## N1

swapL(t0, Rs) = IVj2 B(t0, Tn + l)[L(t0, Tn) — Rs] (4.2)

swapR(t0, Rs) = IV £ B(tQ, Tn + l)[Rs — L(tQ, Tn)] (4.3)

Eq. (4.4) shows that a swap contract is a zero sum game, with the gain of one party being exactly equal to the loss of the other party. The value of a swap is taken

CM -10000-

CM -10000-

20000-

10000-

-20000-

100 200 300 400 500 Time series (2003.1.29-2005.1.28) (a)

100 200 300 400 500 Time series (2003.1.29-2005.1.28) (a)

Figure 4.5 (a) Daily market value of a deferred swapL(t0, T) on a notional principal of $1 million with floating taken to be Libor and fixed interest rate RS = 2.8%. The swap is Libor 2by2: the swap matures two years in the future, that is T - to = 2 years and runs for another two years. The market values are given for T - to e [29.1.2003-28.1.2005]. (b) Figure shows the time variation of Rp(to), the par value for the fixed yearly interest payments, for a 2by10 swapL, with T - t0 e [29.1.2003-28.1.2005].

30000-

20000-

10000-

-20000-

to be the difference between the floating and fixed interest rate receiver swaps and is given by

where Eq. (4.4) yields the last line.

The swaplets combine together to form a swap contract that consists of a portfolio of swaplets. A swap that is initiated at time t0 and runs from T0 to TN and is shown in Figure 4.4(b).

In contracts between parties with equally good credit ratings, the value of the swap for both parties, receiving floating or fixed payments, must have equal value.4 Note from Eq. (4.4) that swapL = -swapR; hence, when the swap contract is initiated, both forward swaps are equal to zero.

It is important to note that although a forward swap contract starts at time t0 with zero value, swap(t, T0) has nonzero values during the time interval t e [t0, T0] -that is, until it matures at time T0. The market value of a swapL for the Libor market is shown in Figure 4.5(a); the swap matures in two years and payments continue for another two years; the fixed interest rate is RS = 2.8%.

4 Since, otherwise, the party with the unfavorable price will not enter into the swap contract.

4.2 Interest rate swaps

One can simplify the expression for the swaps. The Libor zero coupon yield curve represents Libor in terms of Libor zero coupon bonds, as in Eq. (2.21)

and yields the following

¿VJ2 B(t0, Tn + i)L(t0, Tn) = VJ2 [B(t0, Tn) - B(t0, Tn + i)]

swapL(t0, Rs) = V[B(t0, T0) - B(t0, Tn) - iRsJ^ B(t0, Tn + i)] (4.5)

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