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

As the number of steps n in a CRR binomial tree (with u = exp(sigma*sqrt(T/n)), d = 1/u, and risk-neutral probabilities) tends to infinity, what happens to the price of a European call option on a non-dividend-paying share?

The binomial price converges to the Black-Scholes call price. Under risk-neutral probabilities and CRR factors, the log of the terminal share price tends to a normal distribution with variance sigma squared times T, which is exactly the Black-Scholes lognormal model.

  1. AIt converges to the Black-Scholes call priceCorrect
  2. BIt converges to the intrinsic value max(S0 - K, 0)
  3. CIt converges to the share price S0
  4. DIt diverges because the tree becomes non-recombining
  5. It converges to the discounted expected payoff under real-world probabilities

Explanation

With the CRR calibration, log share price at maturity under the risk-neutral measure tends to a normal distribution with mean (r - sigma^2/2)T and variance sigma^2 T, which is the Black-Scholes assumption. The tree price therefore converges to the Black-Scholes formula. Real-world probabilities are not used.

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