# Implied Volatility and Vega

The volatility component of option values is called implied volatility (IV).(For more on implied volatility and how it relates to vega, see Chapter 3.)IV is a percentage, although in practice the percent sign is often omitted.This is the value entered into a pricing model, in conjunction with the other variables, that returns the option’s theoretical value. The higher the volatility input, the higher the theoretical value, holding all other variables constant. The IV level can change and often does—sometimes dramatically.

When IV rises or falls, option prices rise and fall in line with it.But by how much?The relationship between changes in IV and changes in an option’s value is measured by the option’s vega. Vega is the rate of change of an option’s theoretical value relative to a change in implied volatility. Specifically, if the IV rises or declines by one percentage point, the theoretical value of the option rises or declines by the amount of the option’s vega, respectively.

For example, if a call with a theoretical value of 1.82 has a vega of 0.06 and IV rises one percentage point from, say, 17 percent to 18 percent, the amount of the vega. If, conversely, the IV declines 1 percentage point, from 17 percent to 16 percent, the call value will drop to 1.76—that is, it would decline by the vega.A put with the same expiration month and the same strike on the same underlying will have the same vega value as its corresponding call.

In this example, raising or lowering IV by one percentage point would cause the corresponding put value to rise or decline by \$0.06, just like the call. An increase in IV and the consequent increase in option value helps the P&(L) of long option positions and hurts short option positions. Buying a call or a put establishes a long vega position. For short options, the opposite is true. Rising IV adversely affects P&(L), whereas falling IV helps.

Shorting a call or put establishes a short vega position.The Effect of Moneyness on Vega:Like the other greeks, vega is a snapshot that is a function of multiple facets of determinants influencing option value. The stock price’s relationship to the strike price is a major determining factor of an option’s vega. IV affects only the time value portion of an option. Because ATM options have the greatest amount of time value, they will naturally have higher vegas.

ITM and OTM options have lower vega values than those of the ATM options.Exhibit 2.13 shows an example of 186-day options on AT&T Inc. (T),their time value, and the corresponding vegas.Note that the 30-strike calls and puts have the highest time values. This strike boasts the highest vega value, at 0.085. The lower the time premium,the lower the vega—therefore, the less incremental IV changes affect the option.

Since higher-priced stocks have higher time premium (in absolute terms, not necessarily in percentage terms) they will have higher vega. Incidentally, if this were a \$300 stock instead of a \$30 stock, the 186-day ATMs would have a 0.850 vega, if all other model inputs remain the same.The Effect of Implied Volatility on Vega The distribution of vega values among the strike prices shown in Exhibit 2.13 holds for a specific IV level.

The vegas in Exhibit 2.13 were calculated using a 20 percent IV. If a different IV were used in the calculation, the relationship of the vegas to one another might change. Exhibit 2.14 shows what the vegas would be at different IV levels.Note in Exhibit 2.14 that at all three IV levels, the ATM strike maintains a similar vega value. But the vegas of the ITM and OTM options can be significantly different. Lower IV inputs tend to cause ITM and OTM vegas to decline. Higher IV inputs tend to cause vegas to increase for ITMs and OTMs.

The Effect of Time on Vega:As time passes, there is less time premium in the option that can be affected by changes in IV. Consequently, vega gets smaller as expiration approaches.Exhibit 2.15 shows the decreasing vega of a 50-strike call on a \$50 stock with a 25 percent IV as time to expiration decreases. Notice that as the value of this ATM option decreases at its nonlinear rate of decay, the vega decreases in a similar fashion.

One of my early jobs in the options business was clerking on the floor of the Chicago Board of Trade in what was called the bond room. On one of my first days on the job, the trader I worked for asked me what his position was in a certain strike. I told him he was long 200 calls and long 300 puts.“I’m long 500 puts?” he asked. “No,” I corrected, “you’re long 200 calls and 300 puts.” At this point, he looked at me like I was from another planet and said, “That’s 500. A put is a call; a call is a put.” That lesson was the beginning of my journey into truly understanding options.