Pricing Rainbow Options

A previous paper (West 2005) tackled the issue of calculating accurate uni-, biand trivariate normal probabilities. This has important applications in the pricing of multi-asset 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. exotic option, Black-Scholes model, exchange option, rainbow option, equivalent martingale measure, change of numeraire, trivariate normal. 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) • “Put 2 and call 1”, an exchange option to put a predefined risky asset and call the other risky asset, (Margrabe 1978). Thus, asset 1 is called with the ‘strike’ being asset 2. Thus, the payoffs at expiry for rainbow European options are: Best of assets or cash max(S1, S2, . . . , Sn,K) Call on max max(max(S1, S2, . . . , Sn)−K, 0) Call on min max(min(S1, S2, . . . , Sn)−K, 0) Put on max max(K −max(S1, S2, . . . , Sn), 0) Put on min max(K −min(S1, S2, . . . , Sn), 0) Put 2 and Call 1 max(S1 − S2, 0) Date: January 2006. Thanks to the 2005 Mathematics of Finance class at the University of the Witwatersrand for alert feedback in lectures.