Answer: CNY 0.1171

## Explanation This is a currency option valuation problem using a two-step binomial tree. The option is a European-style call option to buy USD (foreign currency) using CNY (domestic currency). ### Given Parameters: - Spot exchange rate (S) = CNY 6.7355 per USD 1 - Strike price (K) = CNY 6.8665 per USD 1 - Domestic risk-free rate (r) = 1.75% (China) - Foreign risk-free rate (r_f) = 3.25% (US) - Time step (Δt) = 0.5 years (since it's a two-step tree for 12 months) - Up factor (u) = 1.0582 - Down factor (d) = 0.9450 ### Step 1: Calculate Risk-Neutral Probability The risk-neutral probability of an up move is calculated as: \[ p = \frac{e^{(r - r_f)\Delta t} - d}{u - d} = \frac{e^{(0.0175 - 0.0325) \times 0.5} - 0.9450}{1.0582 - 0.9450} \] \[ p = \frac{e^{-0.0075} - 0.9450}{0.1132} = \frac{0.9925 - 0.9450}{0.1132} = \frac{0.0475}{0.1132} = 0.4199 \] Thus, the probability of a down move is: \[ 1 - p = 1 - 0.4199 = 0.5801 \] ### Step 2: Build the Exchange Rate Tree **Time 0:** S = 6.7355 **Time 0.5 years:** - Up: S × u = 6.7355 × 1.0582 = 7.127 - Down: S × d = 6.7355 × 0.9450 = 6.365 **Time 1 year:** - Up-Up: 7.127 × 1.0582 = 7.542 - Up-Down: 7.127 × 0.9450 = 6.735 - Down-Down: 6.365 × 0.9450 = 6.015 ### Step 3: Calculate Option Payoffs at Expiration Call option payoff = max(S - K, 0) - Up-Up: max(7.542 - 6.8665, 0) = 0.6755 - Up-Down: max(6.735 - 6.8665, 0) = 0 - Down-Down: max(6.015 - 6.8665, 0) = 0 ### Step 4: Backward Induction **At Time 0.5 years:** - Up node: Expected value = [p × 0.6755 + (1-p) × 0] × e^(-r×Δt) = [0.4199 × 0.6755 + 0] × e^(-0.0175×0.5) = 0.2836 × 0.9913 = 0.2812 - Down node: Expected value = [p × 0 + (1-p) × 0] × e^(-r×Δt) = 0 **At Time 0:** Option value = [p × 0.2812 + (1-p) × 0] × e^(-r×Δt) = [0.4199 × 0.2812 + 0] × 0.9913 = 0.1181 × 0.9913 = 0.1171 Therefore, the value of the currency call option is **CNY 0.1171**.

Author: LeetQuiz .