Managing Directional Risk with Delta

Table 10-2 demonstrates how a trader might use delta to manage a long option position to both increase profit and decrease risk. This technique is based on the behavior of stock prices, which generally do not make large price changes in a straight line; rather, stock prices typically rise for a few days and then fall back before resuming an up

The goal of this managing technique, therefore, is to benefit from normal up-and-down stock-price action by maintaining a relatively constant delta. Since long options have positive gamma, the delta of a long call will increase as the stock price rises and decrease as the stock price falls. This technique therefore involves selling a portion of owned calls when a stock rallies and buying them back when the stock declines.

In the example that follows, a hypothetical trader named Grace implements this trading technique.The top section of Table 10-2 lists the rules that Grace created to govern when to purchase and sell 70 Calls. Grace’s initial position of long 20 of the 70 Calls has a delta of approximately 1,100 (actually,1,060).

Her goal is to approximately maintain this delta as the stock price rises and falls, and she has chosen two triggers for action, deltas of 1,500 and 900. Consequently, when the position delta rises above 1,500, Grace will sell a sufficient number of 70 Calls so that the position delta is reduced to approximately 1,100. Conversely,when the position delta falls below 900, Grace will buy a sufficient number of 70 Calls so that she increases the position delta.

Using 1,100 and 900 as triggers for buying and selling is a subjective decision. Traders can use Op-Eval Pro to experiment with stock-price scenarios and levels of delta based on the number of contracts traded.The middle section of Table 10-2 has six columns and 13 rows that detail how Grace implements her strategy over a 16-day period from 35 days to expiration to 19 days to expiration.

Rows 1 and 2 list stock prices and days to expiration. In column 1, for example, the stock price is 70.00 at 35 days to expiration. Row 3 lists the price of the 70 Call, row 4 lists its delta, and row 5 lists the initial position (“Beg. position”). Rows 6 and 7 hold the total delta and total value of the initial position,respectively. Row 8 indicates the action, buy or sell, and the quantity of calls, and the ending position is listed in row 9.

Row 10 indicates the delta of the ending position, which should be approximately 1,100,and row 11 indicates the value of the ending position. Row 12 lists the cash flow from the trade in row 8, which is the product of quantity of calls in row 8 and the price in row 3 and the multiplier, which is 100 and assumed. After Grace makes the final trade and closes the position,the “Final profit” is listed in column 6, row 13.

The final profit is the total of positive and negative cash flows in row 12.This exercise starts in column 1 of Table 10-2 when the stock price is 70.00 (row 1), the price of the 70 Call is 2.82 (row 2), and its delta is 0.53 (row 4). There is no beginning position (row 5), so there is no beginning delta or value (rows 6 and 7).When Grace buys 20 of these calls (row 8), she creates a position with a total delta of 1,060 (row 10) and a value of $5,640 (row 11).

The purchase is a negative cash flow (line 12). Parentheses indicate option purchases, which are negative cash flows. Cash inflows, from option sales, are numbers without parentheses.In column 2 of Table 10-2, the stock price has risen to 76.00 (row 1) at 32 days to expiration (row 2). The price of the 70 Call has increased to 6.88 (row 3), and its delta is 0.84. Grace’s position of 20 long calls (row 5) therefore has a total delta of 1,680 (row 6), which exceeds Grace’s trigger limit and spurs her to act.

She sells seven calls (row 8) in order to reduce the delta to approximately 1,100. Grace calculated this quantity by subtracting the desired delta from the beginning delta and dividing the quotient by the delta of the call in row 4; that is,(1,680 – 1,100)  (0.84  100)  6.90 ≈ 7. The actual ending delta is 1,092 (row 10). Selling seven calls resulted in a positive cash flow of $4,816 (row 12).

This trading exercise continues in column 3 of Table 10-2 when the stock price falls to 72.00 at 28 days (rows 1 and 2). As a result, the position delta falls to 858 (row 6). To raise the delta to approximately 1,100, Grace must buy four of the 70 Calls (row 8).

She calculates this quantity by subtracting the beginning delta from the desired delta and dividing the quotient by the delta of the call; that is, (1,100 – 858)  (0.66  100)  3.7 ≈ 4. The actual ending delta on this day is 1,122. In columns 4 and 5 in Table 10-2, the stock price rises to 77.00 at 24 days and falls to 73.00 at 21 days, respectively. To adjust the delta to the desired level, Grace sells five of the 70 Calls in column 4 and
buys three in column 5.