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.