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.