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Solving Perpetual American Option Problems with a Free Boundary

Article Quant Q&A · Author: Wolfy
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Summary

The document outlines a method for formulating perpetual American option valuation as a free-boundary problem. In the continuation region, where holding the option is more valuable than exercising, the value satisfies the perpetual Black–Scholes differential equation. Its power-function solutions are combined with exercise conditions to determine the value function and the optimal exercise boundary.

The discussion uses payoffs that add a positive constant to calls and puts, as well as a straddle, but it does not complete the boundary analysis or give solutions for those specific cases. It suggests determining the shape of the continuation region by comparing the candidate value with intrinsic value. A second answer frames the derivation through Dynkin’s formula and the stock-price generator, then imposes the requirement that an American option be worth at least its exercise payoff. The setup assumes a positive interest rate; the supplied material is an outline rather than a full worked derivation.

Key ideas

  • In the continuation region, a perpetual American option satisfies the Black–Scholes differential equation.
  • Power functions provide candidate solutions to the resulting ordinary differential equation.
  • The optimal exercise boundary separates continuation from immediate exercise.
  • The option value must be at least as large as its intrinsic payoff.
  • Dynkin’s formula and the price process generator provide another route to the valuation equation.

Tags

OptionsDerivatives pricingVolatility
Full text
# Perpetual American options


# Perpetual American options












> Formulate and solve the free boundary problem for the perpetual American options with the following payoffs. a.) $(S - K)_{+} + a$ where $a > 0$ b.) $(K - S)_{+} + a$ where $a > 0$ c.) Straddle

I have no idea how to begin, I have some notes on perpetual American options but I am not sure how to solve these questions. I do not have much ode or pde experience but I am sure I would understand it if I saw the solution.

## Answer by M. Jeunesse (score 2, accepted)

https://quant.stackexchange.com/a/25562

I assume $r>0$.

Let look at a)

Let $v$ be the solution.

$v$ is increasing (easy to prove, take $x<y$ and show that $v(x)<v(y)$ due $(S^x_t-K)^++a<(S^y_t-K)^++a$

on the continuity region $C$, i.e $x:v(x)>(x-K)^++a$, you have : $$\text{Black Scholes PDE perpetual case : }\frac{1}{2}\sigma^2x^2v''(x)+rxv'(x)-rv(x)=0$$

solutions are of the form :

$$C_1x^{\frac{-2r}{\sigma^2}}+C_2x$$

Now you have to find out if $C=[0,x^\star)$ or $(x^\star,+\infty)$ or something more complicated $(x^\star_1,x^\star_2)$...

So study $x\to v(x)-(x-K)^+-a$,

## Answer by user9403 (score 4)

https://quant.stackexchange.com/a/25561

- Use Dynkin's formula to write the expectation: $\mathbb{E}[e^{-r\tau} \phi(S_\tau)]= g(S_0)+\mathbb{E}[\int_ 0 ^ \tau (A g -rg) dt]$ where $\phi$ is the payoff.

- Use the infinitismal generator $A$ to derive an ODE which describes the solution

- Use the fact that American options must be equal to or greater than their intrinsic value to derive boundary conditions for the ODE

- Solve the ODE (its pretty straightforward, but if you don't have any background in ODEs try linear combinations of a power of the stock price)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.