Put–call parity with a dividend

European options share strike K=100K=100 and expiry T=1T=1 year on a stock with spot S=100S=100. The continuously-compounded rate is r=5%r=5\%. Unlike the textbook case, the stock pays a known cash dividend of $3 in exactly six months. Compute CPC-P (call minus put price). (to 4 decimals)

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  1. Dividend-adjusted parity: CP=(SPV(div))KerTC-P = (S - \mathrm{PV(div)}) - Ke^{-rT}.
  2. PV(div)=3e0.050.5\mathrm{PV(div)} = 3e^{-0.05\cdot 0.5} and KerT=100e0.05Ke^{-rT}=100e^{-0.05}.

Answer

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1.9511 (± 0.01)

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Asked at: Jane Street, Citadel

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