Why the Numbers Don’t Always Add Up

There will be many times when studying the rho of options in an option chain will reveal seemingly counterintuitive results. To be sure, the numbers don’t always add up to what appears logical. One reason for this is rounding. Another is that traders are more likely to use simple interest in calculating value, whereas the model uses compound interest.

Hard-to-borrow stocks and stocks involved in mergers and acquisitions may have put-call parities that don’t work out right. But another, more common and more significant fly in the ointment is early exercise.Since the interest input in put-call parity is a function of the strike price,it is reasonable to expect that the higher the strike price, the greater the effect of interest on option prices will be.

For European options, this is true to a large extent, in terms of aggregate impact of interest on the call and put pair.Strikes below the price where the stock is trading have a higher rho associated with the call relative to the put, whereas strikes above the stock price have a higher rho associated with the put relative to the call. Essentially, the more in-the-money an option is, the higher its rho. But with European options, observing the aggregate of the absolute values of the call and put rhos would show a higher combined rho the higher the strike.

With American options, the put can be exercised early. A trader will exercise a put before expiration if the alternative—being short stock and receiving a short stock rebate—is a wiser choice based on the price of the put.Professional traders may own stock as a hedge against a put. They may exercise deep ITM puts (1.00-delta puts) to avoid paying interest on capital charges related to the stock.

The potential for early exercise is factored into models that price American options. Here, when puts get deeper in-themoney—that is, more apt to be exercised—the rho decreases. When the strike price is very high relative to the stock price—meaning the put is very deep ITM—and there is little or no time value left to the call or the put, the aggregate put-call rho can be zero. Rho is discussed in greater detail in Chapter 7.Where to Find Option Greeks:There are many sources from which to obtain greeks.

The Internet is an excellent resource. Googling “option greeks” will display links to over four million web pages, many of which have real-time greeks or an option calculator.An option calculator is a simple interface that accepts the input of the six variables to the model and yields a theoretical value and the greeks for a single option.Some web sites devoted to option education, such as MarketTaker/option_modeling, have free calculators that can be used for modeling positions and using the greeks.

In practice, many of the option-trading platforms commonly in use have sophisticated analytics that involve greeks. Most options-friendly online brokers provide trading platforms that enable traders to conduct comprehensive manipulations of the greeks. For example, traders can look at the greeks for their positions up or down one, two, or three standard deviations. Or they can see what happens to their position greeks if IV or time changes.

With many trading platforms, position greeks are updated in real time with changes in the stock price—an invaluable feature for active traders.Caveats with Regard to Online: Greeks Often, online greeks are one click away, requiring little effort on the part of the trader. Having greeks calculated automatically online is a quick and convenient way to eyeball greeks for an option. But there is one major problem with online greeks: reliability.

For active option traders, greeks are essential. There is no point in using these figures if their accuracy cannot be assured. Experienced traders can often spot these inaccuracies a proverbial mile away.When looking at greeks from an online source that does not require you to enter parameters into a model (as would be the case with professional option-trading platforms), special attention needs to be paid to the relationship of the option’s theoretical values to the bid and offer.

One must be cautious if the theoretical value of the option lies outside the bid-ask spread.This scenario can exist for brief periods of time, but arbitrageurs tend to prevent this from occurring routinely. If several options in a chain all have theoretical values below the bid or above the offer, there is probably a problem with one or more of the inputs used in the model. Remember, an option-pricing model is just that: a model.

It reflects what is occurring in the market. It doesn’t tell where an option should be trading.The complex changes that occur intraday in the market—taking the day or weekend out, changes in stock price, volatility, and the interest rate—are not always kept current. The user of the model must keep close watch.It’s not reasonable to expect the computer to do the thinking for you.