Answer: USD 5.11

**1. Determine the price of the European call option ($c$):** The problem provides prices for an Up-and-In Call (UIC) and an Up-and-Out Call (UOC) with the same barrier level (USD 95), strike (USD 100), and maturity. A standard European call option can be decomposed into a combination of barrier options. Specifically, holding an "in" option and an "out" option with the same barrier, strike, and maturity creates a synthetic standard option. This is because if the barrier is hit, the "in" option activates and the "out" option ceases; if the barrier is not hit, the "in" option is worthless and the "out" option pays off. Together, they cover all possible outcomes. $$ \text{Standard Call Price} = \text{Up-and-In Call Price} + \text{Up-and-Out Call Price} $$ $$ c = \text{USD } 5.21 + \text{USD } 1.40 = \text{USD } 6.61 $$ **2. Determine the relationship between the call, put, and forward prices:** According to Put-Call Parity, the relationship between a call ($c$) and a put ($p$) with the same strike ($K$) and maturity ($T$) is: $$ c - p = S_0 - K e^{-rT} $$ Where $S_0$ is the current spot price. The term $S_0 - K e^{-rT}$ represents the present value of the forward price difference, which is equivalent to the value of a forward contract with a delivery price $K$. The problem states that a forward contract with a delivery price (forward price in the contract) of USD 100 is available for USD 1.50. This "availability price" represents the market value of the forward contract ($V_{forward}$). $$ V_{forward} = S_0 - K e^{-rT} $$ Since the strike price for the options is USD 100 and the delivery price for the forward contract is USD 100, we can substitute the forward contract value directly into the put-call parity equation: $$ c - p = V_{forward} $$ $$ \text{USD } 6.61 - p = \text{USD } 1.50 $$ **3. Calculate the price of the European put option ($p$):** Rearranging the equation to solve for $p$: $$ p = c - 1.50 $$ $$ p = 6.61 - 1.50 = \text{USD } 5.11 $$ **Analysis of Options:** * **A: USD 2.00.** Incorrect. This would imply a call-put spread of roughly 4.61, which does not match the forward value of 1.50. * **B: USD 4.90.** Incorrect. * **C: USD 5.11.** Correct. This is derived by summing the barrier call prices to get the standard call price (6.61) and then subtracting the forward contract value (1.50) based on put-call parity. * **D: USD 6.61.** Incorrect. This is the price of the standard European call option, not the put. It would only be the put price if the forward contract had zero value (at-the-money forward), which contradicts the given information. ### Answer C

Author: LeetQuiz Editorial Team