FX Put-Call Parity and Arbitrage

An FX spot rate $S_t$ is quoted as units of domestic currency per one unit of foreign currency. The domestic continuously compounded risk-free rate is $r_d$ and the foreign continuously compounded risk-free rate is $r_f$. European call and put options on this FX rate have maturity $T$ and strike $K$. 1. Derive the put-call parity relationship between $C_0(K, T)$ and $P_0(K, T)$ for FX options. Be careful -- the foreign currency pays a continuous "dividend" at rate $r_f$, so the standard equity parity needs adjustment. 2. Suppose observed market prices violate your parity by $\varepsilon > 0$, meaning the call side is overpriced relative to the put side by $\varepsilon$. Construct an explicit static arbitrage portfolio (state every position and its notional) that locks in a riskless profit at time $0$. 3. Compute the locked-in time-$0$ profit in terms of $\varepsilon$.