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

In a one-period binomial model for a non-dividend-paying share, the risk-neutral probability of an up-move is q. Which statement about q is correct?

The risk-neutral probability is the up-move probability under which the share's expected discounted future price equals its current price. It is a pricing device derived from no-arbitrage, not a real-world probability, and it does not depend on the option's strike.

  1. AIt is the real-world probability that the share price rises, adjusted for investors' risk aversion
  2. BIt is the probability under which the discounted share price at the risk-free rate has expected value equal to the current priceCorrect
  3. CIt equals the probability that the option finishes in the money
  4. DIt must always exceed 0.5 if the share has a positive expected return
  5. It depends on the option's strike price

Explanation

The risk-neutral probability q is chosen so that S0 = e^{-r}[q Su + (1-q) Sd], i.e. the discounted share price is a martingale. It is not the real-world probability, does not depend on the strike, and need not exceed 0.5.

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