Put-call parity with a dividend

Question

Stock at 100, zero interest rates, but the stock will pay a \3 cash dividend before the 1-year expiry. The 100-strike European call trades at \5 and the 100-strike put at \5. Is there an arbitrage? State the parity relation with dividends, find the violation, and give the exact trade and riskless profit.

Accepted Answer

Put-call parity with dividends (zero rates): C - P = S - PV(	ext{divs}) - K. A dividend lowers the forward price, because a holder of the stock collects the cash the option holder does not. Here S - PV(	ext{divs}) - K = 100 - 3 - 100 = -3, so the fair value of C - P is -3: the put should be \3 richer than the call. But the market has C - P = 5 - 5 = 0. The call is \3 too expensive relative to the put. The trade (a conversion): sell the rich call, buy the put, buy the stock. - Sell the call: +5 - Buy the put: -5 - Buy the stock: -100 Net outlay today: \100. Now you hold the stock through the…

Put-call parity with a dividend

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