The statement is false.

Here’s the reasoning:

1. Call Option:

   • A call option gives the holder the right to buy the underlying asset at the strike price ($55 in this case).
   • At expiration (time−1), the stock price is $50, which is lower than the strike price of $55.
   • Since the stock price is less than the strike price, the call option is out-of-the-money and would expire worthless.

2. Put Option:

   • A put option gives the holder the right to sell the underlying asset at the strike price ($45 in this case).
   • At expiration, the stock price is $50, which is higher than the strike price of $45.
   • Since the stock price is greater than the strike price, the put option is in-the-money (the holder could sell at $45, but would prefer to sell at the market price of $50, so it has intrinsic value).

Thus, the put option is in-the-money, making the statement that both options are out-of-the-money false.

Would you like more details or have any questions?

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Here are 5 related questions to expand on this:

1. What happens to the payoff of a European call option if it is in-the-money at expiration?
2. How is the payoff of a European put option calculated at expiration?
3. What role does time play in the value of European options?
4. How would the option payoffs change if the strike prices were different?
5. What factors influence whether an option is in-the-money or out-of-the-money?

Tip: An option is considered "in-the-money" if exercising it would result in a positive cash flow for the holder.