Putcall parity describes the relationship that must exist
between the prices of puts and calls to eliminate the possibility of
arbitrage. If putcall parity doesn’t
hold, then arbitrage will occur until parity is established.
Putcall parity is described by the equation:
P = C + PV(X) – S0 + PV(dividends)
Where:
P = put price
C = call price
PV(X) = present value of the strike price
S0 = current stock price
PV(dividends) = present value of any dividends to be paid on
the stock during the life of the option.
For the purposes of this demonstration, I’ll use at the
money options on SPY that expire on June 27, 2013. It isn’t difficult to build a spreadsheet
model to facilitate comparing different maturities and different strike prices.
The first difficulty that I see is figuring what the
dividends will be. In December, SPY paid
over $1.00, but that was significantly higher than the historical amount,
between $0.60 and $0.70. The reason for
this is, apparently, that some companies moved their dividend payments forward
due to the fiscal cliff. I suspect that
we’ll see a lower than normal dividend from SPY in March, but it will likely
end up about the same after that. So,
I’ll use $0.60 (the low end of the range) for the dividend in March.
One other oddity concerning the dividends is that the ex
dates are long before the payment date.
Consequently, the July 31 dividend, which will be paid after the
expiration of the option, actually belongs to whoever holds the stock before
expiration of the option. So, if I owned
the stock and bought a put to give me downside protection, I would gain the
value of that last dividend, and therefore, the July 31 dividend should be
included in the calculation.
One other thing to note is that we need to compare the bid
price of each to the ask price of the other, because most of us are price
takers, meaning we have to buy at the ask and sell at the bid.
Finally, I’ll use the 3 and 6month Tbill rates as the
discount rates. The 3month rate is
close to the time of the first dividend while the 6month rate is close to the
expiration date. I suppose an argument
could be made that the dividends should be discounted at a higher rate since
they are actually not known, but it won’t make much difference. We won’t count the dividend at the end of
January since the exdividend date has already passed.
So, currently we have this for our options:
Bid

Ask


Put

6.37

6.39

Call

5.3

5.34

6 month riskfree rate

0.08%

3 month riskfree rate

0.05%

S0

148

X

148

Div1

0.65

Div2

0.65

Days to div1

102

Days to div2

194

Days to expiration

160

I won’t show the math here; it’s just a matter of plugging
the numbers into the equation. The
result is that the put bid price should be 6.55, while the put ask price should
be 6.59, a bit higher than the actual prices.
But, it’s only about $0.20, which isn’t enough to offer an arbitrage
opportunity when we factor in transaction costs.
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