To solve this problem, we need to apply the fundamental principles of Put-Call Parity and the relationship between Barrier Options and vanilla European options.

1. Identify the Relationship Between Barrier and Vanilla Options

A key concept in exotic options is that a vanilla option can be decomposed into its "in" and "out" barrier components. Specifically, for a call option with the same strike and maturity: $\text{Vanilla Call} = \text{Up-and-in Call} + \text{Up-and-out Call}$

Using the values provided in the table:

• Up-and-in Call: USD 5.21
• Up-and-out Call: USD 1.40

$\text{Vanilla Call} = 5.21 + 1.40 = \text{USD 6.61}$

2. Apply Put-Call Parity

Once we have the price of the vanilla call, we use the Put-Call Parity formula to find the price of the vanilla European put. The formula is: $C - P = S_0 - Ke^{-rT}$ Where:

• $C = \text{Price of the European call}$
• $P = \text{Price of the European put}$
• $S_0 = \text{Current stock price}$
• $K = \text{Strike price (USD 100)}$
• $r = \text{Risk-free rate (5\%)}$
• $T = \text{Time to maturity (1 year)}$

Note on Forward Price: The relationship can also be expressed using the forward price ($F$): $C - P = (F - K)e^{-rT}$ The problem states the 1-year forward price is USD 100 and the strike price is USD 100. This simplifies the equation significantly: $C - P = (100 - 100)e^{-0.05 \times 1} = 0$ Therefore: $C = P$

3. Final Calculation

Since we determined the vanilla call price ($C$) is USD 6.61 and the Put-Call Parity under these specific conditions (where Forward = Strike) dictates that $C = P$:

$\text{Price of European Put} = \text{USD 6.61}$

Key Takeaway: The "Down-and-in barrier put" and the "Forward contract price" provided in the prompt are distractors designed to test your ability to identify the necessary components of the in-out parity and the put-call parity.

Correct Answer: D: USD 6.61