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Compound Options are options on other options.
European Style Compound Options || American Style Compound Options || References || Other Known Names
Compound Options are options on options. They are constructed in one of the following four ways:
1. Call on Call (CoC)
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.
European-Style Compound Options
Where S* is the value of the stock underlying the underlying option,
The variables
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)
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.
American-Style Compound Options 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
Additional/Useful List of resources Papers:
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73) |
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.
and 
are binary variables in that they take either values of 0 or 1. 
