Rainbow Option

Rainbow option is a term used to describe certain derivatives linked to two or more underliers. Usage varies with regard to what structures the term encompasses. Basket options and quantos are linked to multiple underliers, but are generally not referred to as rainbows.

Some standard forms of rainbow options are:

• A maximum option is a bundle of vanilla options with a variety of features—different strikes, different underliers, some may be puts, others calls, but they generally have the same expiration date. Only one of these may be exercised, and this is chosen in the holder’s favor at expiration.
• A minimum option is a bundle of vanilla options—like a maximum option. Only one of the options can be exercised, and this is chosen in the issuer’s favour at expiration.
• A better-of option is a bundle of long forwards. All mature on the option’s expiration date but have different underliers. At expiration, only one settles, and this is chosen in the holder’s favour.
• A worst-of option is a bundle of long forwards. All mature on the option’s expiration date but have different underliers. At expiration, only one settles, and this is chosen in the issuer’s favour.
• A two-asset correlation option is linked to two underliers. It pays off like a vanilla option on one underlier if the expiration value of the other underlier is in a specified range. The vanilla option can be either a put or a call.
• A spread option is a derivative with a spread as an underlier. The spread might be a price spread, credit spread, calendar spread, etc.

Together, maximum options and minimum options are referred to as min-max options. Better-of or worst-of options are referred to collectively as alternative options

An alternative option can result in the holder having to make a payment to the issuer at expiration. Consider a better-of option on three-month USD/EUR and USD/JPY forwards, both with a USD 100MM notional. If both exchange rates move against the holder, he will have to make a payment to the issue to settle whichever forward has declined least in value.

Worst-of options blur the distinction between option issuers and option holders. Certainly, someone would require a premium or other compensation for holding a worst-of option. Worst-of options arise with bond futures that grant the short party the right to deliver any of several qualifying bonds.

An outperformance option (or Margrabe option) is an option that grants the right to exchange one asset for another. Essentially, it is a spread option with a strike price equal to zero.

Pricing of rainbow options depends upon the particular structure, but it is generally sensitive to correlations between the underliers. The classic paper on analytic solutions for pricing two-factor min-max options is Stulz (1982). Johnson (1987) extends these results to more than two factors. For numerical solutions, see Boyle and Tse (1990). See Margrabe (1978) for outperformance options. Kirk (1995) and Pearson (1995) provide approximate solutions for pricing spread options. Haug (1997) covers many of the above formulas.

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