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.