Explanation

Let's analyze each option:

A. Down-and-out call with barrier at $90 and strike at $110

• This is a call option that becomes worthless if the stock price falls to $90
• Current price: $100, Barrier: $90 (not crossed yet)
• Strike: $110 (out-of-the-money)
• If stock price increases above $110, the call becomes valuable
• BENEFITS from stock price increase

B. Down-and-in call with barrier at $90 and strike at $110

• This is a call option that only becomes active if the stock price falls to $90
• Current price: $100, Barrier: $90 (not crossed yet)
• Strike: $110 (out-of-the-money)
• If stock price increases, the barrier is less likely to be crossed, but the option still has value if it eventually becomes active
• BENEFITS from stock price increase (though indirectly)

C. Up-and-in put with barrier at $110 and strike at $100

• This is a put option that only becomes active if the stock price rises to $110
• Current price: $100, Barrier: $110 (not crossed yet)
• Strike: $100 (at-the-money)
• A put option benefits from stock price DECREASE, not increase
• If stock price increases, the put loses value
• DOES NOT BENEFIT from stock price increase

D. Up-and-in call with barrier at $110 and strike at $100

• This is a call option that only becomes active if the stock price rises to $110
• Current price: $100, Barrier: $110 (not crossed yet)
• Strike: $100 (at-the-money)
• If stock price increases above $110, the call becomes active and valuable
• BENEFITS from stock price increase

Therefore, option C is the only one that does NOT benefit from an increase in stock price because it's a put option, which gains value when stock prices fall.