Calculating Position Risks

Quantifying the delta, gamma, vega, and theta risks of option positions is a straightforward task. Table 10-1 has five columns and five rows that calculate the Greeks of 20 long 70 Calls that were purchased for 2.82 per share each. Assumptions about the current stock price,days to expiration, volatility, interest rate, and dividends are listed at the bottom of the table.Column 1 in Table 10-1 lists five risk factors—price and four Greeks—and column 2 quantifies them on a per-share basis.

Since the underlying is 100 shares for each option, column 3 has the number 100 in every row. Similarly, every row in column 4 has the number 20 because that is the number of options in the position. Column 5 contains the risk factor of the entire position, which is the product of the three numbers in columns 2, 3, and 4.The risk factor “Price” appears in column 1 of the first row of Table 10-1, and the per-share price of 2.82 is listed in column 2.

The multiplier in column 3 and the number of options in column 4 are 100 and 20, respectively. The price risk of the 20-contract position therefore is $5,640 (2.82  100  20), as shown in column 5. The position Greeks in column 5, rows 2 through 5, are calculated in a similar manner to the price of the position, but the risks are stated differently depending on the Greek.

The position delta of 1,070, for example, indicates that this position of 20 long 70 Calls will behave like 1,070 shares of long stock over small stock-price changes. If the stock price rises by $1.00, this position will profit by approximately $1,070, and if the stock price declines by $1.00, this position will lose approximately this same amount.The position gamma of 118 in row 3 of Table 10-1 indicates that a $1.00 move in stock price will change the position delta by 118 shares in the same direction as the change in stock price.

If the stock price rises by $1.00, for example, the position delta will increase by 118 from 1,070 to 1,188. Similarly, if the stock price falls by $1.00,the position delta will decrease by 118 from 1,070 to 952.The position vega of 174 in row 4 of Table 10-1 indicates that a one percentage point change in volatility will change the position value by $174.

If volatility rises from 31 to 32 percent and other factors remain constant, the price of one option will rise by 8.7 cents (0.087 in column 2), which would raise the value of the 20-option position by $174 from $5,640 to $5,814. Similarly, if volatility declines by 1 percent, the position value would decrease by $174 from $5,640 to $5,466.The position theta in Table 10-1 estimates the impact of “one unit”of time decay. In this example, “one unit” is seven days.

The position theta of –620 indicates that the passing of seven days will cause the position value to decrease by $620 if other factors remain constant.A trader can use the information in Table 10-1 to ask—and answer—several questions about risk. First, can I withstand a $1,070 loss if the stock price declines $1.00? What about a $2.00 or $3.00 stock-price decline? Where will I take my loss if the stock price declines?

These are questions that traders must answer individually.One comment needs to be made on volatility risk. The position vega in Table 10-1 tells a trader that a one percentage point change in implied volatility will change the position value by $174. The vega, however, does not estimate the likelihood of implied volatility changing or by how much it might change. Historical data, such as that provided at cboe or ativolatility (see Figure 7-7), can assist, but forecasting volatility is an art, not a science.

In contrast to vega, the position theta provides a much firmer estimate of the risk of time decay. In Table 10-1, if the stock price and other factors are unchanged in seven days, this position will lose $620. A trader can use this estimate with a dollar risk limit to determine how long a position will be held before a loss is taken.Risks of Short Options:Although the Greeks of long and short options are opposite, the risks of short options are not simply opposite the risks of long options.

Positions with uncovered short options have unlimited risk in the case of short calls and substantial risk in the case of short puts. An uncovered short option is a short option that has no offsetting stock or option position that truly limits risk. Although, in practice, stock-price changes are never really unlimited, they can be very large.

As experienced traders know, unexpected events can cause prices to change by 30 percent, 50 percent, or more overnight or in very short periods of time. The risk of short option positions, therefore, must be considered differently than the known maximum price risk of long options. Unfortunately, there is no uniform method of determining the suitability of short option risk. Are 50 short options too many?