A short (negative) put is equal to a short (negative) call plus long stock, after the basis adjustment. Consider that if the put is sold instead of buying stock and selling a call, the interest that would otherwise be paid on the cost of the stock up to the strike price is a savings to the put seller. To balance the equation, the interest benefit of the short put must be added to the call side (or subtracted from the put side). It is the same with dividends.
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The dividend benefit of owning the stock must be subtracted from the call side to make it equal to the short put side (or added to the put side to make it equal the call side). The same delta concept applies here. The short 50-strike put in our example would have a 0.45 delta. The short call would have a –0.55 delta. Buying 100 shares along with selling the call gives the synthetic short put a net delta of 0.45 (–0.55 plus 1.00).
Short Call vs. Short Put + Short Stock
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Similarly, a synthetic short call can be created by selling a put and selling (short) 100 shares of stock. FIGURE 6.4 shows a conceptual overview of these two positions at expiration. Put-call parity can be manipulated as shown here to illustrate the composition of the synthetic short call.
- –Call = –Put –Stock + Strike –Interest + Dividend
Most professional traders earn a short stock rebate on the proceeds they receive when they short stock—an advantage to the short-put/short-stock side of the equation. Additionally, short-stock sellers must pay dividends on the shares they are short—a liability to the married put seller. To make all things equal, one subtracts interest and adds dividends to the put side of the equation.
Comparing Synthetic Calls and Puts
The common thread among the synthetic positions explained above is that, for a put-call pair, long options have synthetic equivalents involving long options, and short options have synthetic equivalents involving short options. After accounting for the basis, the four basic synthetic option positions are:
- Long Call = Long Put + Long Stock
- Short Call = Short Put + Short Stock
- Long Put = Long Call + Short Stock
- Short Put = Short Call + Long Stock
Because a call or put position is interchangeable with its synthetic position, an efficient market will ensure that the implied volatility is closely related for both. For example, if a long call has an IV of 25%, the corresponding put should have an IV of about 25% because the long put can easily be converted to a synthetic long call, and vice versa.
The “Greeks” will be similar for synthetically identical positions, too. The long options and their synthetic equivalents will have positive gamma and vega with negative theta. The short options and their synthetics will have negative gamma and vega with positive theta.
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|pagebreak|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 the same, too. 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. FIGURE 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.
Figure 5: Greeks for a 50-Strike Put-Call Pair on a $50 Stock
The examples used earlier in this article 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, Figure 5 shows that the call has closer to a –.554 delta. The put struck at the same price, then has a .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 minus 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:
This holds true whether the options are in-, at-, or out-of-the-money. For example, with a stock at $54, the 50 put would have a –.205 delta and the call would have a .799 delta. Selling 100 shares against the call to create the synthetic put yields a net delta of –.201.
If long or short stock is added to a call or put to create a synthetic, delta will be the only Greek affected. With that in mind, note the other Greeks displayed in Figure 5—especially theta. Proportionally, the biggest difference in the table is in theta. The disparity is due in part to interest. When the effects of the interest component outweigh the effects of the dividend, the time value of the call can be higher than the time value of the put.
Because the call must lose more premium than the put by expiration, the theta of the call must be higher than the theta of the put. American exercise can also cause the option prices in put-call parity to not add up. Deep-in-the-money puts can trade at parity while the corresponding call still has time value. The put-call equation can be unbalanced. The same applies to calls on dividend-paying stocks as the dividend date approaches. When the date is imminent, calls can trade close to parity while the puts still have time value.
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By Dan Passarelli of Market Taker Mentoring LLC
