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

A non-dividend-paying stock trades at 100. A European call has strike 90 and one year to expiry, and the continuously compounded risk-free rate is 5%. As the volatility used in the Black-Scholes-Merton formula approaches zero, which value does the call price approach?

The call price approaches 14.39. With zero volatility the stock's future price is certain, so the call is worth the stock price minus the discounted strike, or 100 minus 90 times e^(-0.05), which is 85.61. Plain intrinsic value of 10 ignores the interest on the strike.

  1. A14.39Correct
  2. B10.00
  3. C5.39
  4. D0.00

Explanation

As volatility goes to zero the stock grows deterministically at the risk-free rate, so the call is worth max(S0 - K e^(-rT), 0). That is 100 - 90 e^(-0.05) = 100 - 85.61 = 14.39. The value 10.00 is undiscounted intrinsic value. The value 5.39 results from compounding the strike forward instead of discounting it. The value 0.00 would apply only if the call were out of the money on a forward basis.

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