Answer: decreases.

## Explanation In the binomial option pricing model, the value of a put option is affected by the probability of upward movements in the underlying asset's price. ### Key Concepts: 1. **Put Option Basics**: A put option gives the holder the right to sell the underlying asset at a specified strike price. Put options increase in value when the underlying asset price decreases. 2. **Binomial Model Framework**: The binomial model values options by creating a risk-neutral probability framework where: - p = risk-neutral probability of an upward movement - (1-p) = risk-neutral probability of a downward movement 3. **Relationship between Upward Probability and Put Value**: - When the probability of an upward movement (p) increases, the expected future price of the underlying asset increases - Higher expected future prices make it less likely that the put option will be in-the-money (i.e., the underlying price will be below the strike price) - Therefore, the value of the put option decreases ### Mathematical Explanation: In the binomial model, the value of a put option is calculated as: \[ Put = e^{-rT} [p \times P_u + (1-p) \times P_d] \] Where: - r = risk-free rate - T = time to expiration - p = risk-neutral probability of upward movement - P_u = put value if price moves up - P_d = put value if price moves down Since P_u (put value when price goes up) is typically lower than P_d (put value when price goes down), increasing p gives more weight to the lower value P_u, thus decreasing the overall put value. ### Intuitive Explanation: If the probability of the underlying asset price going up increases, the put option becomes less valuable because: - The option holder benefits when prices fall - Higher upward probability means lower probability of price declines - Lower probability of favorable price movements reduces the option's expected payoff Therefore, the correct answer is **A. decreases.**

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