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] |
Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751 |
| [2] |
Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004 |
| [3] |
Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial & Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435 |
| [4] |
Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial & Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783 |
| [5] |
T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 |
| [6] |
Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial & Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365 |
| [7] |
Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019 |
| [8] |
Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413 |
| [9] |
Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241 |
| [10] |
María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004 |
| [11] |
Xiaoyu Xing, Hailiang Yang. American type geometric step options. Journal of Industrial & Management Optimization, 2013, 9 (3) : 549-560. doi: 10.3934/jimo.2013.9.549 |
| [12] |
S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 |
| [13] |
Walter Allegretto, Yanping Lin, Ningning Yan. A posteriori error analysis for FEM of American options. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 957-978. doi: 10.3934/dcdsb.2006.6.957 |
| [14] |
Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 |
| [15] |
Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158 |
| [16] |
Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673 |
| [17] |
Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091 |
| [18] |
Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045 |
| [19] |
Pierangelo Ciurlia. On a general class of free boundary problems for European-style installment options with continuous payment plan. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1205-1224. doi: 10.3934/cpaa.2011.10.1205 |
| [20] |
Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431 |
2018 Impact Factor: 1.008
Tools
Metrics
Other articles
by authors
[Back to Top]