Basket Options

Instead of buying separate insurance policies on five different stocks in a portfolio, you could buy one policy that pays off based on the portfolio's combined value — cheaper, because the portfolio as a whole rarely moves as violently as its most volatile piece, since the pieces don't all lose value at the same time. A basket option is exactly that single policy: its underlying isn't one stock but a weighted average of several, bundled into one number the way a fruit basket's total weight is one number even though it's made of apples, oranges, and pears in different quantities.

The payoff and the diversification discount

A basket call on $n$ assets with weights $w_i$, struck at $K$, pays:

$$\text{Payoff} = \max\!\left(\sum_{i=1}^{n} w_i S_i(T) - K,\ 0\right).$$

In words: each $S_i(T)$ is the price of underlying $i$ at expiry, each $w_i$ is that asset's weight in the basket (weights typically sum to 1, or represent fixed share counts), and the weighted sum is the basket's total value, compared against strike $K$. The volatility of that weighted sum is always less than or equal to the weighted average of the individual volatilities, because of diversification — assets rarely all move in the same direction by the same amount at the same time. This is the same math as portfolio variance: $\sigma_{\text{basket}}^2 = \sum_i w_i^2 \sigma_i^2 + \sum_{i\ne j} w_i w_j \rho_{ij}\sigma_i \sigma_j$, and unless every pairwise correlation $\rho_{ij}$ equals exactly 1, the cross terms shrink the total below what you'd get by just averaging the individual vols.

Worked example 1 — the diversification discount in numbers

A basket is 50% Stock A (vol 30%) and 50% Stock B (vol 30%), correlation $\rho=0.4$. Basket variance $= 0.5^2(0.30)^2 + 0.5^2(0.30)^2 + 2(0.5)(0.5)(0.4)(0.30)(0.30) = 0.0225+0.0225+0.018=0.063$, so $\sigma_{\text{basket}} = \sqrt{0.063}\approx25.1\%$ — noticeably below the simple average of 30% and 30%. An option on this basket is priced using roughly 25.1% vol, not 30%, so it costs meaningfully less than the average of two identical single-name options on A and B alone would cost, purely because the two stocks don't move in perfect lockstep.

Worked example 2 — when correlation approaches one

Same two stocks, same weights and vols, but now $\rho = 0.95$ (they're driven almost entirely by the same factor — say, two banks in the same regulatory regime). Basket variance $=0.0225+0.0225+2(0.5)(0.5)(0.95)(0.30)(0.30)=0.0225+0.0225+0.04275=0.08775$, so $\sigma_{\text{basket}}\approx29.6\%$ — almost back to the individual 30% vol, because near-perfect correlation means there's almost no diversification benefit left: when one stock moves, the other moves almost the same amount in the same direction, so the basket behaves almost like a single stock.

Drag the correlation and weight sliders above — the same curve that shows a portfolio's risk shrinking below the weighted average of its two assets' individual risks is exactly the mechanism that makes a basket option cheaper than the weighted average of separate single-name options on the same names.

What this means in practice

Basket options are cheaper than a strip of individual options on the same names with the same weights, which is exactly why they're used for portfolio-level hedges — an asset manager hedging a whole sector or a currency basket buys one basket option rather than many single-name ones. The trade-off is customization and liquidity: a basket option is bespoke, over-the-counter, and requires the seller to actually estimate the pairwise correlations, which are harder to observe reliably than any single asset's own volatility.