Perpetual American Option Values and the Time Derivative

Summary

The document sets up a question about valuing a perpetual American call under a geometric Ito stock-price model with continuous dividends. It defines the value as the supremum, over exercise times, of the expected discounted payoff, then asks how to interpret or derive a time derivative that a referenced treatment gives as negative interest rate times the value.

The question proposes expressing the stopping-time expectation as an integral over possible exercise times weighted by the probability that exercise occurs at each time. It does not provide a derivation or an answer, and that proposed expression is not examined in the text. In particular, the material does not clarify the distinction between time-to-maturity and calendar time, or the conditions under which the perpetual-option value satisfies a differential equation. It is therefore a useful statement of a modeling and calculus issue, but not a complete method for solving the optimal stopping problem.

Key ideas

• The option value is framed as a supremum of discounted expected exercise payoffs over stopping times.
• The underlying stock follows a diffusion with drift adjusted for continuous dividends.
• The question asks why a time derivative of the value is represented as negative interest rate times value.
• An integral over stopping times is proposed, but the document does not validate or derive that representation.
• No solution or conditions for the perpetual-option differential equation are supplied.

Tags

# Differentiation of value function in perpetual american option


# Differentiation of value function in perpetual american option












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)] $

I'm stuck in deriving the partial derivative of $V(S)$ w.r.t. time.

In my opinion $V(S)$ can be written as follows (where we integrate over all the values of time considering the probability density function of time $u$ being the stopping time $\tau$)

> $V(S) =sup_{\tau \in T}\ \int_0^\infty\ e^{-ru}max(S_u-K,\ 0)\ \mathbb{P}(\tau = u)\ du$

According to the reference (slide 9), the partial derivative of $V(S)$ w.r.t. time comes out to be $-rV(S)$ which is not very clear to me. So please help me out with in-depth proof, if possible.

Thanks :)

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