So far all strikes of a range of options, with the same maturity date, have been treated as all having the same volatility, but in reality one could find differences in volatility between or around at the money options and out of the money options. One could also find discrepancies in volatility among different maturities, as discussed with relation to vega bucketing in the chapter on vega. The combination of these discrepancies is one of the shortcomings of the model, as mentioned in the introduction: the model assumes a stable volatility over different maturities and different strikes.
The difference in volatility in a range of strikes of options with the same maturity on the same underlying is called vertical skew, or volatility surface. The difference in volatility with regards to options with different maturities is called the term structure of volatility.
An example of the term structure has been shown in the chapter on vega, as shown below in Table 11.1:
|Maturity||Initial Volatility||Vega Position (total 8,000)||New volatility||Change in vol × Vega position||P&L|
|Up to 3 months||23%||−$5,000||20%||−3 × −$5,000||$15,000|
|3–6 months||23%||$3,000||21%||−2 × $3,000||−$6,000|
|6–12 months||23%||$1,000||22%||−1 × $1,000||−$1,000|
|More than 12 months||23%||$9,000||23%||0 × $9,000||$0|
As shown for the different buckets, volatility can behave differently per period of time to maturity. There are no fixed rules for how different buckets should ...