# How the Greeks Change

Trying to measure something when the measure itself changes poses obvious problems. Consequently, estimating changes in option values is complicated by the fact that the Greeks change when market conditions change. For example, when the stock price, time to expiration,volatility, or any combination of these factors change, so do the delta,gamma, theta, and vega.

Sometimes an individual Greek will change significantly and have a great impact on an option’s value, but sometimes the change will have little impact. Both graphs and tables are effective ways to illustrate changing Greeks, so the following discussion uses both to make several points about each Greek. Changes in underlying price, time to expiration, and volatility matter most to traders, so Tables 4-7 and 4-8 focus on changes in these three inputs.

Table 4-7 depicts a grid of 100 Call values, 100 Put values,and corresponding Greeks at three stock prices and various days to expiration. A study of this table reveals how delta, gamma, vega, and theta change as stock price, time to expiration, or both change.A study of this table reveals how the Greeks of in the-money, at-the-money, and out-of-the-money options change as volatility, time to expiration, or both change.

The concepts in Tables 4-7 and 4-8 help option traders to analyze the impact of changing market conditions on their positions.How Delta Changes:Delta, as stated earlier, estimates how much an option value changes when the underlying stock price changes and other factors remain constant.Because calls have positive deltas and puts have negative deltas, the five rules will be stated using the absolute values of the deltas.

Deltas and Stock Price:The first rule describes how deltas change as the price of the underlying stock changes. Deltas of both calls and puts increase as the stock price rises and decrease as the stock price falls. Table 4-7 and Figures 4-1A and 4-1B illustrate this concept. At first glance, the graphs may appear identical, but they are not. Moving up the x axis, the call delta rises from 0 to 1.00, whereas the put delta rises from 1.00 to 0.

Column 1 in Table 4-7 shows that as the stock price rises from 100 to 105 to 110 at 56 days, the delta of the 100 Call increases from 0.55 (row E) to 0.71 (row C) to 0.82 (row A), and the delta of the 100 Put rises from 0.45 (row F) to 0.30 (row D) to 0.18 (row B). Remember to keep increases and decreases straight when minus signs are involved! The same concept—deltas rising with an increasing stock price and falling with a decreasing stock price—holds true for any column in Table 4-7.

Deltas and Strike Price:The second rule on deltas concerns the relative level of deltas of in-themoney,at-the-money, and out-of-the-money options. In-the-money options have deltas with absolute values greater than 0.50. At-themoney options have deltas with absolute values of approximately 0.50,and out-of-the-money options have deltas with absolute values less than 0.50 regardless of time to expiration. Table 4-7 illustrates this rule.With a stock Price of 110, the 100 Call is in the money, and the 100 Put is out of the money.

Row A shows that the absolute value of the delta of the 100 Call is always above 0.50, and row B shows that the absolute value of the delta of the 100 Put is always below 0.50.Deltas and Time to Expiration:The third rule on delta concerns how deltas change as expiration approaches. The absolute values of deltas of in-the-money options increase toward 1.00 as expiration approaches.

This rule is illustrated graphically in Figure 4-2A and numerically in Table 4-7. In row A of the table, for example, the stock price is 110, which means that the 100 Call is in the money, and its delta increases from 0.82 at 56 days (column 1) to 0.89 at 28 days (column 3) to 1.00 at expiration (column 5).Note that “Time to Expiry” on the x axis in Figures 4-2A, 4-2B, and 4-2C decreases from right to left. While it is generally intuitive to have lower numbers on the left and higher numbers on the right (see Figures 4-1, 4-3, 4-4, and others), this is not true with time to expiration and options.

Take a few moments, therefore, to accustom yourself to reading and understanding these three graphs.The absolute values of deltas of at-the-money options remain near 0.50. In rows E and F of Table 4-7, where the stock price is 100,both the 100 Call and 100 Put are at the money. In these rows, the absolute value of the deltas remain near 0.50 in all columns. This concept is illustrated graphically in Figure 4-2B.The absolute values of deltas of out-of-the-money options decrease toward zero as expiration approaches.