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In the two graphs above, the  y- axis represents the value of the portfolio, not the profit or loss, because we're assuming that traders are giving options away. They are not, however, and the prices of European put and call options are ultimately governed by put-call parity. In a theoretical, perfectly efficient market, the prices for European put and call options would be governed by the equation:

## Option Put-Call Parity Relations When the Underlying

Observation. The put-call parity relation allows us to express the value of a European put in terms of the value of an European call (and other variables). Hence, when, in a later chapter, we meet the Black-Scholes formula for the value of a European call we shall be able to write down immediately a formula for the value of a European put.

### The Put-Call Parity Theorem - Norstad

For example, an investor is looking to sell a one-year call option on a \$75 stock at the \$75 strike price. If the one-year interest rate is 5%, the cost of borrowing \$7,555 for one year is: \$7,555 x 5% = \$875. Therefore, the call option on this non-dividend paying stock would have to be sold (at a minimum) for \$ just to cover the cost of carrying the position for one year.

#### Handout 20: Arbitrage Proofs for Put-Call Parity and

Put-call parity applies only to European options, which can only be exercised on the expiration date, and not American ones, which can be exercised before.

Synthetic Long Stock = Long Call and Short Put (same expiration & strike)
*Synthetic Long Call = Long Put and Long Stock
Synthetic Long Put = Long Call and Short Stock

The profit or loss on these positions for different TCKR stock prices is graphed below. Notice that if you add the profit or loss on the long call to that of the short put, you make or lose exactly what you would have if you had simply signed a forward contract for TCKR stock at \$65, expiring in one year. If shares are going for less than \$65, you lose money. If they are going for more, you gain. Again, this scenario ignores all transaction fees.

In our interest free, commission free, hypothetical world, the timing of the assignment does not matter, however the exercise would only occur after an assignment. Note too that if XYZ falls below the \$55 strike price, it does not impact the trade as a result of the \$ credit received when the positions were opened. If both options expire worthless, the net result is still a profit of \$.

Say that you purchase a European call option for TCKR stock. The expiration date is one year from now, the strike price is \$65, and purchasing the call costs you \$5. This contract gives you the right—but not the obligation—to purchase TCKR stock on the expiration date for \$65, whatever the market price might be. If one year from now, TCKR is trading at \$65, you will not exercise the option. If, on the other hand, TCKR is trading at \$75 per share, you will exercise the option, buy TCKR at \$65 and break even, since you paid \$5 for the option initially. Any amount TCKR goes above \$75 is pure profit, assuming zero transaction fees.

68 A portfolio comprising a call option and an amount of cash equal to the present value of the option's strike price has the same expiration value as a portfolio comprising the corresponding put option and the underlier. For European options, early exercise is not possible. If the expiration values of the two portfolios are the same, their present values must also be the same. This equivalence is put-call parity. If the two portfolios are going to have the same value at expiration, they must have the same value today, otherwise an investor could make an arbitrage profit by purchasing the less expensive portfolio, selling the more expensive one and holding the long-short position to expiration.

PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-free rate