Valuing Options with No Expiration Under Geometric Brownian Motion

Summary

The document poses a theoretical question about options with no expiration when the underlying stock follows geometric Brownian motion. It asks what differential equation their values satisfy and what the most general solution looks like, then considers whether that solution applies to initially out-of-the-money American calls and puts.

The author suggests taking the Black–Scholes time to expiry toward infinity but offers no derivation or answer. The post therefore identifies questions about boundary conditions, exercise rights, and whether a mathematical solution applies across all stock prices and times, rather than presenting a complete valuation method. It provides no numerical evidence or worked example, so readers would need additional material to resolve the questions.

Key ideas

• The post asks how option values behave when there is no finite expiration date.
• It frames the valuation problem under geometric Brownian motion for the underlying stock.
• It asks for the governing differential equation and its general solution.
• It highlights American exercise and initial moneyness as possible limits on applying that solution.

Tags

# Option that never expires


# Option that never expires












I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, but that would only leave the stock price, $S_t$, on the right-hand side, if I am not mistaken. This seems a bit too simplistic in my opinion, especially since questions 2 and 3 are asking to elaborate on the answer from part 1. Any help/hints would be much appreciated. Thanks!

A stock whose price S follows geometric Brownian motion, $\frac{dS}{S} = \mu dt+ \sigma dB$, has options that never expire.

- What differential equation do the option values satisfy?

- What is the most general solution of the differential equation?

- Consider the cases of American-style call and put options of strike K that are initially out of the money. Does the general solution hold in each case? Does it hold for all values of S and t?

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