Question

Consider a UK-based importer of bicycles, Italian bicycles, who has a €160,000 payable due in one year. He wants to use options to hedge the cost of his payable. One-year at-the-money put and call options on €10,000 exist. The spot exchange rates are €1.00 = $1.120 and £1.00 = $1.400, which makes the spot cross rate €1.00 = £0.80. In the next period, the euro can increase in pound value to £1.00/€1.00 or fall to £0.64/€1.00. The interest rate in dollars is i$ = 0%; the interest rate in euro is i€ = 5.00%; the interest rate in pounds is i£ = 7.10%. -How many at-the-money puts (on €10,000 strike price in £8,000) should he sell today?

HINT: You should be able to verify that your hedge "works" if the exchange rate at time 1 is £1.00/€ or £0.64/€ i.e. you should have the same cash flow in either state.

Answer 1

At the money put option means, strike price = spot price

Strike price of put option = 0.8 £/€

Size of 1 put contract = €10,000

total payment to be maid = €160,000

Therefore short =  €160,000/€10,000 = 16 put contracts

But also, long 16 call contracts with strike price = 0.8 £/€

If exchange rate = £1.00/€

Cost of €160,000 = €160,000 *  £1.00/€ =  £160,000

Pay-off from long call option = max(£1.00/€ - £0.8/€, 0)*€160,000 = £32,000

Pay-off from short put option = - max(£0.8/€ - £1.00/€, 0)*€160,000 = 0

Total cost = £160,000 - £32,000 = £128,000

If exchange rate = £0.64/€

Cost of €160,000 = €160,000 *  £64.00/€ =  £102,400

Pay-off from long call option = max(£0.64/€ - £0.8/€, 0)*€160,000 =0

Pay-off from short put option = - max(£0.8/€ - £0.64/€, 0)*€160,000 = -€25,600

Total cost = £102,400 + €25,600 = £128,000

In both the cases, the total cost of buying €160,000 in pounds is the same. This is how the hedge works