Figure 1.
The change of the option value function $V(t, s)$ with respect to stock price $s$ where $r = 0.05, \;q = 0.1, \;\sigma = 0.3, \;K_1 = 1$ and $K_2 = 1.5$
-
Previous Article
Global dynamics of a latent HIV infection model with general incidence function and multiple delays
- DCDS-B Home
- This Issue
-
Next Article
Dirac-concentrations in an integro-pde model from evolutionary game theory
February
2019, 24(2): 755-781.
doi: 10.3934/dcdsb.2018206
Valuation of American strangle option: Variational inequality approach
| 1. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea |
| 2. | Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany |
In this paper, we investigate a parabolic variational inequality problem associated with the American strangle option pricing. We obtain the existence and uniqueness of $W^{2, 1}_{p, \rm{loc}}$ solution to the problem. Also, we analyze the smoothness and monotonicity of two free boundaries. Finally, numerical results of the model based on this problem are described and used to show the boundary properties and the price behavior.
Mathematics Subject Classification:
Primary: 35R35; Secondary: 91G80.
Citation:
Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B,
2019, 24
(2)
: 755-781.
doi: 10.3934/dcdsb.2018206
![]()
References:
| [1] |
J. S. Chaput and L. H. Ederington,
Volatility trade design, Journal of Futures Markets, 25 (2005), 243-279.
doi: 10.1002/fut.20142. |
| [2] |
X. Chen, F. Yi and L. Wang,
American lookback option with fixed strike price-2-D parabolic variational inequality, J. Differential Equations, 251 (2011), 3063-3089.
doi: 10.1016/j.jde.2011.07.027. |
| [3] |
C. Chiarella and A. Ziogas,
Evaluation of American strangles, Journal of Economic Dynamics and Control, 29 (2005), 31-62.
doi: 10.1016/j.jedc.2003.04.010. |
| [4] |
A. Friedman,
Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Funct. Anal., 18 (1975), 151-176.
doi: 10.1016/0022-1236(75)90022-1. |
| [5] |
A. Friedman, Variational Principles and Free-Boundary Problems, John Wiley & Sons, Inc., New York, 1982. |
| [6] |
L. Jiang,
Existence and differentiability of the solution of a two-phase Stefan problem for quasilinear parabolic equations, Acta Math. Sinica, 15 (1965), 749-764.
|
| [7] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. |
| [8] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
| [9] |
J. Ma, W. Li and Z. Cui,
Valuation of American strangles through an optimized lower-upper bound approach, Journal of Operations Research Society of China, 6 (2018), 25-47.
doi: 10.1007/s40305-017-0174-2. |
| [10] |
S. Qiu, American Strangle Options, Research Report, School of Mathematics, The University of Manchester, 2014. Google Scholar |
| [11] |
K. Tso,
On an Aleksandrov-Bakel'man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations, 10 (1985), 543-553.
doi: 10.1080/03605308508820388. |
| [12] |
Z. Yang and F. Yi,
Valuation of European installment put option: Variational inequality approach, Communications in Contemporary Mathematics, 11 (2009), 279-307.
doi: 10.1142/S0219199709003363. |
| [13] |
Z. Yang and F. Yi,
A variational inequality arising from American installment call options pricing, J. Math. Anal. Appl., 357 (2009), 54-68.
doi: 10.1016/j.jmaa.2009.03.045. |
| [14] |
Z. Yang, F. Yi and M. Dai,
A parabolic variational inequality arising from the valuation of strike reset options, J. Differential Equations, 230 (2006), 481-501.
doi: 10.1016/j.jde.2006.07.026. |
| [15] |
Z. Yang, F. Yi and X. Wang,
A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326.
doi: 10.1137/060670353. |
show all references
References:
| [1] |
J. S. Chaput and L. H. Ederington,
Volatility trade design, Journal of Futures Markets, 25 (2005), 243-279.
doi: 10.1002/fut.20142. |
| [2] |
X. Chen, F. Yi and L. Wang,
American lookback option with fixed strike price-2-D parabolic variational inequality, J. Differential Equations, 251 (2011), 3063-3089.
doi: 10.1016/j.jde.2011.07.027. |
| [3] |
C. Chiarella and A. Ziogas,
Evaluation of American strangles, Journal of Economic Dynamics and Control, 29 (2005), 31-62.
doi: 10.1016/j.jedc.2003.04.010. |
| [4] |
A. Friedman,
Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Funct. Anal., 18 (1975), 151-176.
doi: 10.1016/0022-1236(75)90022-1. |
| [5] |
A. Friedman, Variational Principles and Free-Boundary Problems, John Wiley & Sons, Inc., New York, 1982. |
| [6] |
L. Jiang,
Existence and differentiability of the solution of a two-phase Stefan problem for quasilinear parabolic equations, Acta Math. Sinica, 15 (1965), 749-764.
|
| [7] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. |
| [8] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
| [9] |
J. Ma, W. Li and Z. Cui,
Valuation of American strangles through an optimized lower-upper bound approach, Journal of Operations Research Society of China, 6 (2018), 25-47.
doi: 10.1007/s40305-017-0174-2. |
| [10] |
S. Qiu, American Strangle Options, Research Report, School of Mathematics, The University of Manchester, 2014. Google Scholar |
| [11] |
K. Tso,
On an Aleksandrov-Bakel'man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations, 10 (1985), 543-553.
doi: 10.1080/03605308508820388. |
| [12] |
Z. Yang and F. Yi,
Valuation of European installment put option: Variational inequality approach, Communications in Contemporary Mathematics, 11 (2009), 279-307.
doi: 10.1142/S0219199709003363. |
| [13] |
Z. Yang and F. Yi,
A variational inequality arising from American installment call options pricing, J. Math. Anal. Appl., 357 (2009), 54-68.
doi: 10.1016/j.jmaa.2009.03.045. |
| [14] |
Z. Yang, F. Yi and M. Dai,
A parabolic variational inequality arising from the valuation of strike reset options, J. Differential Equations, 230 (2006), 481-501.
doi: 10.1016/j.jde.2006.07.026. |
| [15] |
Z. Yang, F. Yi and X. Wang,
A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326.
doi: 10.1137/060670353. |
Figure 2.
The change of the free boundaries $A(\tau)$ and $B(\tau)$ with respect to $\sigma$ where $r = 0.05, \;q = 0.05, \;K_1 = 1$ and $K_2 = 1.1$
Figure 3.
Compare the free boundary $B(\tau)$ and the free boundary $F_{c}(\tau)$ with $r = 0.05, \;q = 0.05, \;\sigma = 0.2, \;K_1 = 1$ and $K_2 = 1.1$
Figure 4.
Compare the free boundary $A(\tau)$ and the free boundary $F_{p}(\tau)$ with $r = 0.05, \;q = 0.05, \;\sigma = 0.2, \;K_1 = 1$ and $K_2 = 1.1$
Figure 5.
Upper and lower bounds of $A(\tau)$ and the free boundary $B(\tau)$ , respectively, with $r = 0.05, \;q = 0.05, \;\sigma = 0.2, \;K_1 = 1$ and $K_2 = 1.1$
| [1] |
Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 |
| [2] |
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 |
| [3] |
Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021028 |
| [4] |
Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084 |
| [5] |
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065 |
| [6] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
| [7] |
Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 |
| [8] |
Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 |
| [9] |
Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154 |
| [10] |
Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 |
| [11] |
Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 |
| [12] |
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020453 |
| [13] |
Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083 |
| [14] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
| [15] |
Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 |
| [16] |
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398 |
| [17] |
Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052 |
| [18] |
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380 |
| [19] |
Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 |
| [20] |
Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]