**2.9 Put-Call Parity: European
Options
**

T |

hroughout this chapter we have analyzed the no-arbitrage relationships among
stock, bond and option prices that must hold in the single period binomial model
(see topic 2.4, __Call Option Valuation: A Riskless Hedge
Approach__, topic 2.5, __Put Option Valuation:
A Riskless Hedge Approach__), and topic 2.6 __Option
Valuation: A Synthetic Option Approach__).

Besides
these relationships, an additional arbitrage relationship must hold between the
stock price and put and call option prices.
This relationship is called *put-call
parity*.

In
its simplest form, when the interest rate is zero and both the put P and the
call *C* have the same strike price *X,*
and *S *is the stock price, this
relationship is:

*P - C + S =
X.
*

That
is, a portfolio that consists of long one put with strike*
X,* short one call with strike* X,*
and long one stock is riskless. Furthermore
its value equals the strike price *X*.

Why
must this be the case? First,
observe that if both the put and the call have the same strike price, then only
one of them can finish in-the-money. If
the put is in-the-money, then* S *<*
X,* so the call is out-of-the-money. Similarly,
if the call is in-the-money, then *S>X*,
so the put is out-of-the-money. As
a result, *P - C* equals *X - S* if *S
< X*, and equals S-X if
S>X. In either case, we get

*P - C = X - S
*

The
only other possibility is that* S = X,*
in which case both* P* and *C* are zero, and the relationship then holds trivially.

How
would you take advantage of this relationship to make an arbitrage profit if it
did not hold? The answer depends
upon whether the appropriate bids and asks are available.

If
this relationship does not hold, it implies that either the portfolio trades at
more than *X* or less than *X.*

Suppose
*P - C + S > X* and consider the following trading strategy.
Sell a stock, sell a put, and buy a call.
If you undertake these trades, you receive the amount *P
- C + S *in cash. If the stock
price is greater than* X,* the put is
worthless, and the call can be exercised to obtain a stock at price X. This stock covers your short position, so at the end of the
period, you have to pay X.

If
the stock price is less than* X,* the
call is worthless, but the put will be exercised
against you, so you will
again have to buy a stock at price X. This
stock will cover your short position.

Thus,
independent of whether the stock price is less than X or more than*
X,* you will end up paying *X* at
the end
of the period. But you
received *P - C + S > X* earlier on, so you have locked in a
sure profit. Hence these prices
cannot persist.

Similarly,
if *P - C + S < X,* you can make a sure profit by buying a stock,
buying a put, and selling a call.
Together, these imply that *P - C +
S = X,* which is the parity
relationship.

If
1 plus the risk-free interest rate is r >
1, then the parity relationship for European options says the current
prices must equal the present value of the strike price discounted at the
risk-free interest rate (i.e., *P
- C + S = X/r*).

A
further implication of put-call parity is that if you have solved for the value
of a European call option, you can always determine the value of the European
put option (having the same strike price) by applying the put-call parity
relationship.

In
Chapter 3, __Binomial Model: Two-Period
Analysis__, we extend the one-period binomial option pricing model to two
periods.