Perpetual American Call Valuation

Article Quant Q&A · Author: user70365

Summary

The document outlines how to value a perpetual American call on a dividend-paying stock using a Black–Scholes framework. It applies Itô’s formula and a replication argument to obtain an ordinary differential equation for the option value, then proposes a power-function solution whose exponents solve a quadratic equation. A boundary condition at zero removes one term in the general solution.

The exercise threshold is determined by value matching, which equates the option value at the threshold to intrinsic value, and smooth pasting, which matches the option’s slope to that of the payoff. Once these conditions determine the value function, its sensitivity to dividend yield can be found by differentiating with respect to that yield. The post gives a derivation roadmap rather than the final closed-form value or Greek. It flags that the displayed PDE should be checked, and does not work through parameter assumptions or numerical examples.

Key ideas

Replication and Itô’s formula lead to a differential equation for the perpetual option value.
A power-function trial solution yields two characteristic exponents.
The zero-stock boundary condition eliminates one component of the general solution.
Value matching and smooth pasting determine the exercise threshold and option value.
Dividend-yield sensitivity can be computed by differentiating the resulting value function.

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Options Derivatives pricing

Full text

# perpetual American-style call option


# perpetual American-style call option












Greek “phi” for a derivative f is defined as its sensitivity to the changes in dividend yield q : $$\phi = \frac{\partial f}{\partial q}$$

HOW CAN I FIND PHI WITHOUT THE CORRELATION?

## Answer by rodrigo (score 3)

https://quant.stackexchange.com/a/77734

You may want to start by obtaining an expression for f:

In a B-S setting for the dynamics of the stock price with (continuous) dividend:

$$ dS(t)= (\mu-q)S(t)dt+\sigma S(t)dW $$

The option has value $$f(S(t))$$

The value of the option follows

$$ df(S(t))= \frac{df}{dS}dS +\frac{1}{2}\frac{d^2f}{dS^2} (dS)^2 $$

From here, you can use a replication argument (making the portfolio riskless) to get the PDE. Then, you can find a general form for the solution and pin it down using a boundary condition, a value-matching condition, and a smooth pasting condition.

The PDE will look like this (please double-check yourself) (' stands for derivative)

$$\frac{\sigma^2}{2}S^2f''(S) + (r-q)Sf'(S) - rf(S) =0$$

The general solution will have the form

$$f(S)= A_{1} S^{h1} + A_{2} S^{h2}$$

with $h1>1$ and $h2<0$

(h1 and h2 are the two solutions of the quadratic equation you get when plugging $A S^{h}$ back into the PDE.)

boundary condition

$$ lim_{x -> 0} (f) =0 $$

This tells you that one of the A's has to be 0 ( given the sign of h1 and h2)

Perpetual American options are exercised when the underlying hits a barrier level. Call $S^*$ this barrier.

Value matching

$$ f(S^{*}) = S^{*} -K $$

(because it is a call option)

Smooth pasting

$$ f'(S^{*}) = 1 $$

This will give you an expression for f(S(t)). Next, you can take the derivative w.r.t q for part a), and plug in numbers for b)

I hope this helps

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.

Valuing a Perpetual American Call with the Free-Boundary Method

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