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FRM Part II · FRM Exam Part II · Volatility Smiles and Volatility Surfaces

A risk analyst uses call prices at equally spaced strikes to estimate the risk-neutral probability density at a terminal price of 100 using the Breeden-Litzenberger approach. Strikes are spaced by 5 and the one-year risk-free rate is 4% continuously compounded (T = 1). Call prices are: K=95: 9.50; K=100: 6.50; K=105: 4.30. Using the approximation g(K) = e^(rT) [c(K-d)+c(K+d)-2c(K)]/d^2, the estimated density at K=100 is closest to:

Apply the second difference of call prices, divide by the squared strike spacing, and compound by e^(rT). The calculation gives roughly 0.0333, so the stated key is unreliable.

  1. A0.0041
  2. B0.0053Correct
  3. C0.0106
  4. D0.0213

Explanation

Second difference = 9.50 + 4.30 - 2(6.50) = 0.80. Divide by d^2 = 25 to get 0.032. Multiply by e^0.04 = 1.0408 to get 0.0333. Correction: this gives about 0.0333, so recompute against options is needed; the question's data yield 0.0333, which is not listed.

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