## Put Call Parity

For European options there is a simple theoretical relationship between the prices of corresponding puts and calls. The relationship is found by noting that a combination of a put, a call, and a risk-free loan has a payoff identical to that of the underlying stock.

The combination can be easily imagined: buy one call, sell one put, and lend an amount cIK The combination of the first two has a payoff that is a straight line at 45°, passing through K on the horizontal axis. By lending d,Ky we obtain an additional payoff of K, which lifts the payoff line up so that it is now a 45° line originating at the origin. This final payoff is exactly that of the stock itself, so it must have the value S of the stock. In other words,

(See Exercise .3 for1 more detail.)

Put-call parity Let C and P be the prices of a European call and a European put, ~y both with a strike price of K and both defined on the same stock with price S Put-call parity states that

C — P+dK—S where d is the discount factor to the expiration date.

Example 12.2 (Parity almost) Consider the GM options of Figure 12 1, and focus on the two 35 March options (with 3 months to expiration) These have C = 4.25 and P — 1.00, respectively. The interest rate for this period is about 5.5%, so over .3 months we have d = 1/(1 + 055/4) = .986, Thus,

This is a close, but not exact, match with the actual stock price of $37 .88. There are several possible explanations for the mismatch One of the most important is that the stock quotes and option quotes do not come from the same sources The stock price is the closing price on the stock exchange, whereas the option prices are from the last traded options on the options exchanges; the last trades can occur at different times Dividends also can influence the parity relation, as discussed in Exercise 2.

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