Remember back in school when students were always asking the math teacher why they needed to learn to do something?

For example, why do we option traders need to understand the relationship termed "put-call parity"? A recent article in "thinkMoney," think-or-swim's monthly publication, cleared up my understanding of the term "conversion." It did so while pointing to our need to understand the basic relationships among stock or other underlyings, calls and puts.

That basic relationship is one that I haven't discussed in a while, that equation describing put-call parity. Jim Bittman outlines that equation as the following (Trading Options as a Professional, 153):

+Stock = +call -put

If you remember back to junior high school or middle school, when you first learned about equations, you'll understand that the equation for put-call parity can be manipulated in many ways. For example, we can add +put to both sides of the equation and find a new relationship.

+Stock +put = +call -put +put
+Stock +put = +call

Therefore, the shape of a profit/loss chart of a call should be the same as stock + put. Below, you'll find some sample charts snapped a week or two ago when MSFT was trading just above \$25.00.

Expiration Graph of 100 Shares of MSFT Plus One 25-Strike Put:

Expiration Graph of One 25-Strike Call:

You can see that small differences exist but that the two charts show the same shape and nearly identical expiration breakeven levels and maximum loss levels. The small differences that do exist relate to dividends, interest rates and commission costs. Conversions attempt to capitalize on those differences, but I'm getting ahead of myself here.

Bittman describes the conversion as "a three-part strategy consisting of long stock, long puts, and short calls on a share-for-share basis" (165). Later Bittman specifies, as does the think-or-swim article (20), that the conversation's short call and long put have the same expiration and strike. What kind of position is this? How would it look on a profit-loss chart? If we manipulate that put-call parity formula mentioned above, we can visualize how that chart might look. This is a valuable exercise for all traders, even if we retail traders are unlikely to ever trade a conversion, for reasons I'll discuss later.

We know that long stock is a long position, so what's the equivalent of a short call and long put position at the same expiration and strike? We start with the equation for put-call parity.

+Stock = +call -put
Then we multiply both sides by -1: -1(+Stock) = -1(+call -put)
This results in the following: -Stock = -call +put

Therefore, the option portion of the conversion is the equivalent of a short stock position at the strike price. Therefore, the conversion consists of a balanced position that is long stock and also has a synthetic short stock position. Over a short period of time and with relatively steady volatilities, it's not going to be helped nor hurt by a change in price. Therefore, it's easy to visualize the graph. It should be a straight horizontal line, neither helped nor hurt by a change in price.

What's the benefit to a conversion? As both Bittman and think-or-swim point out, it's a "'cost of carry' trade" (thinkMoney, 20). The professional trader who enters this trade expects that the dividend payout will exceed the interest on the margin required to establish the position and the commissions. While that might be possible for professional traders with their much smaller commission costs, we retail traders can't realistically benefit from such a trade.

We can, however, benefit from knowing about the synthetic relationships that make the conversion possible for professional traders. Savvy, somewhat conservative investors benefit all the time from a similar relationship by collaring their long stock position. I've periodically written about collars throughout the years, and I've certainly encouraged and sometimes pleaded with friends to at least partially collar their long stock positions. Until I had read the thinkMoney article, however, I'd never thought of the collar as the kissing cousin of the conversion. The conversion seemed to be an abstruse construction that was as difficult for the normal trader to understand as it was for them to capitalize on the trade.

We all understand how collars work for us, however, without even knowing the synthetic positions involved. We own stock, Leaps or perhaps a call position with an expiration date several months out. We're willing to accept a certain amount of loss but want to protect the position against a bigger decline. We want to buy an OTM put position to protect that long position. We sell a call to at least partially finance the cost of that position.

We know now from our manipulations of the put-call parity equation that selling a call and buying a put creates a short stock equivalent at some price or another, depending on where they're placed. Because we've moved those strikes apart, however, this kissing cousin of the conversion changes its nature. We're not short stock at the same price as we are long that stock, as we would be with the conversion, and instead are accepting some degree of risk. In addition, this isn't a cost-of-carry trade any longer, although that cost of carry will impact our trade, but rather is a trade that seeks to limit the downside risk at a low or zero cost. Instead of the straight horizontal line chart we expect with a conversion, we now should anticipate a chart in which the gains will level off near the strike of the sold call and the losses will level off near the strike of long put. The chart below was snapped at the same time as the conversion chart above.

MSFT Stock Collared with -1 DEC 27 Call + 1 DEC 23 Put:

(Note: I mistakenly collared this stock with DEC options, but can't go back with the freeware that I used and recreate the same position with MSFT at the same price. The premise remains the same, and it's more important that the graphs were created at the same time.) We see the expected leveling off near the strikes of the 27 call and the 23 put.

The two trades--the collared long equity position and the conversion--have some similarities but also some important differences. While we might not ever establish conversions, I do believe it's important for us to understand that many of our options trades are essentially such kissing cousins. We should look for such relationships. Such manipulations of the put-call parity formula helps us to understand what kind of position we actually own.

If two positions are equivalent, is there a benefit to entering one over the other? For example, thinkMoney points out the similarity in a collared stock position and a debit call vertical. Let's look see if we agree. I've constructed such a debit call position, again snapping the charts at the same time I did the others so that MSFT's price was equivalent and option relationships were the same. Because of the strikes available for JAN, I was not able to duplicate the strikes shown in the theoretical collared position above when I mistakenly used the DEC options, but I think you'll see the similarities anyway. Again, we see gains and losses leveling off near where they did for the collar, with the differences here due to the differences in strikes available.

MSFT Debit Call Spread with +1 22.50 JAN Call and -1 JAN 27.50 Call:

Now that we've compared the conversion and the collar, we can compare this bull call debit spread with the collar. What are the pros and cons of a collared stock position versus the debit spread? Obviously, as thinkMoney points out, if you already own MSFT stock and don't want to or can't get rid of it, perhaps for tax reasons, the collar is the right trade for you if you want the protection the collar offers. What if you don't own the stock, but are cautiously bullish the stock while still wanting downside protection? The bull call debit spread consists of two positions, sold calls and long calls, and therefore has lower transaction costs than the three-position collar. Moreover, the margin withheld will be much larger with the collared stock than with the vertical. In the past, locking up that much margin would have meant a much lower interest payment in the cash in your brokerage account because more of that cash would be tied up in the trade. These days, none of us sees much interest payments, do we, but there might come a time when borrowing costs will again be an important factor to consider. Even if it's not a factor yet, you've lost the opportunity to use that larger tied-up margin for other money-making purposes.

It should be obvious that you don't have to buy stock and then collar it if you are moderately bullish a stock but want some downside protection. Of course, drawbacks to the vertical exist, too, when you compare one position to the other. Now that you've thought about these comparisons, you can compare these positions to the simple purchase of a call. I don't pretend to know all the permutations of when you'd like to do one over the other, but I do think that understanding the put-call parity equation helps us to understand the nature of our positions and then to make decisions about which position works best for us and which poses too much of a risk for us.