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The problem concerns a trader who sells a one-year European put option with a strike price of $85 on an underlying asset initially priced at $75. Using the Black-Scholes delta-hedging model, the trader hedges daily, and after one year, the stock returns to its initial price of $75.

### Key Information:
- **Initial stock price,** $ S_0 = 75 $
- **Strike price,** $ K = 85 $
- **Time to maturity,** $ T = 1 $ year
- **Final stock price at maturity,** $ S_T = 75 $

Given this setup, let’s analyze the provided options:

1. **Options involving buying or selling the asset on day 1:**  
   On day 1, the trader will **buy** shares to hedge the sold put option because the option is initially out-of-the-money, and the delta is negative.

2. **Positions in the stock over time:**  
   The trader’s stock position will vary, typically increasing as the stock price moves toward the strike price. If the stock price remains around $75, the position should theoretically be shorter at maturity since a put option becomes less valuable as expiration nears.

3. **Theta and Gamma for the option:**  
   - **Theta** (time decay) is positive for the trader as the option seller since options lose value as they approach expiration.
   - **Gamma** (rate of change of delta) is positive for a put option near-the-money, but the trader is **short** this option, giving them negative gamma.

4. **Impact of volatility on profit/loss:**  
   - **Increased volatility** increases the value of the put option. Since the trader is short the option, they would lose money if volatility rises.
   - **Delta hedging** does not fully protect against volatility changes (vega risk); it only neutralizes small movements in the stock price.

### True Statements:
- **“By time T the trader will have a shorter position in the stock than on day 1.”**  
- **“The trader has negative theta and positive gamma on the option.”**
- **“If the volatility suddenly increases during the life of the option, then the trader loses money.”**

Would you like a detailed explanation for any of these answers?

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#### Additional Practice Questions:
1. How would the trader's hedge change if the option were a call instead of a put?
2. What impact would a decline in volatility have on the trader’s position?
3. How would the gamma of the option change if the stock price rose above the strike price?
4. Why does positive theta benefit the seller of the option?
5. How does delta hedging reduce risk in an option position?

#### Tip:
Remember, delta hedging only protects against small, linear stock price changes, not against volatility or large price moves (gamma risk).

On day 1, a trader sells a 1y put option with strike $85 on an underlying asset priced at $75. Over the year, the trader hedges the option daily using Black-Scholes delta hedging. After one year, the stock returns to $75. Select all true statements regarding the trader's position and the effects of volatility, theta, and gamma.

Explore the principles of delta hedging in a 1-year put option with a strike price of $85. Using the Black-Scholes model, this problem addresses the trader's adjustments to position, theta, gamma, and the impact of volatility. Perfect for finance students and professionals.

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