Explanation

For a European put option:

Option A: inversely related to the exercise price - This is INCORRECT. The value of a put option is directly related to the exercise price. A higher exercise price makes the put more valuable because it gives the holder the right to sell at a higher price.

Option B: inversely related to the risk-free interest rate - This is CORRECT. Higher interest rates decrease the present value of the exercise price (which is received when the put is exercised), thus reducing the value of the put option.

Option C: directly related to the value of the underlying - This is INCORRECT. The value of a put option is inversely related to the value of the underlying asset. As the underlying price increases, the put becomes less valuable because the right to sell at a fixed price becomes less attractive.

Key Relationships for Put Options:

1. Positive relationship with exercise price (higher strike = more valuable put)
2. Negative relationship with underlying price (higher underlying = less valuable put)
3. Negative relationship with interest rates (higher rates = less valuable put)
4. Positive relationship with volatility (higher volatility = more valuable put)
5. Negative relationship with time to expiration (longer time = more valuable put)

Therefore, only option B correctly describes the relationship for a European put option.