# Estimating Option Prices as Volatility Changes

Table 9-5 demonstrates the first skill, which is the ability to quickly estimate a new option price when the implied volatility changes. The formula that accomplishes this estimation is stated at the top of the table. It starts with an initial theoretical value of the option, the volatility assumption of which is known. The option’s vega or a fraction thereof is then added to or subtracted from the initial theoretical value.

The vega, remember, is the change in option theoretical value for a one percentage point change in volatility. For a refresher on vega, refer to Chapter 4.After the formula, Table 9-5 lists the theoretical values of three calls, their vegas, and the volatility assumptions. The 80 Call, the 85 Call, and the 90 Call have theoretical values of 4.00, 1.75, and 0.65, respectively. The volatility assumption is 30 percent, and the stock price is 81.50.

For the 80 Call, the new level of volatility is 31 percent. For the 85 Call, it is 31.5 percent, and for the 90 call, it is 32 percent. The next row contains the estimated new option prices under the new volatility assumptions.The 85 Call increases from 1.75 to 1.87, and the 90 Call increases to 0.77.The bottom section of Table 9-5 shows that the estimated prices are calculated in two steps. First, the percentage change in volatility is related to the vega.

If volatility changes by 1 percent, for example, the option theoretical value changes by one vega. This is what happened to the 80 Call. The increase in volatility of one percentage point from 30 to 31 percent caused the option theoretical value to rise by one vega from 4.00 to 4.10. The increase in volatility for the 85 Call, however, is one and a half percentage points. Its value therefore rises by one and a half vegas.

The vega of the 85 Call is 0.08, so the increase in volatility from 30 to 31.5 percent causes the theoretical value to increase by 0.12 (0.08 vega  1.5) from 1.75 to 1.87. Finally, the two percentage point increase in volatility for the 90 Call causes its theoretical value to rise by 0.12 from 0.65 to 0.77 (twice its vega of 0.06).Expressing Bid and Ask Prices in Volatility Terms The second skill needed to quickly evaluate option prices as stockprices change is the ability to state the market for an option in volatility terms, which is shown in Table 9-6.

Table 9-6 starts with the same three calls and their theoretical values and vegas from Table 9-5. The next line in Table 9-6 states bid and ask prices for each of the calls. For the 80 Call, for example,the bid price is 3.90, and the ask price is 4.10. For the 85 and 90 Calls, the bid-ask prices are 1.75 and 1.83 and 0.68 and 0.77,respectively.The last row in Table 9-6 states the bid and ask prices in volatility terms.

For the 80 Call, this is 29 percent bid and 31 percent ask. These percentages are calculated the same way that prices with a new volatility assumption were calculated in Table 9-5. The vega or a fraction of it is added to or subtracted from the initial theoretical value with the known volatility.Consider the bid and ask prices for the 80 Call. Given the theoretical value of 4.00, the volatility of 30 percent, and the vega of 0.10,a price of 3.90 is one vega less than 4.00, which is an implied volatility level of 29 percent.

Similarly, a price of 4.10 for the 80 Call is one vega greater than a price of 4.00, so its implied volatility is 31 percent.Consequently, bid-ask prices of 3.90 and 4.10, respectively, for the 80 Call can be stated in volatility terms as 29 percent bid and 31 percent ask.The 85 Call has a bid price of 1.75 and an ask price of 1.83. The top of Table 9-6 indicates that 1.75 is the theoretical value, assuming 30 percent volatility. Adding the vega of 0.08 to this price yields a new price of 1.83, which is 1 percent higher in volatility terms, or 31 percent.

The bid and ask for the 85 Call therefore can be stated in volatility terms as 30 percent bid and 31 percent ask.Finally, assuming 30 percent volatility for the 90 Call, its theoretical value of 0.65 and its vega of 0.06 mean that its bid and ask prices of 0.68 and 0.77 can be stated in volatility terms as 30.5 percent bid and 32 percent ask. The price of 0.68 is one-half a vega greater than 0.65, and 0.77 is two vegas greater.

Using vega to estimate new option prices if volatility changes and stating bid and ask prices in volatility terms are essential skills for professional traders to master because trading decisions often must be made quickly. The following exercises in this chapter and those in the next chapter illustrate that these skills are also valuable in creating and closing positions, in managing positions, and in managing risk.