B481/S98/NOTES/Grabbe-5 FORWARDS, SWAPS, AND INTEREST ...

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Borrow Foreign Currency The trader can The trader can borrow 1/[1+ia*(T/360)] units of foreign currency at time t and repay 1 unit of foreign currency at time t+T. Using the foreign currency at time t, the trader can sell the currency forward while placing the domestic currency on deposit at ib. The spot rate Sb(t), the forward rate Fa(t,T) and the interest rate ib determine the value of the hedge via 1 the equation Sb( t )[ 1 + ib( T / 360) ]( 1 / Fa( t , T )). To see this, [ 1 + ia ∗ ( T / 360] simply fill-in the box diagram, as follows: DOMESTIC CURRENCY FOREIGN CURRENCY time t: 1 receive [ 1 + ia ∗ ( T / 360] Sb(t) borrow 1 [ 1 + ia ∗ ( T / 360] Sb(t) $/for. ib ia* Fa(t+T) $/for. time t+T direct: indirect: receive [ 1 + ib( T / 360) ] [ 1 + ia * ( T / 360] owe 1 unit Sb( t ) receive [ 1 + ib( T / 360) ] [ 1 + ia ∗ ( T / 360] Sb( t ) Fa( t , T ) 12
Clearly, arbitrage profits exist if and only if the trader receives more, at time t+T, from the indirect than he owes via the direct route. Therefore, arbitrage is not profitable if and only if [ 1 + ib( T / 360) ] Sb( t ) [ 1 + ia ∗ ( T / 360] Fa( t , T ) < 1. Rearranging terms yields the following: no arbitrage profits exist if and only if [ 1 + ib( T / 360) ] Fa( t , T ) $ Sb( t ) [ 1 + ia ∗ ( T / 360) ] . Note that a trader always has both paths available for arbitraging the market. Therefore, there are no arbitrage profits if and only if both of the foregoing inequalities hold simultaneously. Thus, we have the following: no arbitrage profits exist if and only if and [ 1 + ia( T / 360) ] Fb( t , T ) # Sa( t ) [ 1 + ib ∗ ( T / 360) ] Fa( t , T ) $ Sb( t ) [ 1 + ib( T / 360) ] [ 1 + ia ∗ ( T / 360) ] . Example 5.6 This example puts the two halves of the system together. Suppose we have S(t) = US$.4428-.4438/DM and F(t,30 days) = US$.4450-.4460/DM. The rate on one-month euro-DM is 5 7 ( = 5 . 875) to 6 percent, while the rate on one-month eurodollars is 10 11 percent. Is there an arbitrage possibility 8 16 ( = 10. 6875) to 10 13 16 = ( 10. 8125) 13
Page 1 and 2: B481/S98/NOTES/Grabbe-5 FORWARDS, S
Page 3 and 4: First, we must consider the convent
Page 5 and 6: Graphically, we have the following:
Page 7 and 8: DIVERGENCES FROM INTEREST PARITY Li
Page 9 and 10: INTEREST PARITY WITH BID/ASKED SPRE
Page 11: Clearly, arbitrage profits exist if
Page 15 and 16: The no-arbitrage conditions are [ 1
Page 17 and 18: To see this, consider the box diagr
Page 19 and 20: INTEREST PARITY AND THE COST OF INF

B481/S98/NOTES/Grabbe-5 FORWARDS, SWAPS, AND INTEREST ...

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