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## 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

* Corresponding author: Jehan Oh

Received  August 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author gratefully acknowledges the support of the National Research Foundation of Korea grant funded by the Korea government (Grant No. NRF-2017R1C1B1001811), BK21 PLUS SNU Mathematical Sciences Division and the POSCO Science Fellowship of POSCO TJ Park Foundation

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.

Citation: Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 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. [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.

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##### 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. [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.
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 $K_2 = 1.5$
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 $K_2 = 1.1$
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 $K_2 = 1.1$
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 $K_2 = 1.1$
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 $K_2 = 1.1$
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