Explanation

In a one-period binomial model, the risk-neutral probabilities are determined by the risk-free rate, not by investors' risk aversion or the actual probabilities of price movements.

Key Concepts:

1. Risk-Neutral Valuation: In derivatives pricing, we use risk-neutral probabilities to value options. These probabilities are not the actual probabilities of price movements but rather probabilities that make the expected return on the underlying asset equal to the risk-free rate.

2. Binomial Model Formula: The risk-neutral probability of an upward movement (p) is calculated as:

   $p = \frac{e^{r\Delta t} - d}{u - d}$

   Where:

   • r = risk-free rate
   • Δt = time period
   • u = upward movement factor
   • d = downward movement factor

3. Why Not Other Options:

   • Option B (investors' risk aversion): Risk-neutral probabilities specifically eliminate risk preferences from pricing, so they don't depend on investors' risk aversion.
   • Option C (probabilities of underlying price moving up and down): These are the actual probabilities, which are different from risk-neutral probabilities.

4. Application: Risk-neutral probabilities allow us to discount expected payoffs at the risk-free rate, simplifying derivatives pricing by eliminating the need to estimate risk premiums.

Therefore, the correct answer is A - the risk-free rate determines the risk-neutral probabilities in a one-period binomial model.