Pricing Rainbow Options

Peter Ouwehand, Graeme West derive the Black—Scholes prices of several styles of (multi-asset) rainbow options using change-of-numeraire machinery. Hedging issues and deviations from the Black-Scholes pricing model are also briefly considered

A previous paper (West 2005) tackled the issue of calculating accurate uni-, bi- and trivariate normal probabilities. This has important applications in the pricing of multiasset options, e.g. rainbow options. In this paper, we derive the Black—Scholes prices of several styles of (multi-asset) rainbow options using change-of-numeraire machinery. Hedging issues and deviations from the Black-Scholes pricing model are also briefly considered.

1. Definition of a Rainbow Option

Rainbow Options refer to all options whose payoff depends on more than one underlying risky asset; each asset is referred to as a colour of the rainbow. Examples of these include:
• “Best of assets or cash” option, delivering the maximum of two risky assets and cash at expiry (Stulz 1982), (Johnson 1987), (Rubinstein 1991)
• “Call on max” option, giving the holder the right to purchase the maximum asset at the strike price at expriry, (Stulz 1982), (Johnson 1987)
• “Call on min” option, giving the holder the right to purchase the minimum asset at the strike price at expiry (Stulz 1982), (Johnson 1987)
• “Put on max” option, giving the holder the right to sell the maximum of the risky assets at the strike price at expiry, (Margrabe 1978), (Stulz 1982), (Johnson 1987)
• “Put on min” option, giving the holder the right to sell the minimum of the risky assets at the strike at expiry (Stulz 1982), (Johnson 1987)

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