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 %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
i) Best of 2 Assets Plus Cash ii) Maximum of 2 Assets Call iii) Better of 2 Assets
iv) Maximum of 2 Assets Put v) Minimum of 2 Assets Call vi) Worse of 2 Assets
vii) Minimum of 2 Assets Put
Formula A + B + C A + B + C -
A + B + C (Where X = 0) ii - iii +
- ii - iii
v - vi +
The roman numerals merely denotes the respective rainbow type and EuroCall(S) is the European call value on the asset in brackets.
5，两色彩虹期权定价
Assuming no arbitrage possibilities, the following parameters are needed to price a rainbow option:
1. Tenor of the Option: T 2. Spot price of the underlyings 3. Dividend rates for the underlyings 4. Volatility of the underlyings
5. Correlation between returns of asset A and B
The pricing of a rainbow option is independent of an interest rate.
In this calculator example, we price 3 types of rainbow options which involve two underlyings (two color rainbow option). Interest rates are assumed to be semiannually compounded.