Answer: the risk-free rate.

## 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.

Author: LeetQuiz .