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## Put-Call Option Interest Rate Parity

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**Objective**Determine the international parity relationship between Call, Put, and Forward prices**Outline**• Two arbitrage portfolios • Derivation of parity conditions • Exemplification**Two arbitrage portfolios**Consider: C: the premium of a call option on the Sfr P: the premium of a put option on the Sfr X: the strike price of call and put options niSfr : Sfr nominal interest rate ni$ : $ nominal interest rate s = $/Sfr : spot exchange rate f = forward rate**Parity conditions derived**It follows that C = s0/(1+niSfr) - X/(1+ni$) +P According to interest rate parity we know that s0/(1+niSfr) = f1/(1+ni$) Hence, if we are in Canada, C = (f1 - X)/(1+ni$) + P In general, C = (f1 - X)/(1+nih) + P**Exemplification**e0 = C$0.7143/Sfr ni$ = 3.5% niSfr = 4.4% One-year forward = C$0.70814/Sfr A call on the Sfr struck at C$0.701/Sfr, expiring in one year sells at C$0.035/Sfr A put on the Sfr struck at C$0.701/Sfr, expiring in one year sells at C$0.023/Sfr**Note**C$0.70814/C$0.7143 = (1.035)/(1.044) IRP holds C$0.035 > C$(0.70814-0.701)/(1.035) + C$0.023 Arbitrage opportunity**Analysis**At expiration, the combined payoff from the two portfolios is always zero. However, buying the first portfolio and shorting the second one has produced an arbitrage profit of (C$2,940- C$2,511.25)=C$428.75 up-front.