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IAI Actuarial Core Principles · Economic Modelling · Binomial option-pricing model

In a CRR tree with u = exp(sigma*sqrt(h)), d = 1/u and risk-free growth e^(rh) per step, the risk-neutral probability of an up move is q = (e^(rh) - d)/(u - d). For small h, which approximation of q is correct to first order in sqrt(h)?

q is approximately 1/2 plus (r minus sigma squared over 2) times the square root of h divided by 2 sigma. Expanding the exponentials gives a numerator of x + (r - sigma^2/2)h with x = sigma root h, and the denominator u - d is about 2x.

  1. Aq is about 1/2 + (r - sigma^2/2)*sqrt(h)/(2*sigma)Correct
  2. Bq is about 1/2 + r*sqrt(h)/sigma
  3. Cq is about 1/2 + (r + sigma^2/2)*sqrt(h)/(2*sigma)
  4. Dq is about 1/2 - (r - sigma^2/2)*sqrt(h)/(2*sigma)
  5. q is about 1/2 + (r - sigma^2/2)*h/(2*sigma)

Explanation

Expand with x = sigma*sqrt(h): e^(rh) is about 1 + rh, d about 1 - x + x^2/2, u - d about 2x. Numerator is rh + x - x^2/2 = x + (r - sigma^2/2)h. Dividing by 2x gives q = 1/2 + (r - sigma^2/2)h/(2 sigma sqrt(h)) = 1/2 + (r - sigma^2/2)sqrt(h)/(2 sigma). The sign of the sigma^2/2 term comes from the d expansion.

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