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FRM Part I · FRM Exam Part I · The Black-Scholes-Merton Model

A stock trades at 50 and pays no dividends. A European call with strike 45 and six months to expiry is priced at 7.20 under Black-Scholes-Merton. The continuously compounded risk-free rate is 4%. What is the lower bound on the price of the equivalent American call (the larger of the European lower bound and intrinsic value), and what does it imply?

The lower bound is S minus the discounted strike, 50 - 45e^(-0.02) = 5.89, which exceeds intrinsic value of 5.00. Since 7.20 is above 5.89, the price violates no bound and no arbitrage exists.

  1. A5.00, implying the 7.20 price is consistent with exercising immediately
  2. B5.89, which is below 7.20 so the price is consistent with no arbitrageCorrect
  3. C7.20 only, because an American call cannot have a lower bound
  4. D6.00, which is above 7.20 so there is an arbitrage

Explanation

The European lower bound is S - K e^(-rT) = 50 - 45 e^(-0.02) = 50 - 45(0.980199) = 50 - 44.109 = 5.89. Intrinsic value is 5.00. The larger is 5.89, below 7.20, so no arbitrage. Using only intrinsic value (5.00) ignores the interest on the strike.

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