Stock Options 15-13
If the portfolio had any other value, there would be an arbitrage opportunity
A fun fact that emerges from put-call parity is this: Suppose S = K (and T > 0),
then C > P. That is, when options are alive, and if the options are exactly at the
money, calls are worth more than puts.
Important Note: Put-call parity is a relationship between the price of a call and
the price of a put. That is, one cannot solve for the call option price using put-call
parity—unless one has the put price. Where does one get the put price? From
put-call parity—assuming one has the call price. This circular reasoning shows
that an option pricing model must be employed to provide call option prices, in
the absence of a put option price.
A. Put-Call Parity with Dividends
The put-call parity argument stated above assumes that the underlying stock
paid no dividends before option expiration. Let’s say the stock does pay a
dividend before option expiration. First, we will rewrite the put-call parity
S = C – P + K / (1 + rt)T
If the stock pays a dividend, the holder of the stock will receive a dividend at
some time before option expiration. To get the same payoff, the holder of the
portfolio needs an extra amount today. Because the dividend occurs at a later
date, this extra amount is the present value of the dividend.
If the stock does pay a dividend before option expiration, then we adjust the put-
call parity equation to:
C – P = S – Div – K / (1 + rt)T
where Div is the present value of dividends to be paid during the life of the
B. What Can We Do with Put-Call Parity?
Put-call parity allows us to calculate the price of a call option before it expires. To
calculate the call option price using put-call parity, however, you have to know the
price of a call option with the same strike price. If you use an option-pricing
model to calculate a call option price, you can use put-call parity to calculate a
Lecture Tip: One key use of parity is replication (or creating synthetic securities).
For example, when Facebook went public, many investors wanted to short the
stock. However, the relative lack of share availability meant high borrowing costs.
So, investors could use the option market to create a synthetic short position:
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