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FRM Part I · FRM Exam Part I · Binomial Trees

A risk manager prices a European call on a non-dividend-paying stock with a CRR binomial tree and steadily increases the number of steps. As the number of steps grows large, the tree's delta at the initial node for the call converges to which quantity from the Black-Scholes-Merton model?

The delta converges to N(d1), the Black-Scholes-Merton call delta. As the number of steps increases, the CRR tree approaches lognormal price dynamics, so both the option price and its hedge ratio match the BSM values. N(d2) is an exercise probability, not delta.

  1. AN(d1)Correct
  2. BN(d2)
  3. CN(d1) - 1
  4. De^(-rT)·N(d2)

Explanation

With u = e^(σ√Δt) and d = 1/u, the tree's lognormal limit is the BSM model. The European call price converges to the BSM price, and the initial delta converges to the BSM call delta, N(d1). N(d2) is the risk-neutral exercise probability, and N(d1) - 1 is the put delta.

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