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The Obstacle Problem Arising from the American Chooser Option

Gugyum Ha, Junkee Jeon, Jihoon Ok

Abstract

We study the obstacle problem associated with the American chooser option. The obstacle is given by the maximum of an American call option and an American put option, which, in turn, can be expressed as the maximum of the solutions to the corresponding obstacle problems. This structure makes the obstacle problem particularly challenging and non-trivial. Using theoretical analysis, we overcome these difficulties and establish the existence and uniqueness of a strong solution. Furthermore, we rigorously prove the monotonicity and smoothness of the free boundary arising from the obstacle problem.

The Obstacle Problem Arising from the American Chooser Option

Abstract

We study the obstacle problem associated with the American chooser option. The obstacle is given by the maximum of an American call option and an American put option, which, in turn, can be expressed as the maximum of the solutions to the corresponding obstacle problems. This structure makes the obstacle problem particularly challenging and non-trivial. Using theoretical analysis, we overcome these difficulties and establish the existence and uniqueness of a strong solution. Furthermore, we rigorously prove the monotonicity and smoothness of the free boundary arising from the obstacle problem.

Paper Structure

This paper contains 18 sections, 13 theorems, 166 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Financial Background: American Chooser Option
  5. Change of variables
  6. Properties of American Call and Put Options
  7. Existence and uniqueness of solution
  8. Existence and uniqueness in a bounded region
  9. Solution to the obstacle problem
  10. Estimate for the derivatives of V
  11. Analysis of the free boundary
  12. Representation of the free boundary
  13. Properties of the free boundary
  14. Regularity of the free boundary
  15. Formulation of the model
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\widehat{C}^A$ and $\widehat{P}^A$ be the solution to CO and PO, respectively, and define for each $\lambda\in \mathbb{R}$ that $\Omega_{\lambda}:=(0,\lambda)\times\mathbb{R}$. Then the following properties hold. Moreover, by eq:CA, the corresponding statements hold for $C$ and $P$ in $\Omega_T$.

Figures (3)

  • Figure 1: The free boundaries $\hat{x}_c(\tau)$ and $\hat{x}_p(\tau)$.
  • Figure 2: solution $V(\tau,x)$ and obstacle $\max\{C(\tau,x),P(\tau,x)\}.$
  • Figure 3: The free boundaries $x_c^{\rm ch}(\tau)$ and ${x}_p^{\rm ch}(\tau)$.

Theorems & Definitions (25)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.5
  • ...and 15 more