Accrual Swaps
ACCRUAL SWAPS AND RANGE NOTES
PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 PHAGAN1@BLOOMBERG.NET 212-893-4231 Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes
1. Introduction. 1.1. Notation. In our notation today is always t = 0, and (1.1a) D(T ) = today’s discount factor for maturity T.
For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1.1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t.
These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1.2a) (1.2b) These are defined via (1.2c) D(T ) = e−
T 0
f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t.
f0 (T 0 )dT 0
,
Z(t; T ) = e−
T t
f (t,T 0 )dT 0
.
1.2. Accrual swaps (fixed).
αj t0 t1 t2
…
tj-1
tj
…
tn-1
tn
period j
Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1.3) t0 < t1 < t2 · · · < tn−1 < tn
1
Rfix
Rmin
Rmax
L(τ)
Fig. 1.1. Daily coupon rate
be the schedule of the coupon leg, and let the nominal fixed rate be Rf ix . Also let L(τ st ) represent the k month Libor rate fixed for the interval starting at τ st and ending at τ end (τ st ) = τ st + k months. Then the coupon paid for period j is (1.4a) where (1.4b) and (1.4c) θj = #days τ st in the interval with Rmin ≤
PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 PHAGAN1@BLOOMBERG.NET 212-893-4231 Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes
1. Introduction. 1.1. Notation. In our notation today is always t = 0, and (1.1a) D(T ) = today’s discount factor for maturity T.
For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1.1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t.
These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1.2a) (1.2b) These are defined via (1.2c) D(T ) = e−
T 0
f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t.
f0 (T 0 )dT 0
,
Z(t; T ) = e−
T t
f (t,T 0 )dT 0
.
1.2. Accrual swaps (fixed).
αj t0 t1 t2
…
tj-1
tj
…
tn-1
tn
period j
Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1.3) t0 < t1 < t2 · · · < tn−1 < tn
1
Rfix
Rmin
Rmax
L(τ)
Fig. 1.1. Daily coupon rate
be the schedule of the coupon leg, and let the nominal fixed rate be Rf ix . Also let L(τ st ) represent the k month Libor rate fixed for the interval starting at τ st and ending at τ end (τ st ) = τ st + k months. Then the coupon paid for period j is (1.4a) where (1.4b) and (1.4c) θj = #days τ st in the interval with Rmin ≤