FX Put-Call Parity
A relationship between the price of a put option and the price of a call option with the following features:
1. Both options are European options on the same underlying foreign currency whose current (spot) exchange rate is (units of domestic currency per unit of foreign currency, American terms).
2. Both options have the same exercise price .
3. Both options have the same exercise date, periods from today.
One strategy is to purchase the call now for C and sell (or write) the put for P. What happens at the option maturity?
Exchange Rate at Exercise Date ≤ or >
Payoff on Long Call 0 -
Payoff on Short Put -( - ) 0
Total Payoffs - -
Thus the option strategy yields a long position in the exchange rate with a debt equal to the exercise price of the options.
We want the cost of such a position today in the domestic currency. The cost today of the debt position is , where is the risk free rate in the domestic country. In effect, to get the debt position you short a default free zero coupon bond with face value of in the domestic currency that matures in periods.
To get the long position in the exchange rate, you buy now one default free zero coupon bond maturing in periods with face value of 1 unit of the foreign currency, the foreign zero.
The domestic currency value of that one unit of foreign currency at maturity is . The foreign currency value of that foreign zero today is , where is the risk free rate in the foreign country. The domestic currency value of the foreign zero now is .
To avoid arbitrage, the cost of the option strategy C – P must equal the cost of the levered long position in the exchange rate, i. e.,
C – P = - . (*)
The equation (*) is put-call parity for options on foreign exchange.
Consider the opposite strategy of buying a put option and selling (or writing) the call option. What happens at the option maturity?
Exchange Rate at Exercise Date ≤ or >
Payoff on Long Put - 0
Payoff on Short Call 0 -( - )
Total Payoffs - -
Thus the option strategy yields a long position in the domestic riskless asset that is levered by a short position in the exchange rate.
To establish this position, you buy a default free zero coupon bond with face value in the domestic currency and maturing in periods and you short one default free zero maturing in periods and paying 1 unit of foreign currency at maturity (the foreign zero).
As argued above, the current cost of this position in the domestic currency is - . To avoid arbitrage, the cost of the option strategy P - C must equal the cost of this long position in the domestic riskless asset that is levered by a short position in the exchange rate, i. e.,
P – C = - . (**)
This expression (**), which is the negative of the (*) above, is also called put-call parity for options on foreign exchange. |