Final answer:
To calculate the value of a one-year call option using the binomial approach, we asses the payoffs from exercising the option in up and down scenarios, estimate risk-neutral probabilities, and discount the expected payoff back to present value using the given interest rate. However, due to lack of specific details like up and down factors for risk-neutral probabilities, assumptions have to be made which simplifies the calculations. With assumed parameters, the present value of the option is USD 1,250.
Step-by-step explanation:
To calculate the value of a one-year call option using the binomial approach, we first determine the outcomes at the expiration of the option. In the up scenario, where the future spot rate is USD 2.10/EUR, the option will be exercised because the spot rate is higher than the strike price (USD 1.80/EUR). The payoff in this case is (USD 2.10 - USD 1.80) × EUR 10,000 = USD 3,000. In the down scenario, where the future spot rate is USD 1.10/EUR, the option will not be exercised since the spot rate is lower than the strike price, leading to a payoff of USD 0.
Next, we need to calculate the risk-neutral probabilities. Since the one-year forward rate equals the spot rate (USD 1.50/EUR), it indicates there is no arbitrage opportunity, and we can assume a risk-neutral world where the expected return on the underlying asset is the risk-free rate. However, the problem is missing the necessary up and down factors (u and d) or volatility information to calculate the true risk-neutral probabilities. Assuming that these probabilities are 50%, which is a common simplification when the exact parameters are unknown, the expected value of the option at expiry is then 0.5 × USD 3,000 (up scenario) + 0.5 × USD 0 (down scenario) = USD 1,500.
Considering the one-year US interest rate is 20%, we discount the expected value back to the present using the formula PV = FV / (1 + r). Thus, PV = USD 1,500 / (1 + 0.20) = USD 1,250. This gives us the present value of the call option in terms of USD.
Note: Typically, an exact risk-neutral probability would be derived using the Cox-Ross-Rubinstein model or a similar binomial model, which would require additional details such as the up and down factors and the volatility of the underlying asset. Also, the discounting should be done using the risk-free rate corresponding to the currency of the payoff; since we are calculating the option price in USD, we should use the USD interest rate.