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

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AcadiFi · Accepted Answer

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. **Best-of Call Payoff (2 assets):** Payoff = max(max(S1_T, S2_T) - K, 0) The holder benefits from whichever asset finishes higher, then applies the strike. ```mermaid graph TD A["Rainbow Option
2 Assets: S1, S2"] --> B{"At Expiration"} B --> C["S1_T = $115
S2_T = $108"] B --> D["S1_T = $92
S2_T = $103"] B --> E["S1_T = $88
S2_T = $91"] C --> F["Best-of: $115
Payoff: max(115-100,0) = $15"] D --> G["Best-of: $103
Payoff: max(103-100,0) = $3"] E --> H["Best-of: $91
Payoff: max(91-100,0) = $0"] F --> I["Worst-of: $108
Payoff: max(108-100,0) = $8"] D --> J["Worst-of: $92
Payoff: max(92-100,0) = $0"] E --> K["Worst-of: $88
Payoff: max(88-100,0) = $0"] ``` **Correlation Impact:** | Correlation (rho) | Best-of Value | Worst-of Value | |---|---|---| | +1.0 | Equals vanilla on either | Equals vanilla on either | | +0.5 | Higher than vanilla | Lower than vanilla | | 0.0 | Significantly higher | Significantly lower | | -0.5 | Much higher | Near zero | | -1.0 | Maximum possible | Approaches zero | When 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. **Worked Example:** Northbridge Capital buys a 1-year best-of call on Elysium Energy and Cobalt Mining shares, both starting at $50, strike $50. - Elysium volatility: 35% - Cobalt volatility: 40% - Correlation: 0.3 - Risk-free rate: 4% Using the Stulz (1982) two-asset pricing model: - Vanilla call on Elysium alone: $8.12 - Vanilla call on Cobalt alone: $9.25 - Best-of call: **$12.80** (premium over the better single-asset option) - Worst-of call: **$4.57** (discount to the cheaper single-asset option) Note: Best-of + Worst-of = Call_on_S1 + Call_on_S2 (Margrabe-type decomposition) Check: $12.80 + $4.57 = $17.37 versus $8.12 + $9.25 = $17.37 **Applications:** - Outperformance strategies in asset management - Structured products linked to baskets of indices - Executive compensation tied to relative stock performance Practice multi-asset derivative problems in our FRM question bank.

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

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Correlation (rho)	Best-of Value	Worst-of Value
+1.0	Equals vanilla on either	Equals vanilla on either
+0.5	Higher than vanilla	Lower than vanilla
0.0	Significantly higher	Significantly lower
-0.5	Much higher	Near zero
-1.0	Maximum possible	Approaches zero