How do rainbow options on max/min of multiple assets price?

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AcadiFi · Accepted Answer

Rainbow options pay off based on the maximum or minimum of multiple underlyings. Stulz (1982) derived closed forms for bivariate max/min options using bivariate normal CDFs. A call on max(S1, S2) with strike K is: S1*M(d1+sigma1*sqrt(T), y1; rho1) + S2*M(d2+sigma2*sqrt(T), y2; rho2) - K*e^(-rT)*[M(d1,d2;rho) - ...], where cross-terms account for which asset exceeds the other.

Key intuition: a call on the max is more valuable than either individual call (you get the best of two chances), and the price increases as correlation decreases (lower correlation means more upside diversification). A call on the min is worth less than either individual call and increases with correlation.

For Sable's best-of-two on stocks A ($80) and B ($75), K = $82, T = 0.5, sigma1 = 30%, sigma2 = 35%, rho = 0.6, r = 4%: individual call values are $4.20 and $4.85. Best-of-two call with rho=0.6 is $6.80; with rho=0.3 it rises to $7.45; with rho=0.9 it falls to $5.80.

Worst-of calls (on min) are popular in structured products because they're cheaper—issuers sell downside in exchange for yield enhancement. Best-of calls cost more and suit investors wanting exposure to sector winners. Correlation risk is the primary hedging challenge; practitioners use correlation swaps or delta-hedge in each asset separately.

How do rainbow options on max/min of multiple assets price?

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