Why the Numbers Don’t 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-the-
money—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.