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# Chapter 16ATM Volatility and Correlation

Correlation is an important measure within FX derivatives. The ATM volatility and correlation framework is often used to calculate ATM volatility in cross-currency pairs. Dephased vega exposures are also calculated within the same framework.

## ATM Volatility Triangles

In trigonometry, the cosine rule relates the lengths of sides of a triangle to the cosine of one of its angles as shown in Exhibit 16.1. If the length between point A and point B is denoted AB, the cosine rule states that:  Exhibit 16.1 Cosine rule triangle

The cosine rule can also be applied to ATM implied volatility in three currency pairs at the same tenor as shown in Exhibit 16.2. The distance between EUR and USD represents EUR/USD ATM volatility; the longer the length, the higher the ATM volatility. The angle θ is the inverse cosine ( ) of the correlation (ρ) between spot log returns in EUR/USD and GBP/USD. Going forward, ρ is described as the correlation between the major currency pairs (EUR/USD and GBP/USD in this example) and the output pair (EUR/GBP) is called the cross-currency pair. Beware of the overlap in symbols between correlation and rho exposure (see Chapter ...

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