Put-call parity describes how a call and put at each strike
are related. Look at this formula:
Call price – put price + strike price = Stock price
+ the cost of carry
Cost of carry refers to any net interest cost (or gain) that
comes when you buy the stock, which has its own formula:
Cost of carry = days to expiration/360 x interest
rate x strike price – dividends
For an extended
option relationships made
easy, go to
little bit of knowledge can be dangerous when it comes
to trading. All too often, I’ve seen retail traders who took
a few option classes, maybe read a few books, and
thought they had spotted some money just lying on the
proverbial table, left there by incautious market makers.
They strapped on the “can’t lose” position a little too big,
and got slammed when a detail about the trade emerged
that the market makers knew, but that the beginning
traders didn’t. I hate to see that. Understanding the
relationships that exist between options can be vital
to preventing the same costly mistakes so many retail
traders have made when they thought they saw some-
thing too good to be true.
I'LL TAKE A COMBO, PLEASE
Each option for a particular underlying security relates to
every other option in one of three ways: put-call parity
(between same strikes), boxes (between different strikes)
and “jelly” rolls (between different expirations). The best
way to view these relationships is by looking at the synthetic stock, or the “combo.” A long combo is long a call and
short a put at the same strike price and expiration. A short
combo is short a call and long a put. The combo changes
point for point with the stock price, which is why it’s synthetic stock.
The difference between the combo and the stock
most often has to do with the cost of carry (see
“Capiche?”, thinkMoney/04). What’s important to understand for this discussion is that combos are used to
understand the three relationships: put-call parity, boxes,
and jelly rolls.
Now, that may look complex. But it’s actually quite simple
when stripped down to its shorts. You buy the stock, and
you spend cash, which means you either borrow money
and are charged interest or you reduce your cash and
stop earning interest on it. You also are entitled to any dividends payable when you own the stock. But, wait! There’s
a typo! The formula uses the strike price, not the stock
price! Ahh, good catch, but the formula is correct as written.
This position is called a conversion, and it buys the
stock and sells the combo (sells the call and buys the put)
based entirely on put-call parity. If I buy actual stock and
sell synthetic stock, I will have a position that is not sensitive to changes in the stock price. In that sense, the position is “flat.” But the net amount of cash I spend on the
trade is based on the price I pay for the stock, the cash
taken in by selling the call, and the cash spent buying the
put. That net amount comes very close to the strike price,
which is why the cost of carry uses the strike price, not the
Let’s look at AAPL, for example. With AAPL, which
doesn’t pay dividends, at $120.50, the 120 call with 25
days to expiration was 6.80. What’s the 120 put worth?
First, the cost of carry equals 25/360 X .25% (
current Fed funds rate) X 120 = 0.02.
Then, 6.80 - put price + 120 = 120.50 + 0.02.
Therefore, put price = 6. 28.
When the bid/ask on the 120 calls was 6.75 – 6.85,
the bid/ask on the 120 puts was 6. 25 – 6.30, which is
wrapped around 6.80. Hey! Put-call parity works!
Caveat emptor: Under certain circumstances, put-call
parity doesn’t work, and you could be lured into thinking
you’ve ingeniously uncovered an arb situation that a market maker missed somehow. Take a look at Citicorp (C),
which was trading at $2.94 recently. At one point, the
three calls with 25 days to expiration were 0.36 – 0.37,
and the three puts were 0.60 – 0.62. Citigroup doesn’t pay
a dividend anymore, so the cost of carry is 25/360 X 0.25%
X 3 = 0.0005—almost nothing. Let’s see here, 0.037 – 0.62
+ 3 = 2.94 – 0.0005. Wait a second. No, it doesn’t. Hey! The
combo is 0.20 too cheap! There’s an arb! I’m a genius! I can
short the stock and buy the combo, hang on until expiration, and that 0.20 is mine! Slow down, Sparky.
Put-call parity doesn’t work in C options at this point
because C is really, really hard to borrow. You think Citigroup is going out of business? Join the club. But you can’t
short the stock. Instead, you can buy puts, sell calls, or sell