Put-call parity

Put-call parity describes the relationship that must exist between the prices of puts and calls to eliminate the possibility of arbitrage.  If put-call parity doesn’t hold, then arbitrage will occur until parity is established.

Put-call 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 6-month T-bill rates as the discount rates.  The 3-month rate is close to the time of the first dividend while the 6-month 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 ex-dividend 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 risk-free rate	0.08%
3 month risk-free 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.