Question

3. The two-month interest rates in Switzerland and the United States are, respectively, 1% and 2% per annum with continuous compounding. The spot price of the Swiss franc is $1.0500. The futures price for a contract deliverable in two months is also $1.0500.
   1. What arbitrage opportunities does this create?
   2. What would be the risk-free profit made by the arbitrageur?

Answer 1

Spot Price of Swiss franc=$1.0500

Interest rate in USA=2% per year =0.02(Continuously compounded)

Interest rate in Switzerland=1% per year=0.01(Continuously compounded)

Future value of $1.0500 invested in USA for two months(1/6 year)=$1.0500*(e^(0.02*(1/6)))=$1.0535

Future value of 1 Swiss franc invested in Switzerland for two months(1/6 year)=1*(e^(0.01*(1/6)))=1.0017 Swiss franc

Future Price based on interest rate parity: $1.0535=1.0017 Swiss franc

1 Swiss franc=$(1.0535/1.0017)=$1.0518

2 months future price =$1.0500

There is arbitrage opportunity since future price is less than fair price based on interest rate parity.

If you buy futures at $1.0500 , the profit would be (1.0518-1.0500)=$0018 per Switzerland franc.

Other way is :

Buy Futures at $1.0500

Step1: borrow Swiss franc =(1/1.0017)=0.998303 at 1%

Step 2: Convert into US $ at spot rate of1.0500)=$(0.998303*1.0500)=$1.048218

Step3: Invest at interest rate of 2% in USA

Step4. On maturity you will get =$1.048218*(e^(0.02*(1/6)))=$1.0518

Step5:Pay contracted price of $1.0500 to buy 1 Swiss franc to pay back the loan with interest in Swiss franc

Step6: The difference (1.0518-1.0500)=$0018 is the risk free profit