Compound Option
A compound option is an option on an option — a derivative contract that gives the holder the right (but not the obligation) to buy or sell another option at a specified price on or before a specified date. Compound options are used primarily to hedge situations where the need for protection is itself uncertain, reducing upfront cost compared to buying the underlying option outright.
Key takeaways
- There are four types: call on call (CoC), call on put (CoP), put on call (PoC), and put on put (PoP) — each with two strike prices and two expiration dates.
- Compound options are cheaper than outright options because the holder pays premium only for uncertainty over whether the underlying option will be needed.
- Common uses include hedging a bid for a foreign currency-denominated contract (where the currency exposure only crystallizes if the bid wins), and hedging callable bond portfolios.
- Valuation requires a two-dimensional integration over the bivariate normal distribution, first derived analytically by Robert Geske (1979).
- Compound options exhibit higher-order Greeks (such as sensitivity to the first expiry, the second expiry, and both strikes) that require careful risk management.
Explanation
The compound option structure addresses a fundamental problem in risk management: sometimes you need protection against a risk that may not materialize. A corporation bidding for a foreign project knows it will need to hedge FX risk if it wins the contract — but buying a full-sized FX option outright wastes premium if the bid fails. A call on a put (CoP) solves this: pay a small upfront premium for the right to buy a put option at a predetermined strike at a specific future date (e.g., the contract award date). If the bid succeeds, exercise the CoP and acquire the put; if the bid fails, let the CoP expire.
Geske's 1979 model provides the analytical pricing formula for a European call on a European call (CoC):
CoC = S·e^(−qT₂)·M(a₁, b₁; √(T₁/T₂)) − K₂·e^(−rT₂)·M(a₂, b₂; √(T₁/T₂)) − K₁·e^(−rT₁)·N(a₂)
where T₁ is the first expiration date, T₂ > T₁ is the underlying option's expiration, K₁ is the price to buy the underlying option at T₁, K₂ is the underlying option's strike, S is the current asset price, M(·,·;ρ) is the bivariate standard normal CDF, and a₁, a₂, b₁, b₂ are standardized parameters incorporating the critical asset price at T₁ (the price at which the holder is indifferent between exercising and not).
The two key features distinguishing compound option risk from vanilla option risk are: (1) the compound option's delta is lower than an equivalent vanilla's because the holder is one step removed from the underlying asset, and (2) the compound option has two sources of time decay — the first option loses time value as T₁ approaches, but the underlying option's time value also affects CoC value through the Geske model's bivariate structure. For risk desks, the interaction between the two expiration dates and strikes creates a complex multi-dimensional Greek surface that demands careful scenario analysis.
Formula
Geske CoC = S·e^(−qT₂)·M(a₁,b₁;√(T₁/T₂)) − K₂·e^(−rT₂)·M(a₂,b₂;√(T₁/T₂)) − K₁·e^(−rT₁)·N(a₂)
Example
An oil company is bidding on a deepwater exploration contract in Brazil denominated in BRL. If awarded (decision date: 3 months), the company will need to hedge its BRL/USD exposure over a 12-month development period using a 12-month BRL put option (right to sell BRL). Instead of buying the 12-month put outright at a cost of 3.8% of notional, the company buys a 3-month call on the 12-month put (a call on put) for 1.2% of notional. If it wins the bid, it exercises the compound option and acquires the BRL put at the predetermined strike, locking in its FX hedge. If it loses the bid, its maximum loss is the 1.2% compound option premium — far less than the 3.8% it would have spent on the outright put.
Related terms
Backwardation Contract Month Delta Expiration Date Implied Volatility Margin Mark To Market Option Premium Put Option Scenario Analysis Time Decay