American-Exercise Options
Put-call parity was designed for European-style options. The early exercise
possibility of American-style options gums up the works a bit. Because a
call (put) and a synthetic call (put) are functionally the same, it is logical to
assume that the implied volatility and the greeks for both will be exactly the
same. This is not necessarily true with American-style options. However,
put-call parity may still be useful with American options when the
limitations of the equation are understood. With at-the-money American-
exercise options, the differences in the greeks for a put-call pair are subtle.
Exhibit 6.5 is a comparison of the greeks for the 50-strike call and the 50-
strike put with the underlying at $50 and 66 days until expiration.
EXHIBIT 6.5 Greeks for a 50-strike put-call pair on a $50 stock.
Call Put
Delta 0.5540.457
Gamma0.0750.078
Theta 0.0200.013
Vega 0.0840.084
The examples used earlier in this chapter in describing the deltas of
synthetics were predicated on the rule of thumb that the absolute values of
call and put deltas add up to 1.00. To be a bit more realistic, consider that
because of American exercise, the absolute delta values of put-call pairs
don’t always add up to 1.00. In fact, Exhibit 6.5 shows that the call has
closer to a 0.554 delta. The put struck at the same price then has a 0.457
delta. By selling 100 shares against the long call, we can create a combined-
position delta (call delta plus stock delta) that is very close to the put’s
delta. The delta of this synthetic put is −0.446 (0.554 − 1.00). The delta of a
put will always be similar to the delta of its corresponding synthetic put.
This is also true with call–synthetic-call deltas. This relationship
mathematically is