Rhos and Stock Price

The second rule governing rhos describes the relative level of rhos as the underlying stock price changes. Table 4-7 and Figures 4-13A and 4-13B show that rhos increase as the stock price rises. Column 1 in Table 4-7, for example, shows that as the stock price rises from 100 to 105 to 110 at 56 days (column 1), the rho of the 100 Call rises from 0.08 (row E) to 0.10 (row C) to 0.12 (row A), and the rho of the 100 Put rises from 0.06 (row F) to 0.04 (row D) to 0.03 (row B).Remember to keep increases and decreases straight when minus signs are involved!

The same concept—that rhos rise with a rising stock price and fall with a falling stock price—holds true for any column in Table 4-7. Figures 4-13A and 4-13B demonstrate this concept graphically.At first glance, the graphs may appear identical, but they are not.That rhos increase with rising stock prices is another result of the cost-of-carry concept. Higher-priced stocks are more expensive to finance than lower-priced stocks, and changes in interest rates have a greater impact on the absolute cost of financing high-priced stocks than on low-priced stocks.

Rhos and Time to Expiration:Figures 4-14A, 4-14B, and 4-14C show the third rule. Rhos increase in an almost linear manner with increases in time. The cost-of-carry concept also explains this consequence. At a given interest rate, financing costs are higher—in a linear relationship—for longer time periods than for shorter periods. And if interest rates change, the absolute impact will be higher—linearly—for the longer time period than for the shorter.

Rhos and Volatility:The fourth rule describes the impact of volatility on rho. This concept is complicated. Table 4-8 and Figures 4-15A, 4-15B, and 4-15C show that volatility has different effects on rhos of in the money, at the money, and out of the money options. The difficult aspect to grasp is that volatility only affects rho indirectly through its impact on option prices.Consider rows D and A in column 1 of Table 4-8, which show that an increase in volatility from 25 to 50 percent increases the rho of the 110 Call from 0.03 to 0.05.

Note also that the call price rises from 1.02 to 4.39. In addition to the change in cost of carry of the underlying stock, a change in the interest rate also would affect the cost (foregone interest) of owning the call. The foregone interest on 4.39 is 400 percent that of the foregone interest on 1.02 regardless of the level of interest rates. Therefore, the effect of a change in the interest rate must be greater when volatility is higher than when it is lower.

Figure 4-15C shows that rising volatility has an exponential impact on the rho of an out-of-the-money call from approximately 10 to 50 percent volatility.Above 50 percent, the impact of rising volatility levels off, and rho approaches its limit, just as the option price approaches its limit at high levels of volatility (see Figure 2-6).Figure 4-15B shows that the impact of volatility on rhos of at-the money calls is more linear—and actually declining—than for out-ofthe-money calls, and Figure 4-15A shows that rising volatility causes rhos of in-the-money calls to decrease.

Fortunately, this complicated interaction of volatility and rho is of little importance to the vast majority of traders, even full-time ones.Position Greeks:The term position refers to whether an option is purchased (i.e., long) or written (i.e., short). For example, if Adam buys 25 XYZ November 100 Calls, his “position” is long 25. If Matthew buys 15 QRS April 45 Puts and sells 15 QRS April 40 Puts, his position is long 15 April 45 Puts and short 15 April 40 Puts.

What Adam and Matthew and all traders need is a method of estimating how their position will perform if market conditions change,that is, if one or more of the inputs to the option-pricing formula changes. Position Greeks indicate whether an entire position will experience profit or loss when a particular input to the option-pricing formula is changed.

You will learn how to calculate and interpret position Greeks after the following discussion about the use of positive and negative signs.In options trading, plus and minus signs can have three different meanings.“+”And “-” Have Three Different Meanings First, when associated with a quantity of options, plus signs mean “long,” and minus signs mean “short.”

The position description “+3 NDX January 2200 Puts at 12.50” is read as “long 3 NDX 2200 Puts at 12.50 each.”The position description “-15 XSP November 145 Calls at 9.10” is read as “short 15 XSP November 145 Calls at 9.10 each.”Second, when associated with an option’s delta, vega, theta, or rho, plus and minus signs mean that the option value is positively or negatively correlated with changes in the respective input.