Compound Option

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Compound Option

A compound option is an option on an option. The exercise payoff of a compound option involves the value of another option. Therefore, a compound option then has two expiration dates and two strike prices. Take the example of a European style call on a call. On the first expiration date T1, the holder has the right to buy a new call using the strike price X1. The new call has expiration date T2 and strike price X2.

There are also two pieces of (exchange rate) volatilities corresponding to the option and its underlying options in the model. All of them come from one (spot) volatility surface

A generalization of compound option is the chooser option where the holder on the first expiration date T1 can choose whether the option is a call or a put. Both the call and put have the same expiration date T2 and strike price X. The underlying options can be comprised of a linear combination of options with different strikes, notional amounts and maturities. Thus, the option on strategy has very broad coverage.

The pricing of many other derivative instruments can be modeled as compound options. By visualizing the underlying stock as an option on the firm value, an option on stock of a levered firm that expires earlier than the maturity date of the debt issued by the firm can be regarded as a compound option on the firm value.

On the expiration of the option (the first expiration date of the compound option), the holder chooses to acquire the stock or otherwise. The decision depends on whether the stock as a call on the firm value is more valuable than the strike price. Another example is the pricing of American options with discrete dividends

At time right before a dividend payment date, the holder chooses to exercise the American option to receive the stock (plus the dividend payment right after the dividend date) or continue to hold the option. This is like a chooser option with the dividend payment date as the choose date. The holder makes the choice of holding the American call option or receiving the exercise payoff of the American call.

The decision depends on the relative magnitude of the dividend payment and the insurance value associated with the continued holding of the American call. Bonds with extended maturities can be modeled as compound options. On the first maturity date, the bondholder chooses to extend the maturity of the bond or receive the face value. The financial decision depends on the relative magnitude of the face value and the present value of the extended bond on the first maturity date

Lastly, in real options applications, multi-stage investment project can be modeled as compound American options. The exercise of a presently available real option opens the opportunity for the implementation of future real options.

There are various types of compound options:

  • compound options (4 products: call/put on call/put);
  • call and put options on collar (2 products);
  • call and put options on bull spread between calls (2 products);
  • call and put options on bull spread between puts (2 products);
  • call and put options on straddle/strangle (4 products);
  • call and put options on butterfly (2 products).

Each of the 16 options have 8 Greeks (delta, gamma, vega, rho, rhof, theta, volga, vanna). For an option on purchasing 100 units of the (foreign) currency, We use the following approximations (V stands for option price).


Delta calculation
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