Multi-asset exotics: baskets, best-of, spreads and quantos

Add a second underlying and correlation becomes a price input, not just a risk measure. The useful skill is signing the exposure without computing anything: does more correlation make this payoff more or less valuable? That one question sorts out baskets, best-of, spreads and quantos in a sentence each.

Structure	Payoff	Correlation exposure
Basket call	$(\,\bar{S} - K)^+$	Long — more correlation means a more volatile basket
Spread option	$(S_1 - S_2 - K)^+$	Short — correlation damps the spread
Best-of call	$(\max(S_1, S_2) - K)^+$	Short — you want them to diverge
Worst-of put	$(K - \min(S_1, S_2))^+$	Short — you want one of them to fall
Dispersion (short index, long names)	—	Short

Table 3.1 · Correlation exposure by structure. The pattern: anything that aggregates the assets is long correlation, and anything that selects between them is short it. That is enough to sign almost any structure you are shown.

Equation 3.2

Basket and spread volatility

A basket adds the cross term; a spread subtracts it. Setting $w_2 = -1$ turns the first into the second, which is the whole distinction.

\sigma_{\text{basket}}^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2

$+2\rho\sigma_1\sigma_2$: Basket: correlation raises the volatility.
$-2\rho\sigma_1\sigma_2$: Spread: correlation lowers it.

Example 3.3

Two assets each with $30\%$ volatility and correlation $0.8$ . What is the volatility of an equally weighted basket, and of the spread between them?

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Worked solution

Formula $\sigma^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 \pm 2w_1w_2\rho\sigma_1\sigma_2$
Substitute $\sigma_1 = \sigma_2 = 0.30, \ \rho = 0.8$
Solve $\text{basket } (w = \tfrac12): \ 0.0225 + 0.0225 + 2(0.25)(0.8)(0.09) = 0.081 \Rightarrow 28.5\%$
$\text{spread } (w = 1, -1): \ 0.09 + 0.09 - 2(0.8)(0.09) = 0.036 \Rightarrow 19.0\%$
Answer $\text{basket } 28.5\%, \ \text{spread } 19.0\%$

Sanity check. High correlation barely diversifies the basket — $28.5\%$ against $30\%$ — while it collapses the spread from a hypothetical $42.4\%$ at zero correlation down to $19\%$ . That asymmetry is why spread options are so much more sensitive to the correlation input than baskets are.

Correlation is the input you cannot observe. Volatilities are implied from liquid options, so they are read off the market. Correlation is not: there is no liquid instrument quoting the correlation between two specific stocks, so it has to be estimated from history or backed out of an index. That makes it the parameter most likely to be wrong, and structures whose value is most sensitive to it — spread options above all — carry a model risk that does not appear in any Greek. Desks reserve against it explicitly rather than pretending the estimate is a price.

Multi-asset exotics: baskets, best-of, spreads and quantos

After this lesson you should be able to

Basket and spread volatility

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