Put-call parity with interest rates

Question

Stock at 100, no dividends, but rates are positive: the 1-year discount factor is 0.95, so a certain \100 paid in one year is worth \95 today. The 1-year 100-strike European call trades at \9 and the 100-strike put at \3. Is there an arbitrage? State the parity relation with rates, find the violation, and give the exact trade and riskless profit.

Accepted Answer

Put-call parity with rates: C - P = S - K e^{-rT}, where Ke^{-rT} is the present value of the strike. Here Ke^{-rT} = 100 	imes 0.95 = 95, so the fair value of C - P is S - 95 = 100 - 95 = 5. But the market has C - P = 9 - 3 = 6. The synthetic forward is \1 rich, so the call is expensive relative to the put. The trade (a conversion): sell the rich synthetic, buy the replica. - Sell the call: +9 - Buy the put: -3 - Buy the stock: -100 Net outlay today: 9 - 3 - 100 = -\94, so you pay \94 now. As in the zero-rate case, this locks in exactly \100 at expiry (called away or exercised, you sell you…

Put-call parity with interest rates

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