Rainbows on the horizon after a rainy day comprise of various colours of the light spectrum, and although rainbow options aren't as colourful as their atmospheric counterparts, they get their name from the fact that their underlying is two or more assets rather than one.

With numerous types of rainbow options, we define five of the main types:

a) Best of n Assets Plus Cash

- This type of rainbow effectively has n + 1 payoff possibilities. If we consider a 2 asset "best of plus cash", the payoff at expiry is a choice between Asset 1, Asset 2, or the predetermined cash amount. There is no strike price and the payoff is given as:

Stulz (1982) presented the first significant paper on rainbow options, and from there, we can find a set of analytical formulae for best of 2 assets plus cash.

Where:

Where N(x) is the cumulative normal distribution and B(a, b, rho) is the bivariate cumulative normal distribution. is the correlation between the 2 assets, and other variables are defined under the standard notation.

The effects of varying the cash amount are shown in the below graph:

b) Better of n Assets

This type of rainbow is similar to the best of n assets plus cash but with the exception that there is no possible cash payoff, and X is set to 0. With this in mind, a better of 2 assets rainbow is essentially a two-asset call option, with a payoff being:

c) Worse of n Assets

Essentially the opposite to the better of n assets, with the payoff being on the asset with the lower value. We can give the payoff for this option on 2 assets as:

d) Maximum of n Assets

This rainbow is similar to the best of n assets plus cash we referred to in part a, with the exception that no cash payoff is possible and there is a strike price for this type of option. The payoff of a call and put are given as:

We can generalise the formulae to the call and put, which we show later.

e) Minimum of n Assets

The counterpart to a maximum of n assets, this rainbow pays out the value of the underperformer of the n assets. The payoff for minimum of 2 asset rainbow calls and puts are given as:

Although rainbows often focus on two or three underliers, we have generalised it to being n underliers.

Generalisation of Formulae:

In part a) we produced a set of formulae for the best of 2 assets plus cash. Here we tabulise the set of equations which can be used to value all types of 2 asset rainbow options.

Considering our initial formula for the best of 2 assets plus cash, we break it up into 3 parts:

Where the variables are defined as previous, but we reiterate them for reference purposes:

By using these formulae, we show that the various rainbows can be priced: (The term "X" is used to define the cash amount for the Best of 2 Assets Plus Cash, but denotes the strike price for the other options)

Rainbow Type	Formula
i) Best of 2 Assets Plus Cash	A + B + C
ii) Maximum of 2 Assets Call	A + B + C -
iii) Better of 2 Assets	A + B + C (Where X = 0)
iv) Maximum of 2 Assets Put	ii - iii +
v) Minimum of 2 Assets Call	- ii
vi) Worse of 2 Assets	- iii
vii) Minimum of 2 Assets Put	v - vi +

The roman numerals merely denotes the respective rainbow type and EuroCall(S) is the European call value on the asset in brackets.

Other Known Names / Variants:

Basket Options
Best of Options
Better of Options
Exchange Options
Maximum Of Options
Minimum Options
Minimum Of Option
Multi-Asset Options
Quanto Options
Spread Options
Three-Asset Options
Two-Asset Options
Worse of Options

Additional/Useful List of resources

Papers:

Andreas, A., Engelmann, B., Schwendner, P., Wystup, U., "Fast Fourier Method for the Valuation of Options on Several Correlated Currencies", Working Paper, (2001)
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Cox, J., Ross, S., & Rubinstein M., “Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sept '79).
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002 - Chapter 12
Kolb, R., "Futures, Options, Swaps", Blackwell Publishing 3rd Edition, 2001 - Chapter 18
Lindset, S., "Options on the Maximum or the Minimum of Several Assets Revisited" Norwegian University of Science & Technology Working Paper, 2003
Merton, R.,"Theory of Rational Option Pricing", Bell Journal of Economics & Management (June '73)
Stulz, R. M., "Options on the Minimum or the Maximum of Two Risky Assets", Journal of Financial Economics, 10(2), 161-185 (1982)