The Obstacle Problem Arising from the American Chooser Option

1. Problem Statement and Motivation

The Financial Context:
The paper addresses the mathematical pricing of the American Chooser Option. This financial derivative grants the holder the right to choose, at any time prior to expiration $T$, whether to exercise the option as an American Call or an American Put. The payoff at the exercise time $\theta$ is $\max\{C^A(\theta, S_\theta), P^A(\theta, S_\theta)\}$, where $C^A$ and $P^A$ are the values of the corresponding American Call and Put options.

The Mathematical Challenge:
Pricing this option leads to a parabolic obstacle problem. The core difficulty lies in the nature of the obstacle function.

• In standard American options, the obstacle is an explicit payoff function (e.g., $(S-K)^+$).
• In the American Chooser option, the obstacle is $J(t, s) = \max\{C^A(t, s), P^A(t, s)\}$.
• Crucially, $C^A$ and $P^A$ are themselves solutions to separate obstacle problems. Therefore, the obstacle for the chooser option is the maximum of two unknown strong solutions, making it implicit, non-smooth, and highly complex to analyze using standard PDE techniques.

Objective:
The authors aim to establish the existence and uniqueness of a strong solution to this specific obstacle problem and rigorously prove the monotonicity and smoothness ($C^\infty$) of the resulting free boundaries.

---

2. Methodology

The authors employ a rigorous Partial Differential Equation (PDE) framework, distinguishing their work from previous probabilistic approaches. The methodology involves several key steps:

A. Transformation to Forward Problems

To simplify the analysis, the authors perform a change of variables:

• Time reversal: $\tau = T - t$.
• Log-price transformation: $x = \ln s$.
  This converts the backward parabolic problems into forward non-degenerate parabolic obstacle problems defined on domains $\Omega_T = (0, T) \times \mathbb{R}$. The solutions are denoted as $V(\tau, x)$, $C(\tau, x)$, and $P(\tau, x)$.

B. Penalty Method for Existence

To handle the inequality constraints of the obstacle problem, the authors use the penalty method:

1. Approximation: They introduce a penalized problem where the obstacle constraint is relaxed using a smooth penalty function $\beta_\epsilon$.
2. Regularization: Since the obstacle $J = \max\{C, P\}$ is not differentiable at the intersection point of $C$ and $P$, they approximate $J$ using a smooth function $J_\epsilon = \phi_\epsilon(C-P) + P$, where $\phi_\epsilon$ is a smooth approximation of the max function.
3. Fixed Point Argument: They prove the existence of a solution to the penalized problem on a bounded domain using the Schauder fixed point theorem.
4. Convergence: They establish uniform bounds (a priori estimates) for the penalized solutions and their derivatives. By taking limits as the penalty parameter $\epsilon \to 0$ and the domain size $n \to \infty$, they construct a solution to the original obstacle problem.

C. Analysis of Free Boundaries

The paper investigates the structure of the free boundaries (the optimal exercise boundaries).

• Decomposition: The authors prove that the exercise region of the chooser option is the disjoint union of the regions where the solution touches the Call payoff ($e^x - K_c$) and the Put payoff ($K_p - e^x$).
• Key Insight: They demonstrate that the solution $V$ only contacts the obstacle along the smooth payoff parts of the underlying options, not in their continuation regions. This is critical because it ensures the obstacle behaves like a smooth function near the free boundary.
• Stefan-Type Reformulation: By establishing that the contact occurs on smooth payoffs, they reformulate the problem as a Stefan-type free boundary problem. This allows them to apply standard regularity arguments (bootstrap arguments) to prove the smoothness of the boundaries.

---

3. Key Contributions and Results

A. Existence and Uniqueness

• Theorem 3.4: The authors prove the existence and uniqueness of a strong solution $V \in W^{1,2}_{p, loc}(\Omega_T) \cap C(\Omega_T)$ to the American Chooser obstacle problem.
• They overcome the non-standard nature of the obstacle (being the max of two solutions) by carefully analyzing the interaction between the Call and Put solutions and utilizing comparison principles.

B. Properties of the Solution

• Monotonicity: The solution $V(\tau, x)$ is non-decreasing with respect to time $\tau$ (Lemma 3.5).
• Derivative Bounds: The spatial derivative is bounded: $-e^x \leq \partial_x V \leq e^x$.

C. Structure of Free Boundaries

The paper identifies two distinct free boundaries:

1. $x^c_{ch}(\tau)$: The boundary where the holder exercises the Call option.
2. $x^p_{ch}(\tau)$: The boundary where the holder exercises the Put option.

Key Structural Results:

• Disjointness: The exercise regions for the Call and Put components are disjoint. Specifically, $x^p_{ch}(\tau) < x^c_{ch}(\tau)$ for all $\tau$.
• Monotonicity:
  • $x^p_{ch}(\tau)$ is strictly decreasing in time.
  • $x^c_{ch}(\tau)$ is strictly increasing in time.
• Initial Behavior: As $\tau \to 0^+$, the boundaries converge to specific limits determined by the intersection point $\bar{x}$ of the Call and Put values at $t=T$ and the standard American option boundaries.

D. Regularity of Free Boundaries

• Theorem 4.8: The most significant analytical result is the proof that both free boundaries $x^c_{ch}(\tau)$ and $x^p_{ch}(\tau)$ are infinitely differentiable ($C^\infty$) on $(0, T)$.
• This is achieved by deriving a Stefan-type condition (Equation 4.10) relating the velocity of the boundary to the spatial derivative of the solution. The non-vanishing denominator in this condition (guaranteed by the fact that contact occurs only on smooth payoff parts) allows the application of standard regularity theory.

---

4. Significance

1. Theoretical Advancement: This paper provides the first rigorous PDE-based analysis of the American Chooser option. Previous studies relied on probabilistic methods; this work bridges the gap by treating the problem as a complex obstacle problem with an implicit obstacle.
2. Handling Implicit Obstacles: The methodology developed to handle an obstacle defined as the maximum of two other obstacle solutions is novel. It demonstrates how to manage the lack of explicit pointwise knowledge of the obstacle while still proving strong regularity.
3. Financial Implications: The proof of $C^\infty$ smoothness for the free boundaries is crucial for numerical implementation. It validates the use of high-order numerical schemes (like finite difference or finite element methods) for pricing these complex derivatives, ensuring stability and convergence.
4. Structural Insight: The result that the exercise regions are disjoint and the boundaries are monotonic provides deep insight into the optimal exercise strategy: the holder will never be indifferent between exercising a Call and a Put at the same asset price; the decision is strictly separated by the asset price level.

In summary, the paper successfully resolves the mathematical complexities arising from the American Chooser option, establishing a solid theoretical foundation for its pricing and highlighting the smooth, well-behaved nature of its optimal exercise boundaries despite the non-standard structure of the problem.