Chooser options are exactly what their name suggests - it is an option which allows the holder to choose whether their option is a call or a put at a particular date.

Introduction || Simple Chooser || Complex Chooser || American Chooser Options || Quadrature Methods || Known Names & Variants || References || Pricing Models

Chooser options are exactly what their name suggests - it is an option which allows the holder to choose whether their option is a call or a put at a particular date. Chooser options are generally more expensive than standard vanilla options due to its flexibility of choice. Although this is the case, when we consider the structure of a chooser, we find that it is identical to that of constructing a straddle, or a position in a call and a put simultaneously, with the exception that chooser options are comparatively cheaper.

This type of option comes in two distinct forms:

a) Simple Chooser

This type of chooser gives the holder of the option a choice of either a vanilla call option or a vanilla put at a predetermined time t, where the payoff can be given as:

Where

and

denotes the respective European call and put values with maturity date T, where T is the difference between the final maturity and the time to choice, and t is the time to choice. The payoff function represents the maximum value of a European call or European put at choice date t.

These types of options have been traded since July of 1990 with the initial contracts traded by Bankers Trust.

Rubinstein (1991) shows how the above payoff function can be adjusted to give our valuation formula based on the put-call parity relationship.

The payoff function shown above can be shown to equal.

Which becomes:

Where D is the dividend yield (or foreign interest rate) and r is the risk free rate. Rubinstein goes on to show that this is an equivalent position in a long call and short put, giving us a chooser option in terms of European vanilla options.

The value of a chooser option is then:

Where:

Where again, the variables are the same as those defined earlier, with T being the time to maturity and lowercase t being the time to choice.

The graph below shows how time to choice affects the option value.

And the following one shows that chooser options are generally quite expensive; by varying the strike price, we see that even when the asset price = the strike price, the value is still significant.

b) Complex Chooser

A complex chooser is similar to that of a simple chooser with the exception that either or both the strike price and time to maturity for the 'choice' call / put may not be the same, giving rise to the payoff:

Where is the call strike, is the put strike, is the time to maturity for the call and is the maturity for the put.

Because of this property, a complex chooser cannot be broken down in terms of vanilla options.

Rubinstein (1991) shows the value of a complex chooser:

B(a, b, rho) is the bivariate cumulative normal distribution, where:

The value of I must be iteratively solved using the Newton-Raphson method using a search method which satisfies the condition:

Where:

American Chooser Options

Choosers can be American in the sense that the choice of a call and put at the choice date is an American option rather than European in exercise. We can price an American chooser in a similar fashion to the valuation given for European choosers, but replacing the European payoff function with an American one to find an approximate price.

Quadrature Methods

For the complex choosers we have just mentioned, the use of an iterative search, although elegant in its own way, does cause difficulties in certain situations and convergence of the search can break down. Nelken (1993) describes an alternative method which makes use of quadrature methods such as the left-end-point quadrature, Gauss-Kronrod method or the trapezoidal method to approximate the integral governing the complex chooser.

Such a technique is numerically more stable than an iterative search for the value of I, and should converge more rapidly than the linear method given by Rubinstein (1991). For greater details, we recommend the reader to consult Nelken's paper.

Other Known Names / Variants

Choice Options
Compound Chooser
Compound Options
Straddle Option

Additional/Useful List of resources

Papers:

Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Haug, E., "Complete Guide to Option Pricing Formulas", McGraw-Hill, 1998
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002 - Chapter 12
Merton, R.,(a) "Option Pricing when Underlying Stock Returns Are Discontinuous", Journal of Financial Economics 3, pp. 125-144 (Jun '73)
Merton, R.,(b) "Theory of Rational Option Pricing", Bell Journal of Economics & Management (June '73)
Nelken, I., "Square Deals", Risk, 6, 4, (Apr '93)
Rubinstein, M., "Options for the Undecided", Risk, 4, (Apr '91)