Answer: risk free rate.

## 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

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