ATM curves can be constructed in two steps. First, a core ATM curve is established. Then additional parameters are introduced so the correct ATM implied volatility is generated for all possible expiry dates. Note that within this chapter, some calculations are *approximate*.

The key measure for building ATM curves is

where is the ATM implied volatility to time (measured in years). For example, the variance of a 3mth ATM option with 12.0% implied volatility is .

Variance can be thought of as a measure of *cumulative spot movement*. It has two powerful properties:

- Variance over any time period must be nonnegative.
- Variance is additive (i.e., variance over two days = variance on first day + variance on second day).

Variance can be used to calculate the forward ATM implied volatility (usually called **forward implied volatility** or just “forward vol” by traders) between two dates in the future. Given ATM implied volatility to time , ATM implied volatility ...

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