Explain the reasoning behind hedging several currencies at once using options.
The idea
A firm with exposures in several currencies can hedge the combined position with one option on a basket of the currencies, instead of buying an option on each. The reasoning is diversification. Exchange rates are imperfectly correlated, so the basket's volatility is lower than the weighted average of the individual volatilities, and an option's premium rises with volatility. With two currencies and weights w1 and w2, the variance of the basket depends on the correlation ρ. Basket variance = w1² × σ1² + w2² × σ2² + 2 × w1 × w2 × ρ × σ1 × σ2. A lower correlation means lower basket volatility and a cheaper basket option. The payoff depends on the aggregate value of the basket, so gains on some currencies offset losses on others. Separate options would pay on each currency alone and cost more for exposures that already cancel. The saving shrinks when correlations rise, as in stress, and the basket hedge leaves basis risk against the firm's actual cash flows.
Formulas
Variance of a two-currency basket
basket variance = w1² × σ1² + w2² × σ2² + 2 × w1 × w2 × ρ × σ1 × σ2
- w1, w2
- the share of the exposure in each currency
- σ1, σ2
- the volatility of each currency
- ρ
- the correlation between the two currencies
Below a correlation of one the basket's volatility, the square root, is under the weighted average of the two, which is why one basket option costs less than two separate ones. The saving fades as correlations rise in stress.
A practice question
What determines the payoff of a basket (multi-currency) option?
- The currency in the basket that performs best relative to its own strikeWhy not: That describes a best-of option, not a basket option.
- The sum of the payoffs of separate options on each currencyWhy not: A portfolio of separate options pays each leg separately, with no netting.
- The combined value of the basket of currencies relative to the strike, so gains on some currencies offset losses on others - correct
- Only the currency with the largest weight, with other currencies ignoredWhy not: All currencies contribute in proportion to their weights.
A basket option pays on the aggregate value, so offsetting moves across the currencies reduce the payout. That netting is the reason for its lower price.
A harder one
A firm has 60% of its foreign-currency exposure in currency X (annual volatility 12%) and 40% in currency Y (annual volatility 8%). The correlation between X and Y is 0.25. The volatility of the basket is closest to:
- 10.40%Why not: Takes the weighted average of the volatilities, which assumes perfect correlation of 1.
- 7.88%Why not: Omits the covariance term, which is equivalent to assuming zero correlation.
- 0.74%Why not: Reports the basket variance as if it were the standard deviation, forgetting the square root.
- 8.58% - correct
Var = 0.6² x 0.12² + 0.4² x 0.08² + 2 x 0.6 x 0.4 x 0.25 x 0.12 x 0.08 = 0.007360; the square root is 8.58%. This is below the 10.4% weighted average, which is why basket options cost less.
The hardest tier
A firm has two equal exposures of USD 10 million each, in currencies P and Q. Each has annual volatility of 10%, and the correlation between them is 0.3. Using the approximation that a one-year at-the-money option costs about 0.4 x volatility x notional, how much cheaper is one basket option on the USD 20 million total than two separate USD 10 million options? The saving is closest to:
- USD 234,315Why not: Uses a correlation of zero for the basket volatility instead of the stated 0.3.
- USD 155,019 - correct
- USD 0Why not: Assumes the basket volatility equals the individual volatility (perfect correlation), so no saving.
- USD 644,981Why not: Reports the basket option premium rather than the saving compared with separate options.
Separate options: 2 x 0.4 x 0.10 x 10m = USD 800,000. Basket volatility = 10% x sqrt((1+0.3)/2) = 8.06%, so premium = 0.4 x 0.0806 x 20m = USD 644,981. Saving is about USD 155,000.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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