Answer: volatility of the underlying.

## Explanation The correct answer is **C. volatility of the underlying**. ### Understanding Put Option Valuation For European put options, the value is influenced by several factors according to the Black-Scholes model: 1. **Volatility of the underlying (σ)** - Direct relationship 2. **Time to expiration (T)** - Direct relationship 3. **Strike price (K)** - Direct relationship 4. **Risk-free interest rate (r)** - Inverse relationship 5. **Value of the underlying (S)** - Inverse relationship ### Why Option C is Correct - **Volatility (σ)**: Higher volatility increases the probability that the underlying asset will move significantly, potentially making the put option more valuable. This is a direct relationship - as volatility increases, put option value increases. ### Why Other Options are Incorrect - **A. Risk-free interest rate**: This has an inverse relationship with put option value. Higher interest rates decrease the present value of the strike price, reducing put option value. - **B. Value of the underlying**: This has an inverse relationship with put option value. As the underlying asset price increases, put options become less valuable (since they give the right to sell at a fixed strike price). ### Key Concept The relationship between put option value and volatility is captured by the **vega** of an option, which measures sensitivity to volatility changes. For both puts and calls, vega is positive, meaning option values increase with higher volatility. **Note**: The question specifically asks about a "direct" relationship, meaning as the factor increases, the put option value also increases. Only volatility has this direct relationship among the options listed.

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