Compound Options are options on other options.
1. Call on Call (CoC)
2. Call on Put (CoP)
3. Put on Put (PoP)
4. Put on Call (PoC)
These options are highly sensitive to the volatility of the volatility as there is also an underlying option, making it more difficult to hedge, relative to standard single options.
Where S* is the value of the stock underlying the underlying option, is the underlying strike price and X is the compound strike. t is the expiry date of the compound and T is the expiry date of the underlying option.
The variables and are binary variables in that they take either values of 0 or 1. is given as 1 when the underlying option is a call, and -1 when the underlying option is a put option. is given as 1 when the compound is a call and -1 when the compound is a put.
This application of compound options was first considered by Geske (1977), followed similarly by Geske (1979), Selby & Hodges (1987) and Rubinstein (1991). The variables considered when valuing a compound option are:
1. Price of the underlying asset of the underlying option (S)
2. Exercise prices of underlying option and the compound option (X1 & X2)
3. Dividend payments (if any) on the underlying asset (D)
4. Risk free rate (r)
5. Expiry dates for the underlying option (T1) and the compound option (T2)
The 4 formulae for pricing the options are as follows:
For a call on call:
Call on put:
Put on call:
Put on put:
Where the variables are defined as:
Where S* is the critical stock price for which the following criteria holds:
It can be solved iteratively using the Newton-Rhapson method.
For overlapping Brownian increments, we can denote the correlation of the compound and underlying options as:
Also note that in the equations,
Is the bivariate cumulative distribution function.
In a Black & Scholes world without dividends, American style Compound options would not be be valuable to hold as it is never to exercise American style options which pay no dividends. More on this shortly.
1. Mother-and-Daughter options
2. Options on Options
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Geske, R., "The Valuation of Corporate Liabilities as Compound Options” Journal of Financial and Quantitative Analysis, Vol. 12, No. 4, UCLA, pp. 541-552. (Nov 1977)
Geske, R., "The Valuation of Compound Options", Journal of Financial Economics, 7, 63-81 (1979)
Hodges, S. D. & Selby, M. J. P., "On the Evaluation of Compound Options" Management Science, 33 (3), 347-355. (1987)
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002 - Chapter 19
Rubinstein, M., "Double Trouble", Risk 5, p.73 (1973)