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FRM Part I · FRM Exam Part I · Properties of Options

European options on a non-dividend-paying asset expire in 6 months. The continuously compounded risk-free rate is 4%. Prices are: call strike 50 = c1, call strike 60 = c2 with c1 - c2 = 5.50; put strike 60 minus put strike 50 = 4.60. Combining a bull call spread and a bear put spread creates a box spread paying 10 at expiry. Which statement is correct (e^-0.02 = 0.9802)?

Sell the box and gain about 0.30 in present value terms. The box has a certain payoff of 10, worth 9.802 today, yet it trades at 10.10. Selling it collects the excess. Ignoring discounting would give 0.10, and buying the overpriced box would lose money.

  1. ABuy the box for 10.10; arbitrage gain with present value about 0.30
  2. BSell the box, receiving 10.10 now; arbitrage gain with present value about 0.30Correct
  3. CSell the box, receiving 10.10 now; arbitrage gain of 0.10
  4. DNo arbitrage exists because the box costs more than 10

Explanation

The box pays 10 for certain, worth 10 x 0.9802 = 9.802 today. Its market cost is 5.50 + 4.60 = 10.10, which is too high. Sell the box, collect 10.10, and owe 10 at expiry (PV 9.802), gaining about 0.30 today. Answer 0.10 forgets to discount, and buying the box locks in a loss.

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