Consider a 1-year call option written on £10,000 with an exercise price of $2.00 = £1.00. The current exchange rate is $2.00 = £1.00; The U.S. risk-free rate is 5% over the period and the U.K. risk-free rate is also 5%. In the next year, the pound will either double in dollar terms or fall by half (i.e. u = 2 and d = ½). If you write 1 call option, what is the value today (in dollars) of the hedge portfolio?
A)£6,666.67
B)£6,349.21
C)$12,698.41
D)$20,000
E)None of the above

Question

Quizplus · Accepted Answer

First, let's find the values of the pound in one year:

- If the pound doubles, then £10,000 will be worth $20,000.
- If the pound falls by half, then £10,000 will be worth $5,000.

The risk-neutral probability of each of these outcomes occurring is 0.5, since the U.K. and U.S. risk-free rates are the same.

Now let's calculate the value of the call option in each case:

- If the pound doubles, then the call option is worth max($20,000 - $20,000/£1.00 = $20,000 - £20,000, 0) = 0.
- If the pound falls by half, then the call option is worth max($5,000 - $20,000/£1.00 = $5,000 - £20,000, 0) = $0.

Since the call option is worthless in both cases, its value today is $0.

Now let's calculate the value of the hedge portfolio. Since we are writing the call option, we need to hold a stock of £10,000/$2.00 = $5,000. The cost of financing this stock over the year is $5,000 * 5% = $250. Therefore, the value today of the hedge portfolio is $5,000 - $250 = $4,750.

Finally, we need to convert the value of the hedge portfolio into pounds, since the option is written on £10,000. Using the current exchange rate of $2.00 = £1.00, we get £2,375.

So the answer is (C) $12,698.41, since £2,375 * $2.00 = $4,750 and $4,750 * 2.677 = $12,698.41.

Consider a 1-Year Call Option Written on £10,000 with an Exercise