Deltas of Calls and Puts with the Same Strike

The fourth rule on deltas is that the sum of the absolute values of the call delta and the put delta is approximately 1.00. With the stock at 100 at 56 days to expiration (column 1, rows E and F), for example,the delta of the 100 Call is 0.55, and the delta of the 100 Put is 0.45. The sum of the absolute values of these numbers, 0.55 and 0.45, is 1.00. At any point in Table 4-7, this relationship holds true and is another result of put-call parity.

Deltas and Volatility:The fifth rule on deltas explains how deltas change as volatility changes.The rule is that as volatility increases, the absolute value of a delta changes toward 0.50. In other words, deltas of out-of-the-money options increase, and deltas of in-the-money options decrease. This rule is illustrated in Table 4-8, which has two sections.

In both sections, the stock price is 100, so the 90 Call is in the money, the 100 Call is at the money, and the 110 Call is out of the money. The bottom section of Table 4-8 (rows D, E, and F) assumes a volatility of 25 percent, and the upper section (rows A, B and C) assumes a volatility of 50 percent. With volatility of 25 percent and 56 days to expiration, the in-the-money 90 Call has a delta of 0.89 (column 1, row F).

Raising the volatility to 50 percent (column 1, row C) lowers the delta to 0.75. Comparing any two corresponding deltas in rows F and C yields the same result:The increase in volatility causes the delta of the in-the-money 90 Call to decrease.For the delta of the out-of-the-money 110 Call, the change is opposite.In Table 4-8, with volatility of 25 percent and 56 days to expiration,the delta of the 110 Call is 0.30 (column 1, row D).

Raising the volatility to 50 percent increases the delta to 0.36 (column 1, row A). Comparing any two corresponding deltas in rows D and A yields the same result: The increase in volatility causes the delta of the out-of-the-money 110 Call to increase. This concept is illustrated graphically in Figure 4-3C.

Deltas of at-the-money options remain near 0.50 (absolute value of delta) over a wide range of volatility, as illustrated in Figure 4-3B and rows E and B in Table 4-8. As the figure demonstrates, deltas of at-themoney options increase to 1.00 at very low levels of volatility because the option value itself is very low and has a high correlation with stockprice movement.

Consider a hypothetical situation in which the stock price is 100, there are 60 days to expiration, and the volatility is 1 percent.In this example, a 60-day at-the-money 100 Call might have a value of 0.82 and a delta of 0.98. An increase of 10 cents to 100.10 would cause the call value to rise to 0.92 and the delta to rise to 0.99. Although a stock with 1 percent volatility is highly improbable, this exercise can help traders to think through what might happen in unusual market conditions— which actually may occur in a 20- or 30-year trading career.

Deltas change toward 0.50 when volatility increases and away from 0.50 when volatility decreases because a change in volatility changes the size of one standard deviation while the distance from stock price to strike price remains constant. Consider a situation in which the stock price is 100, one standard deviation is 5 percent, or five points, and the 105 Call has a delta of 0.35.

Under these circumstances, the strike price of 105 is one standard deviation (5 percent) away from the current stock price of 100. If volatility were to double to 10 percent with the stock price unchanged, then the 105 Call would be only one-half of a standard deviation from the stock price.

Since the 105 Call is now closer to the stock price—in volatility terms—then it’s delta must be closer to 0.50.Similarly, if volatility were to decrease with the stock price unchanged, then the 105 Call would be further out of the money in volatility terms;that is, it would be more than one standard deviation above the stock price.

Since options that are further out of the money have deltas with lower absolute values, the delta of the 105 Call would decrease given a decrease in volatility and other factors remaining constant.Option Prices and Volatility:In addition to information about the Greeks, Table 4-8 also reveals something significant about the impact of changing volatility on option prices.

When volatility increases, values of out-of-the-money options increase exponentially, whereas values of at-the-money options remain approximately linear. The impact on in-the-money options is less than one for one. In column 1 of Table 4-8, for example, the doubling of volatility from 25 to 50 percent causes the in-the-money 90 Call to increase by 24 percent from 11.26 to 13.94.