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IAI Actuarial Core Principles · Economic Modelling · Principles of option pricing

In a one-period binomial model for a non-dividend-paying share, the risk-neutral probability of an up-move is defined as q = (e^(rΔt) − d)/(u − d). Which statement about q is correct?

The risk-neutral probability is the value of q that makes the share's expected future price, discounted at the risk-free rate, equal today's price. It is a pricing device, not the real-world probability, and it does not depend on investors' risk preferences or expected returns.

  1. AIt is the real-world probability that the share price rises, adjusted for investors' risk aversion
  2. BIt is the probability that makes the discounted expected share price equal to the current share priceCorrect
  3. CIt must always equal 0.5 when u and d are symmetric about 1
  4. DIt depends on the expected return that investors require on the share
  5. It is the probability that the option finishes in the money

Explanation

q is chosen so that q·u·S + (1−q)·d·S = S·e^(rΔt), so the expected share price discounted at the risk-free rate equals today's price. It is not the real-world probability and does not depend on investors' required return. It also does not equal the in-the-money probability.

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