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%

1. Calculate the current €/£ spot exchange rate.
2. Calculate the hedge ratio.

Answer 1

1. Calculate the cross rate between the euro and pound as follows:

€/£ = (€/$)* ($/£) Here, Divided and multiply $, to find the cross rate, because the rates are given in dollars.

Here, €/$ = 1/1.60

$/£ = 2

€/£ = (1/1.6) * 2

€/£=1.25

Therefore, the cross rate is 1.25. 2. 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 = 1.25\euro / 1 \pounds or 0.8 \pounds / 1 \euro (i.e it will take 1.25 \euro to buy 1 \pounds)

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

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

Exchange rate if the euro strengthens = S- = 1.5625 \euro for 1 \pounds or 0.64 \pounds for 1 \euro , 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.