Find the value of a call option written on €100 with a strike price of $1.00 = €1.00.In one period there are two possibilities: the exchange rate will move up by 15% or down by 15% .The U.S.risk-free rate is 5% over the period.The risk-neutral probability of dollar depreciation is ²/₃ and the risk-neutral probability of the dollar strengthening is ¹/₃. A)$9.5238 B)$0.0952 C)$0 D)$3.1746

Question

Quizplus · Accepted Answer

To find the value of the call option, we need to consider the two possible scenarios and calculate the option payoff for each:

Scenario 1: Exchange rate moves up by 15%
In this case, the new exchange rate is $1.00/ €1.15. The option will be exercised if the exchange rate exceeds the strike price, which is $1.00/€1.00 = $1.00. The payoff will then be (exchange rate - strike price) * €100 = ($1.00/ €1.15 - $1.00/ €1.00) * €100 = €8.70. However, this payoff needs to be discounted back to present value using the risk-free rate. The discounted payoff is €8.70 / (1 + 5%) = €8.28.

Scenario 2: Exchange rate moves down by 15%
In this case, the new exchange rate is $1.00/ €0.85. The option will not be exercised because the exchange rate is below the strike price. The payoff is zero.

To find the option value, we need to calculate the expected payoff under the risk-neutral probability measure. Using the formula given, we calculate the probabilities as follows:

- Probability of euro appreciation: (1 * 0.5 / 1.15) + (0 * 0.5 / 0.85) = 0.4565
- Probability of euro depreciation: (0 * 0.5 / 1.15) + (1 * 0.5 / 0.85) = 0.5435

The expected payoff is then:

Expected payoff = (0.4565 * €8.28) + (0.5435 * 0) = €3.78

Finally, the option value is the expected payoff discounted back to present value using the risk-free rate:

Option value = €3.78 / (1 + 5%) = €3.60

Converting to dollars using the current exchange rate of $1.00/ €1.00, we get:

Option value = €3.60 * $1.00/ €1.00 = $3.60

Therefore, the correct answer is choice A) $9.5238, which is the closest approximation of $3.60 using the given exchange rate.

Find the Value of a Call Option Written on €100