Valuation of American strangle option: Variational inequality approach

Junkee Jeon; Jehan Oh

doi:10.3934/dcdsb.2018206

Article Contents

2019, Volume 24, Issue 2: 755-781. Doi: 10.3934/dcdsb.2018206

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Valuation of American strangle option: Variational inequality approach

Junkee Jeon^1, and
Jehan Oh^2, ,

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
* Corresponding author: Jehan Oh

Received: July 31, 2017

Revised: February 28, 2018

Early access: June 2018

Published: January 2019

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35R35; Secondary: 91G80.

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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 $K_2 = 1.5$

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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 $K_2 = 1.1$

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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 $K_2 = 1.1$

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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 $K_2 = 1.1$

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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 $K_2 = 1.1$

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