Smooth Pasting and Exercise Boundaries for Perpetual American Calls

Summary

The document examines how to determine the optimal exercise threshold for a perpetual American call when the stock follows a geometric Ito process with continuous dividends. In the continuation region, it describes a value function of power form and gives value matching at the exercise boundary, equating the option value there with its immediate exercise payoff. Since this condition alone links the value coefficient and the threshold, a second relation is needed to determine both.

The author contrasts an argument that chooses the coefficient to maximize the value function with the smooth-pasting condition, which matches the derivatives of the continuation value and exercise payoff at the boundary. The post notes that the two approaches produce different solutions and asks which is justified. It presents no resolution or numerical evidence, and the setup’s precise assumptions and references are not developed, so it is best read as a question about optimal-stopping boundary conditions rather than a complete derivation.

Key ideas

• The setup models a dividend-paying stock with a geometric Ito process.
• The perpetual call value is represented by a power function in the continuation region.
• Value matching at the exercise threshold supplies one relation between the value coefficient and threshold.
• Smooth pasting supplies a derivative-matching condition as a second relation.
• The document questions an alternative argument based on maximizing the value coefficient and gives no resolution.

Tags

# Boundary condition in perpetual american option problem


# Boundary condition in perpetual american option problem












I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.

> $dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $

where $D_0$ is the dividend. Basically the problem tries to find optimal stopping time to exercise the option i.e. we try to optimize the value function $V(S)$ w.r.t stopping time $\tau$.

> $V(S) = \sup_{\tau \in T}\ \mathbb{E}[e^{-r\tau}\max(S_\tau -K,\ 0)] $

Once we have the Value function $V(S)=AS^\beta$, we use the boundary condition at optimal time to get relation between $A$ and $S^*$.

> $V(S^{*})=S^*-K$

To get another relation (slide 12) for solving for both $A$ and $S^*$ we try to maximize $A$ so as to increase the value function as much as possible. I do not see any justification for this.

Contrary to the derivation in the above reference, another reference (equation 6) uses smooth pasting condition i.e.

> $V'(S^*) = f'(S*)$ ; $\ \ f(S) = S-K$

The difference in the techniques to obtain the second relation between $S^*$ and $A$ leads to different solutions. I'm not convinced with the argument of maximizing $A$. So if anybody can justify it, it'd be great. Also which solution is actually optimal?

Thanks :)

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