Explanation

This is a currency option pricing problem using the Garman-Kohlhagen model (extension of Black-Scholes for currencies). The formula for a European call option on a currency is:

$C = S_0 e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2)$

Where:

• $S_0 = 1.34$ (current exchange rate)
• $K = 1.40$ (strike price)
• $T = 1$ (1 year)
• $r_d = 0.04$ (domestic risk-free rate - USD)
• $r_f = 0.03$ (foreign risk-free rate - EUR)
• $\sigma = 0.30$ (volatility)

Calculate $d_1$ and $d_2$:

$d_1 = \frac{\ln(S_0/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}} = \frac{\ln(1.34/1.40) + (0.04 - 0.03 + 0.30^2/2) \times 1}{0.30 \times \sqrt{1}}$ $d_1 = \frac{-0.0435 + (0.01 + 0.045) \times 1}{0.30} = \frac{-0.0435 + 0.055}{0.30} = \frac{0.0115}{0.30} = 0.0383$

$d_2 = d_1 - \sigma\sqrt{T} = 0.0383 - 0.30 = -0.2617$

Calculate normal CDF values:

• $N(d_1) = N(0.0383) \approx 0.5153$
• $N(d_2) = N(-0.2617) \approx 0.3968$

Compute the call price: $C = 1.34 \times e^{-0.03 \times 1} \times 0.5153 - 1.40 \times e^{-0.04 \times 1} \times 0.3968$ $C = 1.34 \times 0.9704 \times 0.5153 - 1.40 \times 0.9608 \times 0.3968$ $C = 0.670 - 0.533 = 0.137$

The price is approximately $0.136, which matches Option A.