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

An American call option is written on a share that pays no dividends before expiry, and the risk-free interest rate is positive. In a binomial model, which statement about this option is correct?

With no dividends and a positive interest rate, an American call should never be exercised early. Its continuation value is at least S minus the discounted strike, which exceeds the immediate payoff S minus K. The early exercise right is therefore worthless, and the American call has the same value as the European call.

  1. AIt is never optimal to exercise early, so its value equals that of the otherwise identical European callCorrect
  2. BIt should always be exercised at the first node where it is in the money
  3. CIt is worth strictly more than the otherwise identical European call because of the early exercise right
  4. DEarly exercise is optimal whenever the share price falls below the strike price
  5. Early exercise is optimal whenever the risk-neutral probability of an up-move exceeds one half

Explanation

Exercising early gives only the intrinsic value S-K. The option's continuation value is at least S - K/(1+r)^t, which is larger than S-K when r>0 and no dividend is paid, so waiting is better. The early exercise right therefore has no value, and the American call equals the European call. The other options wrongly give the right a positive value or rely on irrelevant conditions.

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