Question

Let us assume that around Dec. 2019, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0485 (USD per SWF). Interest rates in the United States and Switzerland were 0.75% and 0.25% per annum, respectively with continuous compounding. Assume also that the four-month forward exchange rate was 1.0630 (US$ per SWF)

1. Is there an arbitrage opportunity for you to make money? (Hint: Recall how to find no-arbitrage forward price!)

2. If there is NO arbitrage opportunity, explain WHY?

3. If there is an arbitrage opportunity, using 1.0m USD or 1.0m SWF, how much arbitrage profit you can have in four-month in USD?

Answer 1

No arbitrage forward price

Let us first go through the given details in the question

· Spot Exchange Rate = 1.0485 USD per SWF

· Interest Rate in U.S = 0.75% per annum

· Interest Rate in Switzerland = 0.25% per annum

· 4 month Forward exchange rate = 1.0630 (US$ per SWF)

1. To find the arbitrage opportunity, we have to out no-arbitrage forward price

Ans. No-arbitrage forward price formula is as follows:

F=S*(1+id) divided by (1+if​)​ to the power n

Or

F=S*{(1+id)/(1+if)}^n

Where

· N is time to maturity, F is no-arbitrage forward price, S is spot price, id is interest rate in domestic currency and if is interest rate in foreign currency.

Let us now put all the given inputs in the equation:

F = 1.0485 USD per SWF * {(1+.075%)/(1+0.25%)^4/12}

F= 1.050240252 USD per SWF

Therefore, No -arbitrage forward price for this contract is 1.050240252 USD per SWF.

Note the current forward price traded on exchange is given as 1.0630 (US$ per SWF) that shows an opportunity to exploit arbitrage and gain profits.

2. If there is NO arbitrage opportunity, explain WHY?

Ans. The current forward price traded on exchange is given as 1.0630 (US$ per SWF) that shows an opportunity to exploit arbitrage and gain profits.

No -arbitrage forward price for this contract is 1.050240252 USD per SWF.

The difference between the two shows the investor/trader can exploit the opportunity to gain profits.

3. If there is an arbitrage opportunity, using 1.0m USD or 1.0m SWF, how much arbitrage profit you can have in four-month in USD?

Ans. Assuming 1 million USD or 1 million SWF, arbitrage profit you can have in four-month in USD can be calculated as follows:

· No -arbitrage forward price for this contract is 1.050240252 USD per SWF

· Note the current forward price traded on exchange is given as 1.0630 (US$ per SWF)

The forward price is higher than the no-arbitrage forward price. This shows that the investor/trader will take a ‘short position’ or sell the forward price traded on exchange is given as 1.0630 (US$ per SWF) and take a ‘long position’ or buy the current spot price.

The difference to exploit at expiry (after 4 months) is 1.0630 subtracted by 1.050240252 is 0.012759748 (US$ per SWF).

=0.012759748 (US$ per SWF) * 1 million SWF = US$ 12759.748 at expiry after 4 month.

Hence, US$ 12759.748 arbitrage profit you can have in four-month in USD

You can also calculate this profit as on today by simply discounting it back with US interest rate. The question asked to calculated profit in 4 month in USD.

I hope you understood and find this useful.