What is a rainbow option, and how does the correlation between underlying assets affect its pricing?

A rainbow option is a multi-asset derivative whose payoff depends on the relative performance of two or more underlying assets. The 'best-of' variant pays based on the highest-performing asset, while 'worst-of' uses the lowest performer.\n\nBest-of Call Payoff (2 assets):\n\nPayoff = max(max(S1_T, S2_T) - K, 0)\n\nThe holder benefits from whichever asset finishes higher, then applies the strike.\n\n`mermaid\ngraph TD\n A[\"Rainbow Option
2 Assets: S1, S2\"] --> B{\"At Expiration\"}\n B --> C[\"S1_T = $115
S2_T = $108\"]\n B --> D[\"S1_T = $92
S2_T = $103\"]\n B --> E[\"S1_T = $88
S2_T = $91\"]\n C --> F[\"Best-of: $115
Payoff: max(115-100,0) = $15\"]\n D --> G[\"Best-of: $103
Payoff: max(103-100,0) = $3\"]\n E --> H[\"Best-of: $91
Payoff: max(91-100,0) = $0\"]\n F --> I[\"Worst-of: $108
Payoff: max(108-100,0) = $8\"]\n D --> J[\"Worst-of: $92
Payoff: max(92-100,0) = $0\"]\n E --> K[\"Worst-of: $88
Payoff: max(88-100,0) = $0\"]\n`\n\nCorrelation Impact:\n\n| Correlation (rho) | Best-of Value | Worst-of Value |\n|---|---|---|\n| +1.0 | Equals vanilla on either | Equals vanilla on either |\n| +0.5 | Higher than vanilla | Lower than vanilla |\n| 0.0 | Significantly higher | Significantly lower |\n| -0.5 | Much higher | Near zero |\n| -1.0 | Maximum possible | Approaches zero |\n\nWhen correlation is +1, both assets move identically, so the best-of option collapses to a single-asset option. As correlation decreases, the spread between asset paths widens, increasing the probability that at least one asset performs well.\n\nWorked Example:\nNorthbridge Capital buys a 1-year best-of call on Elysium Energy and Cobalt Mining shares, both starting at $50, strike $50.\n\n- Elysium volatility: 35%\n- Cobalt volatility: 40%\n- Correlation: 0.3\n- Risk-free rate: 4%\n\nUsing the Stulz (1982) two-asset pricing model:\n- Vanilla call on Elysium alone: $8.12\n- Vanilla call on Cobalt alone: $9.25\n- Best-of call: $12.80 (premium over the better single-asset option)\n- Worst-of call: $4.57 (discount to the cheaper single-asset option)\n\nNote: Best-of + Worst-of = Call_on_S1 + Call_on_S2 (Margrabe-type decomposition)\nCheck: $12.80 + $4.57 = $17.37 versus $8.12 + $9.25 = $17.37\n\nApplications:\n- Outperformance strategies in asset management\n- Structured products linked to baskets of indices\n- Executive compensation tied to relative stock performance\n\nPractice multi-asset derivative problems in our FRM question bank.