Explanation

For European put options, the value is inversely related to the risk-free rate. This relationship can be understood through the Black-Scholes model and option pricing principles:

Key Relationships for Put Options:

1. Risk-Free Rate (r): When interest rates rise, the present value of the exercise price (which is paid at expiration) decreases. Since put options give the right to sell at the exercise price, a lower present value of the exercise price makes the put less valuable. Therefore, put option values are inversely related to the risk-free rate.

2. Exercise Price (X): Put option values are positively related to the exercise price. A higher exercise price means the put holder can sell the underlying asset at a higher price, making the option more valuable.

3. Volatility (σ): Put option values are positively related to volatility. Higher volatility increases the probability that the underlying asset price will move below the exercise price, making the put option more valuable.

Mathematical Explanation: In the Black-Scholes model for European put options:

$P = Xe^{-rT}N(-d_2) - S_0N(-d_1)$

Where:

• $P$ = Put option price
• $X$ = Exercise price
• $r$ = Risk-free rate
• $T$ = Time to expiration
• $S_0$ = Current stock price
• $N(\cdot)$ = Cumulative standard normal distribution

From this formula, we can see that the term $Xe^{-rT}$ decreases as $r$ increases, reducing the put option value.

Summary of Relationships:

• Risk-free rate: Inverse relationship (correct answer)
• Exercise price: Direct relationship
• Volatility: Direct relationship
• Time to expiration: Direct relationship (for American puts, but ambiguous for European puts)
• Underlying asset price: Inverse relationship