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 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:

   $p = \frac{e^{rT} - d}{u - d}$

   where:

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

3. Why Not Other Options:

   • Option B (investors' risk aversion): Risk-neutral probabilities specifically remove risk preferences from the pricing model.
   • Option C (actual probabilities): The actual probabilities of price movements are not used in risk-neutral valuation; instead, we adjust probabilities to reflect risk neutrality.

Mathematical Derivation:

The risk-neutral probability is derived from the condition that the expected return under the risk-neutral measure equals the risk-free rate:

$S_0 = e^{-rT}[pS_u + (1-p)S_d]$

where $S_0$ is the current price, $S_u$ is the price after upward movement, and $S_d$ is the price after downward movement.

This approach allows derivatives to be priced without considering investors' risk preferences, making the pricing model more objective and consistent across different market participants.