###### 9.4.3 Example: Implied Volatilities

In Section 8.6, we designed a primary mapping for a portfolio of natural gas options. This employed distinct key factors modeling implied volatilities for every strike/expiration pair. There were more than 1,000 key factors! Let’s reduce this number with a variables remapping.

In modeling volatility surfaces, we may either model separate volatility curves for each expiration or directly model an entire surface. The latter approach works well if implied volatilities for different expirations are closely related, which makes interpolation across expirations reasonable. In many markets, volatilities for different expirations are not so related. In agricultural markets, implied volatilities for options expiring before a harvest may have little relationship to implied volatilities for options expiring after the harvest. Natural gas implied volatilities for winter expirations may have little relationship to implied volatilities for spring expirations. For this example, we take the approach of modeling individual volatility curves for each expiration.

In the earlier example, our primary mapping depended upon a number of component vectors for implied volatilities. Let’s focus this example on the one for second nearby options:

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^{1}*R** ^{Vols2}* has 106 key factors. We shall reduce this to 3 with a variables remapping:

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In Section 6.7.2, we discussed constant-delta implied volatilities—implied volatilities tracked over time for specific normalized deltas. Let’s have represent changes in three of these:8

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represents changes in implied volatilities for each of the strikes represented by :

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We obtain from by quadratic interpolation. A complication is the fact that components of correspond to specific deltas whereas components of correspond to specific strikes. We must infer, based upon time-1 market variables, the specific strikes corresponding to normalized call deltas of .25, .50, and .75. Once we have these, we can interpolate to obtain .

Whereas models *changes* in implied volatilities, models implied volatilities. It approximates ^{1}*R** ^{Vols2}* and is defined as

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