CHAPTER 7
Rho
Interest is one of the six inputs of an option-pricing model for American options. Although interest rates can remain constant for long periods, when interest rates do change, call and put values can be positively or negatively affected. Some options are more sensitive to changes in the interest rate than others. To the unaware trader, interest-rate changes can lead to unexpected profits or losses. But interest rates don’t have to be a wild-card risk. They’re one that experienced traders watch closely to avoid unnecessary risk and increase profitability. To monitor the effect of changes in the interest rate, it is important to understand the quiet greek—rho.
Rho and Interest Rates
Rho is a measurement of the sensitivity of an option’s value to a change in the interest rate. To understand how and why the interest rate is important to the value of an option, recall the formula for put-call parity stated in Chapter 6.
Call + Strike − Interest = Put + Stock
1
From this formula, it’s clear that as the interest rate rises, put prices must fall and call prices must rise to keep put-call parity balanced. With a little algebra, the equation can be restated to better illustrate this concept:
and
If interest rates fall,
and
Rho helps quantify this relationship. Calls have positive rho, and puts have negative rho. For example, a call with a rho of +0.08 will gain $0.08 with each one-percentage-point rise in interest rates and fall $0.08 with each one-percentage-point fall in interest rates. A put with a rho of −0.08 will lose $0.08 with each one-point rise and gain $0.08 in value with a one-point fall.
The effect of changes in the interest variable of put-call parity on call and put values is contingent on three factors: the strike price, the interest rate, and the number of days until expiration.
Interest = Strike×Interest Rate×(Days to Expiration/365)
2
Interest, for our purposes, is a function of the strike price. The higher the strike price, the greater the interest and, consequently the more changes in the interest rate will affect the option. The higher the interest rate is, the higher the interest variable will be. Likewise, the more time to expiration, the greater the effect of interest. Rho measures an option’s sensitivity to the end results of these three influences.
To understand how changes in interest affect option prices, consider a typical at-the-money (ATM) conversion on a non-dividend-paying stock.
Short 1 May 50 call at 1.92
Long 1 May 50 put at 1.63
Long 100 shares at $50
With 43 days until expiration at a 5 percent interest rate, the interest on the 50 strike will be about $0.29. Put-call parity ensures that this $0.29 shows up in option prices. After rearranging the equation, we get
In this example, both options are exactly ATM. There is no intrinsic value. Therefore, the difference between the extrinsic values of the call and the put must equal interest. If one option were in-the-money (ITM), the intrinsic value on the left side of the equation would be offset by the Stock − Strike on the right side. Still, it would be the difference in the time value of the call and put that equals the interest variable.
This is shown by the fact that the synthetic stock portion of the conversion is short at $50.29 (call − put + strike). This is $0.29 above the stock price. The synthetic stock equals the Stock + Interest, or
Certainly, if the interest rate were higher, the interest on the synthetic stock would be a higher number. At a 6 percent interest rate, the effective short price of the synthetic stock would be about $50.35. The call would be valued at about 1.95, and the put would be 1.60—a net of $0.35.
A one-percentage-point rise in the interest rate causes the synthetic stock position to be revalued by $0.06—a $0.03 gain in the call value and a $0.03 decline in the put. Therefore, by definition, the call has a +0.03 rho and the put has a −0.03 rho.
Rho and Time
The time component of interest has a big impact on the magnitude of an option’s rho, because the greater the number of days until expiration, the greater the interest. Long-term options will be more sensitive to changes in the interest rate and, therefore, have a higher rho.
Take a stock trading at about $120 per share. The July, October, and January ATM calls have the following rhos with the interest rate at 5.5 percent.
Option
Rho
July (38-day) 120 calls
+0.068
October (130-day) 120 calls
+0.226
January (221-day) 120 calls
+0.385
If interest rates rise 25 basis points, or a quarter of a percentage point, the July calls with only 38 days until expiration will gain very little: only $0.017 (0.068 × 0.25). The October 120 calls with 130 days until expiration gain more: $0.057 (0.226 × 0.25). The January calls that have 221 days until they expire make $0.096 theoretically (0.385 × 0.25). If all else is held constant, the more time to expiration, the higher the option’s rho, and therefore, the more interest will affect the option’s value.
Considering Rho When Planning Trades
Just having an opinion on a stock is only half the battle in options trading. Choosing the best way to trade a forecast can make all the difference to the success of a trade. Options give traders choices. And one of the choices a trader has is the month in which to trade. When trading LEAPS—Long-Term Equity AnticiPation Securities—delta, gamma, theta, and vega are important, as always, but rho is also a valuable part of the strategy.
LEAPS
Options buyers have time working against them. With each passing day, theta erodes the value of their assets. Buying a long-term option, or a LEAPS, helps combat erosion because long-term options can decay at a slower rate. In environments where there is interest rate uncertainty, however, LEAPS traders have to think about more than the rate of decay.
Consider two traders: Jason and Susanne. Both are bullish on XYZ Corp. (XYZ), which is trading at $59.95 per share. Jason decides to buy a May 60 call at 1.60, and Susanne buys a LEAPS 60 call at 7.60. In this example, May options have 44 days until expiration, and the LEAPS have 639 days.
Both of these trades are bullish, but the traders most likely had slightly different ideas about time, volatility, and interest rates when they decided which option to buy.
Exhibit 7.1
compares XYZ short-term at-the-money calls with XYZ LEAPS ATM calls.
EXHIBIT 7.1
XYZ short-term call vs. LEAPS call.
To begin with, it appears that Susanne was allowing quite a bit of time for her forecast to be realized—almost two years. Jason, however, was looking for short-term price appreciation. Concerns about time decay may have been a motivation for Susanne to choose a long-term option—her theta of 0.01 is half Jason’s, which is 0.02. With only 44 days until expiration, the theta of Jason’s May call will begin to rise sharply as expiration draws near.
But the trade-off of lower time decay is lower gamma. At the current stock price, Susanne has a higher delta. If the XYZ stock price rises $2, the gamma of the May call will cause Jason’s delta to creep higher than Susanne’s. At $62, the delta for the May 60s would be about 0.78, whereas the LEAPS 60 call delta is about 0.77. This disparity continues as XYZ moves higher.
Perhaps Susanne had implied volatility (IV) on her mind as well as time decay. These long-term ATM LEAPS options have vegas more than three times the corresponding May’s. If IV for both the May and the LEAPS is at a yearly low, LEAPS might be a better buy. A one- or two-point rise in volatility if IV reverts to its normal level will benefit the LEAPS call much more than the May.
Theta, delta, gamma, and vega are typical considerations with most trades. Because this option is long term, in addition to these typical considerations, Susanne needs to take a good hard look at rho. The LEAPS rho is significantly higher than that of its short-term counterpart. A one-percentage-point change in the interest rate will change Susanne’s P&(L) by $0.64—that’s about 8.5 percent of the value of her option—and she has nearly two years of exposure to interest rate fluctuations. Certainly, when the Federal Reserve Board has great concerns about growth or inflation, rates can rise or fall by more than one percentage point in one year’s time.
It is important to understand that, like the other greeks, rho is a snapshot at a particular price, volatility level, interest rate, and moment in time. If interest rates were to fall by one percentage point today, it would cause Susanne’s call to decline in value by $0.64. If that rate drop occurred over the life of the option, it would have a much smaller effect. Why? Rate changes closer to expiration have less of an effect on option values.
Assume that on the trade date, when the LEAPS has 639 days until expiration, interest rates fall by 25 basis points. The effect will be a decline in the value of the call of 0.16—one-fourth of the 0.638 rho. If the next rate cut occurs six months later, the rho of the LEAPS will be smaller, because it will have less time until expiration. In this case, after six months, the rho will be only 0.46. Another 25-basis-point drop will hurt the call by $0.115. After another six months, the option will have a 0.26 rho. Another quarter-point cut costs Susanne only $0.065. Any subsequent rate cuts in ensuing months will have almost no effect on the now short-term option value.
Pricing in Interest Rate Moves
In the same way that volatility can get priced in to an option’s value, so can the interest rate. When interest rates are expected to rise or fall, those expectations can be reflected in the prices of options. Say current interest rates are at 8 percent, but the Fed has announced that the economy is growing at too fast of a pace and that it may raise interest rates at the next Federal Open Market Committee meeting. Analysts expect more rate hikes to follow. The options with expiration dates falling after the date of the expected rate hikes will have higher interest rates priced in. In this situation, the higher interest rates in the longer-dated options will be evident when entering parameters into the model.
Take options on Already Been Chewed Bubblegum Corp. (ABC). A trader, Kyle, enters parameters into the model for ABC options and notices that the prices don’t line up. To get the theoretical values of the ATM calls for all the expiration months to sit in the middle of the actual market values, Kyle may have to tinker with the interest rate inputs.
Assume the following markets for the ATM 70-strike calls in ABC options:
Calls
Puts
Aug 70 calls
1.75–1.85
1.30–1.40
Sep 70 calls
2.65–2.75
1.75–1.85
Dec 70 calls
4.70–4.90
2.35–2.45
Mar 70 calls
6.50–6.70
2.65–2.75
ABC is at $70 a share, has a 20 percent IV in all months, and pays no dividend. August expiration is one month away.
Entering the known inputs for strike price, stock price, time to expiration, volatility, and dividend and using an 8 percent interest rate yields the following theoretical values for ABC options:
The theoretical values, in bold type, are those that don’t line up in the middle of the call and put markets. These values are wrong. The call theoretical values are too low, and the put theoretical values are too high. They are the product of an interest rate that is too low being applied to the model. To generate values that are indicative of market prices, Kyle must change the interest input to the pricing model to reflect the market’s expectations of future interest rate changes.
Using new values for the interest rate yields the following new values:
After recalculating, the theoretical values line up in the middle of the call and put markets. Using higher interest rates for the longer expirations raises the call values and lowers the put values for these months. These interest rates were inferred from, or backed out of, the option-market prices by use of the option-pricing model. In practice, it may take some trial and error to find the correct interest values to use.
In times of interest rate uncertainty, rho can be an important factor in determining which strategy to select. When rates are generally expected to continue to rise or fall over time, they are normally priced in to the options, as shown in the previous example. When there is no consensus among analysts and traders, the rates that are priced in may change as economic data are made available. This can cause a revision of option values. In long-term options that have higher rhos, this is a bona fide risk. Short-term options are a safer play in this environment. But as all traders know, risk also implies opportunity.
Trading Rho
While it’s possible to trade rho, most traders forgo this niche for more dynamic strategies with greater profitability. The effects of rho are often overshadowed by the more profound effects of the other greeks. The opportunity to profit from rho is outweighed by other risks. For most traders, rho is hardly ever even looked at.
Because LEAPS have higher rho values than corresponding short-term options, it makes sense that these instruments would be appropriate for interest-rate plays. But even with LEAPS, rho exposure usually pales in comparison with that of delta, theta, and vega.
It is not uncommon for the rho of a long-term option to be 5 to 8 percent of the option’s value. For example,
Exhibit 7.2
shows a two-year LEAPS on a $70 stock with the following pricing-model inputs and outputs:
EXHIBIT 7.2
Long 70-strike LEAPS call.
The rho is +0.793, or about 5.8 percent of the call value. That means a 25-basis-point rise in rates contributes to only a 20-cent profit on the call. That’s only about 1.5 percent of the call’s value. On one hand, 1.5 percent is not a very big profit on a trade. On the other hand, if there are more rate rises at following Fed meetings, the trader can expect further gains on rho.
Even if the trader is compelled to wait until the next Fed meeting to make another $0.20—or less, as rho will get smaller as time passes—from a second 25-basis-point rate increase, other influences will diminish rho’s significance. If over the six-week period between Fed meetings, the underlying declines by just $0.60, the $0.40 that the trader hoped to make on rho is wiped out by delta loss. With the share price $0.60 lower, the 0.760 delta costs the trade about $0.46. Furthermore, the passing of six weeks (42 days) will lead to a loss of about $0.55 from time decay because of the −0.013 theta. There is also the risk from the fat vegas associated with LEAPS. A 1.5 percent drop in implied volatility completely negates any hopes of rho profits.
Aside from the possibility that delta, theta, and vega may get in the way of profits, the bid-ask spread with these long-term options tends to be wider than with their short-term counterparts. If the bid-ask spread is more than $0.40 wide, which is often the case with LEAPS, rho profits are canceled out by this cost of doing business. Buying the offer and selling the bid negative scalps away potential profits.
With LEAPS, rho is always a concern. It will contribute to prosperity or peril and needs to be part of the trade plan from forecast to implementation. Buying or selling a LEAPS call or put, however, is not a practical way to speculate on interest rates.
To take a position on interest rates in the options market, risk needs to be distilled down to rho. The other greeks need to be spread off. This is accomplished only through the conversions, reversals, and jelly rolls described in Chapter 6. However, the bid-ask can still be a hurdle to trading these strategies for non–market makers. Generally, rho is a greek that for most traders is important to understand but not practical to trade.
Notes
1
. Please note, for simplification, dividends are not included.
2
. Note, for simplicity, simple interest is used in the calculation.