 Consider a one-year, $45-strike European call option and a one-year, $45-
strike European put option on the same underlying asset. You observe that the time−0 stock price
equals $40 while the time−1 stock price equals $50. Then, both of the options are out-of-the-money
at expiration. 

Let's break down the information provided and analyze the scenario in the context of European options.

### Given Information:

1. **Call Option:**
   - **Strike Price (K)**: $45
   - **Time-0 Stock Price (S₀)**: $40
   - **Time-1 Stock Price (S₁)**: $50

2. **Put Option:**
   - **Strike Price (K)**: $45
   - **Time-0 Stock Price (S₀)**: $40
   - **Time-1 Stock Price (S₁)**: $50

3. **European Option:** 
   - A European option can only be exercised at expiration (time-1).
   
4. **Out-of-the-money at expiration:** 
   - A **call option** is out-of-the-money if the stock price at expiration is **less than or equal** to the strike price.
   - A **put option** is out-of-the-money if the stock price at expiration is **greater than or equal** to the strike price.

### Step-by-step Analysis:

#### 1. European Call Option:

- **Strike Price (K)** = $45
- **Stock Price at expiration (S₁)** = $50

A call option gives the holder the right (but not the obligation) to buy the underlying asset at the strike price. The payoff for a call option is:
\[
\text{Call Payoff} = \max(S_1 - K, 0)
\]
Here, $S_1 = 50$ and $K = 45$. Since $S_1 > K$, the call option is **in-the-money** (not out-of-the-money), and the payoff would be:
\[
\text{Call Payoff} = \max(50 - 45, 0) = 5
\]

Thus, at expiration, the **call option is in-the-money** and not out-of-the-money as mentioned in the question.

#### 2. European Put Option:

- **Strike Price (K)** = $45
- **Stock Price at expiration (S₁)** = $50

A put option gives the holder the right to sell the underlying asset at the strike price. The payoff for a put option is:
\[
\text{Put Payoff} = \max(K - S_1, 0)
\]
Here, $S_1 = 50$ and $K = 45$. Since $S_1 > K$, the put option is **out-of-the-money**, and the payoff would be:
\[
\text{Put Payoff} = \max(45 - 50, 0) = 0
\]

Thus, at expiration, the **put option is indeed out-of-the-money**.

### Conclusion:
- The **call option** should be **in-the-money** at expiration with a payoff of $5, not out-of-the-money as the question suggests.
- The **put option** is **out-of-the-money** at expiration with a payoff of $0, which is correct.

It seems there is a contradiction in the statement regarding the call option being out-of-the-money at expiration.

Would you like a more detailed explanation or clarification on any part of this analysis?

Here are some related questions you might want to explore:
1. How do the intrinsic and time values of options change over time?
2. How does the payoff structure differ between European and American options?
3. What happens to the value of out-of-the-money options as expiration approaches?
4. How would the pricing of these options change if dividends were involved?
5. What is put-call parity, and how does it apply to this scenario?

**Tip:** When analyzing option payoff scenarios, always compare the strike price with the stock price at expiration to determine if the option is in-the-money or out-of-the-money.

Consider a one-year, $45-strike European call option and a one-year, $45-strike European put option on the same underlying asset. You observe that the time−0 stock price equals $40 while the time−1 stock price equals $50. Then, both of the options are out-of-the-money at expiration.

Explore the pricing of a one-year, $45-strike European call and put option with a stock price moving from $40 to $50. Learn how to determine if these options are in-the-money or out-of-the-money at expiration, and understand the key payoffs for each option type.

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