CoachThe implied correlation of an index with weights is the single correlation that makes the members’ implied volatilities consistent with the index’s implied volatility :
Exemples
Example 17.7 (A synthetic 20-stock index)
Take an index of twenty equally weighted stocks with flat smiles and volatilities spaced evenly from 22% to 36%, and let its own one-year smile be chapter 9’s: 21.9% at 90, 19.2% at 100, 17.0% at 110. The implied correlation is 0.549 at 90, 0.408 at 100 and 0.308 at 110 (Figure 17.2, left). Priced with a constant correlation of 0.408, the members produce an index smile that is flat, 19.3% at all three strikes: constant correlation cannot produce an index skew out of flat member smiles. In variance-swap terms, using the index’s strip volatility of 22.1%, the implied correlation is 0.559.