Compound Options

A regular option gives you the right to buy or sell a stock at a fixed price on a future date. A compound option gives you the right to buy or sell another option at a fixed price on a future date — a call on a call, say, where you pay a small premium now for the right to decide, later, whether to pay a second premium to acquire the underlying option itself. It's used when a firm faces a decision that will only become relevant conditional on an earlier decision, such as a company holding the right to license a drug patent (the first option) only if an earlier stage of a trial succeeds (a gate the compound structure captures directly).

Pricing one is harder than pricing a plain option because the payoff at the first expiry depends on the value of the underlying option at that date, which itself depends on the stock's volatility, time to the underlying option's own expiry, and the strike embedded in it — so the compound option's value is sensitive to the volatility of the underlying option's price, a second-order quantity, not just the volatility of the stock. Geske's 1979 formula extends Black-Scholes to this two-stage problem using a bivariate normal distribution in place of the single normal distribution that appears in ordinary option pricing.

In practice, compound options show up less as traded exchange products and more as a modeling tool: employee stock options with vesting conditions, R&D and other staged corporate investments, and convertible bonds with embedded call features are often valued by treating the later decision as a compound option on the earlier one.