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FRM Part I · FRM Exam Part I · Options Markets

A European call and a European put on the same non-dividend-paying stock have the same strike of $80 and expire in one year. The call trades at $9.00, the put at $5.00, and the stock at $82. The continuously compounded risk-free rate is 5% (e^-0.05 = 0.951229). Which strategy locks in an arbitrage profit, and what is the profit at inception in present-value terms?

The call plus the present value of the strike costs 85.10, while the put plus the stock is worth 87.00. Buy the call, lend the present value of the strike, sell the put and short the stock, earning about $1.90 immediately with no risk.

  1. ABuy the call, sell the put, short the stock, and lend $76.10; profit about $0.10 less than zero so no arbitrage
  2. BSell the call, buy the put, buy the stock, and borrow the PV of the strike; profit about $1.90
  3. CBuy the call, sell the put, short the stock, and lend the PV of the strike; profit about $1.90Correct
  4. DSell the call, buy the put, short the stock, and lend the PV of the strike; profit about $3.90

Explanation

Parity: c - p = 4.00, while S0 - K e^{-rT} = 82 - 76.098 = 5.902. The call is cheap relative to the put and stock: c + K e^{-rT} = 85.098 versus p + S0 = 87.00. Buy the call and PV of strike (lend), sell the put and short the stock, receiving 87.00 - 85.098 = $1.90 today. The other combinations trade in the wrong direction.

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