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Extendible options are exactly what their name suggests they are; options which can be extended by the holder or writer of the option. These options have become increasingly popular over recent years, particularly for underlyings which are volatile.
Introduction || Holder Extendible || Writer Extendible || Jump Diffusion || Other Known Names / Variants || References
Extendible options are exactly what their name suggests they are; options which can be extended by the holder or writer of the option. These options have become increasingly popular over recent years, particularly for underlyings which are volatile.
Two types of extendible options exist:
a) Holder Extendible b) Writer Extendible
Each of these options have its primary feature explained by its name, and allow either the holder or writer to extend the option at a predetermined date
For holder extendible options, the option can be extendible by the holder of the option, but an additional premium must be given for the ability to do so from the point of the holder. The extension also incorporates an adjustment in the strike price, and this is up to the discretion of the adjuster.
The payoffs for holder extendible calls and puts are given as:
Where C(S, X, T) and P(S, X, T) denote the European call and put option valuation of S, X and T.
Longstaff (1990) presents a set of pricing formulas these options under a Black-Scholes framework and Haug (1998) treats them via the following formulae:
Where "Value" represents the respective Black-Scholes call and put formulae using The bivariate cumulative normal distribution:
can calculated by breaking the distribution down to:
and similarly, the cumulative normal distribution N(a, b) can be reduced to N(b)-N(a). Where:
Although not a particularly efficient technique, we must solve the values for I using iterative Newton-Raphson technique to satisfy the following conditions:
For a Call:
For a Put:
Haug (1998) also details a number of special rule of thumb cases for the values of I.
This type of extendible option is somewhat easier to price because of the key feature which allows the writer to extend the option without paying a premium. The option is extended at the 'extend date' only if the option is out-of-money, with the simple reason being there to be no point in extending the option if it is already in the money. The payoff of writer extendible calls and puts can be respectively given as:
A set of simplified formulae can be used to give the pricing solution to a writer extendible option.
Where, "Value" is the respective call or put values for
Dias & Rocha (2001) use a similar process to price extendible oil options under a jump diffusion process driven by a Poisson process. Merton (1973) proposed the jump diffusion model to model the random price fluctuations of the underlying stock. His model can be extended to numerous other types of options, and in the paper by Dias & Rocha, application to extendible oil options can be applied to the case of general options.
Buyer Extendible
Additional/Useful List of resources Papers:
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73) |
until a final maturity date T.
is the extended strike price, T is the extended time to maturity date,
.
is set to 1 for a call option and -1 for a put option. 
and 
are the same as defined for a holder extendible option.