Perpetual American options

Question

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 1

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 2

1. 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.
2. Use the infinitismal generator $A$ to derive an ODE which describes the solution
3. Use the fact that American options must be equal to or greater than their intrinsic value to derive boundary conditions for the ODE
4. 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)