Question

Consider an option to buy €12,500 for £10,000. In the next period, the euro can strengthen against the pound by 25 percent (i.e., each euro will buy 25 percent more pounds) or weaken by 20 percent.

Big hint: don't round, keep exchange rates out to at least 4 decimal places.

		Spot Rates			Risk-Free Rates
S0($/€)		$1.60 = €1.00		i$	3.00%
S0($/£)		$2.00 = £1.00		i€	4.00%
S0(€/£)		€1.25 = £1.00		i£	4.00%

Calculate the hedge ratio.

Answer 1

Here we are discussing options on currency exchange.

We first create a portfolio by hedging the risk.

Take a long position to buy pounds from euros and write a call option with exercise price = X = or 0.8 / 1   (i.e it will take 1.25 to buy 1 )

As given in the question, the spot exchange rate of euro and pounds is given as 1.25 = 1

Exchange rate if the euro strengthens = S⁺ = 1   for 1 , i.e euro strengthens by 25%

Exchange rate if the euro strengthens = S^- = 1.5625   for 1 or 0.64 for 1   , i.e euro weakens by 20%

so, if the value of call option in an up move or down move is c ⁺ and c^-,

c⁺ = max(0,S⁺ - X) = max(0, 1-0.8) = 0.2

c^- = max(0,S^- - X) = max(0,0.64-0.8) = 0

Now to hedge the risk it is assumed that the value of the portfolio will be the same in either up move or down move

c⁺ - h*S⁺ = c^- - h* S^-

h =( c⁺- c^- ) / S⁺- S^-

h =( 0.2 - 0.0) / (1 - 0.64)

h = 0.2/0.36

h = 5/9 = 0.5555

Note: All the calculations have been done on one euro and one pound. The size can be taken 10000 by multiplying all the numbers with 10000. The hedge ratio will still come out to be the same.