Basket and Rainbow Options

Every exotic so far has had exactly one underlying. A multi-asset option pays off on several underlyings at once — a currency basket, a handful of tech stocks, a commodity index's components. Two payoff shapes dominate in practice. A basket option pays on a weighted average of the underlyings, as if they had been blended into one synthetic asset. A rainbow option pays on the best or worst performer among them — a genuinely different bet, since it singles one asset out rather than blending them.

\text{basket call: } \Big(\textstyle\sum_i w_i S_i(T) - K\Big)^+ \qquad \text{best-of call: } \Big(\max_i S_i(T) - K\Big)^+

A worst-of structure is the mirror image — often written as a put, (K - \min_i S_i(T))^+, since desks mostly sell worst-of protection, not worst-of upside. The single fact that dominates pricing all of these, far more than it dominated anything earlier in this module, is how the underlyings move together.

Why correlation is everything here

A basket built from n assets with weights w_i, volatilities \sigma_i and pairwise correlations \rho_{ij} behaves — to a standard moment-matching approximation — like a single lognormal asset with its own effective volatility:

\sigma_{\text{basket}}^2 = \sum_i \sum_j w_i w_j\, \rho_{ij}\, \sigma_i \sigma_j.

Push every \rho_{ij} \to 1 (the underlyings move in lockstep) and this collapses to \sigma_{\text{basket}} = \sum_i w_i \sigma_i, the simple weighted average of the individual volatilities — no diversification benefit survives. Push correlation down toward 0 or negative, and the assets' wiggles start to cancel inside the sum, shrinking \sigma_{\text{basket}} well below that weighted average. Since an option's value rises with the volatility of the random variable in its payoff, a basket call gets cheaper as correlation falls — lower correlation means more netting, less spread in the basket's own outcome.

Best-of and worst-of rainbows move the opposite way for a subtler reason: they don't care about netting, they care about the chance that at least one name does something extreme. With many weakly-correlated assets, it becomes far more likely that some asset in the basket has a great year (helping a best-of call) — and, just as easily, that some asset has a terrible one (hurting a worst-of put's seller). Low correlation therefore makes a best-of call more valuable and a worst-of put more valuable too, even though it makes a basket call less valuable. Correlation isn't a single dial that moves every multi-asset option the same way — it moves basket and rainbow structures in opposite directions.

Seeing the correlation effect

Take a basket that is a straight 50/50 blend of two stocks, each starting at 100, with volatilities 25\% and 30\%, struck at-the-money with a year to run. The chart below prices an at-the-money call on that basket, using the moment-matching approximation above, as correlation sweeps from -1 to +1. The value climbs steadily as the two legs become more correlated — exactly the pattern the formula predicts, and a striking reminder that the same two options, on the same two underlyings, can be worth very different amounts depending on a number (correlation) that has nothing to do with either stock's own volatility.

A worst-of put on the same two names would trace the opposite shape — most expensive when correlation is low (plenty of ways for at least one leg to crater independently) and cheapest when correlation is high (the two legs live or die together, so there's little extra "one of them might fail alone" risk to insure against).

In the years before the 2008 crisis, retail-facing "worst-of autocallable" notes — paying an attractive coupon unless the worst performer among, say, three unrelated blue chips breached a barrier — were sold in huge volume, because a worst-of structure lets an issuer offer a headline coupon that looks generous for what seems like modest risk. The catch is exactly the correlation effect above: with three imperfectly correlated stocks, the chance that at least one of them has a bad year is much higher than the chance any single stock does. Many buyers priced the note as if it carried single-stock risk, when it actually carried something closer to "worst of three independent coin flips" risk — understanding correlation wasn't a technicality, it was the entire difference between a fair price and a nasty surprise.

It is tempting to price a basket call by just averaging the prices of individual vanilla calls on each underlying (with strikes chosen so their weighted sum equals K). This is a real and costly mistake, and not merely "approximately fine": a portfolio of options \sum_i w_i (S_i(T) - K_i)^+ floors each asset separately at zero before summing, so a bad leg can never be offset by a good one. A true basket option floors the sum at zero, (\sum_i w_i S_i(T) - K)^+, which lets a strong leg's gains net against a weak leg's losses before the floor is applied. Netting can only reduce the payoff, so a genuine basket option is always worth less than or equal to the corresponding portfolio of individual vanilla options — pricing it as that portfolio systematically overprices it, and ignores correlation entirely.