |
| [6] |
I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York., 1972. Google Scholar |
| [7] |
J. H. Yoon and J. H. Kim,
The pricing of vulnerable options with double Mellin transforms, J. Math. Anal. Appl., 422 (2015), 838-857.
doi: 10.1016/j.jmaa.2014.09.015. |
| [1] |
Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505 |
| [2] |
Kais Hamza, Fima C. Klebaner. On nonexistence of non-constant volatility in the Black-Scholes formula. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 829-834. doi: 10.3934/dcdsb.2006.6.829 |
| [3] |
Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609 |
| [4] |
Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221 |
| [5] |
Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2841-2851. doi: 10.3934/dcdss.2020122 |
| [6] |
Zhenghuan Gao, Peihe Wang. Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021152 |
| [7] |
Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations & Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005 |
| [8] |
B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29 |
| [9] |
Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231 |
| [10] |
Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 |
| [11] |
Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240 |
| [12] |
Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006 |
| [13] |
Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252 |
| [14] |
Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 |
| [15] |
Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 |
| [16] |
Yana Guo, Yan Jia, Bo-Qing Dong. Global stability solution of the 2D MHD equations with mixed partial dissipation. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021141 |
| [17] |
Tak Kuen Siu, Howell Tong, Hailiang Yang. Option pricing under threshold autoregressive models by threshold Esscher transform. Journal of Industrial & Management Optimization, 2006, 2 (2) : 177-197. doi: 10.3934/jimo.2006.2.177 |
| [18] |
Figen Özpinar, Fethi Bin Muhammad Belgacem. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 615-624. doi: 10.3934/dcdss.2019039 |
| [19] |
Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. |
| [20] |
Yang Cao, Qiuting Zhao. Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021064 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]