Question

1. Today is Jan. 31, 2021 and the current exchange rate between US dollar and Canadian dollar is US$0.7946/CAD (U.S. is the home country). The current June 2021 Micro CAD/USD Futures exchange rate is US$0.7949/CAD. Contract size is CAD $10,000. For more information about this futures contract, please go to the website below:

The risk-free rate in Canada is 0.15% per annum with continuously compounding. Assume all futures contracts expire in the last day of the month.

1. What should the risk-free rate in US be assuming no arbitrage in the market? (3 points)

2. If the risk-free rate in US is 0.08% per annum with continuously compounding, is there an arbitrage opportunity? If yes, please specify your arbitrage strategy and cash flow. If no, please explain. (6 points)

3. Following (b), if you need to pay 0.1% of your total value of US dollar as a transaction fee when you convert between US dollar and Canadian dollar in the spot market, (i.e., if you want to buy 1 Canadian dollar, you need to pay US$ (0.7946 + 0.1%*0.7946); if you want to sell 1 Canadian dollar, you will receive US$(0.7946-0.1%*0.7946)), and no transaction fee for futures contracts, is there still an arbitrage opportunity? If yes, please specify your arbitrage strategy and cash flow. If no, please explain. (5 points)

Answer 1

a) The contract expires on 15th June, 2021. No of days till expiry = 28+31+30+31+15 = 135

Let the risk free rate in US be r, then As per Interest rate parity

Futures rate/ Spot rate = exp(r*135/365)/exp(0.0015*35/365)

=> exp(r*135/365) = 0.7949/0.7946*exp(0.0015*135/365)

=> r = 0.002520586 or 0.25% p.a. compounded continuously

b) If the risk free rate in US is 0.08% , there is an arbitrage opportunity as follows

i) Borrow USD 10000 for 135 days at 0.08% p.a. and convert the same to CAD to get 10000/0.7946 = CAD 12584.95

ii) Invest the CAD at 0.15% for 135 days to get 12584.95*exp(0.0015*135/365) =CAD 12591.93 after 135 days

iii) Sell CAD 12591.93 in June Futures at $0.7949/CAD today

iv) In June, get CAD 12591.93 from investment and Sell them using Futures to get 12591.93*0.7949 = $10009.33

v) pay USD10000*EXP(0.0008*135/365) = USD 10002.96  and take the remaining amount of USD 6.37 as arbitrage profit

c) If there is a transaction fee of 0.1% in Spot market but no transaction fee in Futures market

i) Borrow USD 10000 for 135 days at 0.08% p.a. and convert the same to CAD to get 10000/(0.7946+0.1%*0.7946) = CAD 12572.38

ii) Invest the CAD at 0.15% for 135 days to get 12572.38*exp(0.0015*135/365) =CAD 12579.35 after 135 days

iii) Sell CAD 12579.35 in June Futures at $0.7949/CAD today

iv) In June, get CAD 12579.35 from investment and Sell them using Futures to get 12579.35*0.7949 = $9999.33

v) pay USD10000*EXP(0.0008*135/365) = USD 10002.96  

As the amount remained is negative, there is no arbitrage opportunity possible with the transaction charges.