# American Institute of Mathematical Sciences

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

* Corresponding author: Jehan Oh

Received  August 2017 Revised  March 2018 Published  February 2019 Early access  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, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
##### References:

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##### References:
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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