Question

The forward price of a currency is given by f(S, t) = S e^(r−rf ) (T −t) ,

a) Show that if f(S, t) < S e(r−rf ) (T −t) , then arbitrage profits can be made. Hint: because f(S, t) is “too low” and S is “too high,” today’s arbitrage trades are

i) Enter a long position in the forward contract. (That is, agree to buy the foreign currency at time T at the price f(S, t) per unit of foreign currency.)

ii) Borrow one unit of the foreign currency at the rate rf for the period from t to T, and sell the foreign currency for S and invest S at the home rate r, for the period from t to T.

Answer 1

This can be proved by taking the following example,

Suppose current spot exchange rate is 94.10¥ / $

The risk free rate is 0.5%

3-months forward exchange rate is 93.50¥/$

Yen interest rate is 1.2% p.a

Dollar interest rate is 3.5% p.a.

Since, 93.50 < 94.10*e^{(1.2 - 0.5)*1/4}

Or, 93.50 < 100.92

Arbitrage profit can be made through following steps –

Make a long position on dollar forward contract.

Borrow 100 dollar at the rate of 3.5% p.a. for 3-months

Amount to be repaid after 3 months   =          100*(1 + 0.035/4)

                                                            =          $100.875

Convert $100 into Yen at spot exchange rate of 94.10¥ / $

Equivalent Yen           =          100*94.10

                                    =          ¥9,410

Invest this dollar into Japanese market for 3-months at the interest rate of 1.2% p.a.

Amount after 3-months          =          9,410*(1 + 0.012/4)

                                                =          ¥9,438.23

Convert it into dollar at 3-months forward rate of 93.50¥/$

Equivalent dollar         =          9,438.23 / 93.50

                                    =          $100.9436

Thus, arbitrage profit =          $100.9436 - $100.875

                                    =          $0.0686 per $100.