BUILDING CURVES USING AREA PRESERVING QUADRATIC SPLINES PATRICK S. HAGAN
[email protected] Abstract. We compare two interpolation methods which are widely used to construct discount curves, forecast (projection) curves, basis curves, and other financial curves. We find that the area-preserving, quadratic-spline method is superior to the “smart quadratic” method, yielding smoother, more natural looking forward curves with few of the artifacts exhibited by the smart quadratic curves. We also show how to efficiently implement both methods by an iterative bootstrapping scheme.
1. Introduction. Discount curves, basis spread curves, and forecast (projection) curves are normally represented as ( ) = −
(1.1)
(0 )0
0
where () is the instantaneous forward curve. To create these curves, one parses the market data from financial instruments. Setting the theoretical values of these instruments to their current market values yields constraints. For discount curves, these constraints are of of the form[1] (1.2)
X =1
( ) ≡
X =1
−
0
(0 )0
=
for = 1
where is the pay date of the instrument. Spread and projection curves leads to similar systems of equations. These equations depend only on the areas under the instantaneous forward curve between the different paydates: (1.3)
Z
Z
(0 ) 0
0
(0 ) 0
−1
If 0 () is one curve that satisfies all the financial constraints, then any curve (), no matter how crazy, would also satisfy the contraints if it has the same areas between the paydates. So the goal of the interpolation method is to obtain the “most natural” forward curve () which satisfies these constraints. Here we compare two widely used interpolation methods: the area-preserving quadratic spline1 and the “smart quadratic” method. We find that the area-preserving quadratic spline results in smoother, more natural looking forward curves (). In particular, the quadratic spline doesn’t exhibit the “double humps” (local maxima in adjacent intervals) and other artifacts which often arise from the smart quadratic method. 2. Piecewise constant forward interpolation. Bootstrapping. The piecewise constant forward method is the simplest interpolation method. Let us organize the financial instruments by maturity, (2.1)
0 ≡ 0 1 · · ·
where is the maturity (last pay date) of instrument and 0 is today. Let us also assume that the instantaneous forward curve is piecewise constant, (2.2a) (2.2b) 1 This
() ≡ () ≡
for −1 ≤ for
is essentially the “quadratic” method implemented on Bloomberg. 1
= 1 2
Since the maturity date of the first instrument is 1 , all the discount factors for the first instrument depend only on 1 , ( ) = −1 ( −0 )
(2.3)
for 0 ≤ ≤ 1
One chooses 1 numerically so that the first constraint is satisfied, 1 X
(2.4)
1 −1 (1 −0 ) = 1
=1
Usually a global or monotone Newton scheme is used. The second instrument’s maturity is 2 , so it only depends on 1 and 2 , (2.5a) (2.5b)
( ) = −1 ( −0 ) ( ) = −2 ( −1 )−1 (1 −0 )
for 0 ≤ ≤ 1 for 1 ≤ ≤ 2
The first constraint does not depend on 2 , so we can choose 2 numerically to satisfy the second constraint without affecting the first constraint. One continues this “bootstrap scheme” by choosing 3 to satisfy ´ are determined. contraint 3, 4 to satisfy contraint 4, and so on, until all the ’s The piecewise constant forward interpolation method is extremely stable. Of all the interpolation methods, it yields the smallest maximum and the largest minimum. However, the jumps in () at the maturity dates are clearly artifacts of the method, not present in the real curve, which can adversely impact pricing, risks, and hedges. 3. Smart quadratic interpolation. Smart quadratic interpolaton is implemented by combining iteration with bootstrapping. For each iteration , we start with the financial constraints X
(3.1)
−
=1
0
(0 )0
()
=
for = 1 ()
where the right hand side has been modified. For the first step, the original right hand side is used, ≡ . Then, (i) the modified financial constraints are bootstrapped as above to obtain the piecewise constant interpolation () () () function 2.2a, 2.2b with the values 1 , 2 , . . . , ; (ii) the smart quadratic spline is created by first linearly interpolating from the midpoints of each flat segment and determining where these lines cross the domain boundaries: (3.2a)
≡ ( ) =
+1 − () − −1 () + +1 − −1 +1 − −1 +1
for = 1 − 1
The end values 0 and are chosen as (3.2b) (3.2c)
³ ´ () 1 − 1 ³ ´ () () ≡ ( ) = + 12 − −1 ()
0 ≡ (0 ) = 1 −
1 2
for reasons that become apparent below; (iii) for the smart quadratic, the instantaneous forward curve is defined to be ³ ´ () (3.3a) () = −1 (1 − ) + − 3 −1 + − 2 for −1 (1 − ) 2
between −1 and , where () =
(3.3b)
− −1 − −1
Note that () is continuous across the boundaries , since (−1 ) = −1 and ( ) = . For dates before 0 and after the last date , flat extrapolation is used: () = 0 () =
(3.3c) (3.3d)
for 0 , for .
With the choice 3.2b, 3.2c, we have 0 (0) = 0 ( ) = 0, consistent with flat extrapolation; (iv) The quadratics in 3.3a are area-preserving, Z
(3.4)
−1
()
() = ( − −1 )
so the smart quadratic discount factors ( ) are identical to the discount factors obtained from the piecewise constant forward curve at every maturity date . Consequently, the modified financial constraints in 3.1 are very nearly satisfied; they are not exactly satisfied because the discount factors at coupon dates between maturity dates are slightly different than those obtained from the piecewise constant forward curve. So at each iteration, we determine the errors with the original constraints, (3.5a)
X
0 −
=1
0
(0 )0
()
− =
for = 1
and add this to the right hand side, (3.5b)
(+1)
()
=
()
− ()
and iterate. Typically only a few iterations are needed to before the errors are essentially zero. The smart quadratic method is reliable and widely used, but the resulting curves () often exhibit “double humps,” two maxima or two minima in adjacent intervals. See figure 3.1. These are clearly artifacts of the method, not present in the real forward curve. In addition. the slopes of () jump at each maturity . These artifacts introduce errors into our pricing, risks, and hedges, although they are generally smaller than the errors introduced by the piecewise constant forward method. 4. Area preserving quadratic spline interpolation. This method is identical to smart quadratic interpolation except that the values are chosen to ensure that 0 () is continuous at the maturity points . For each iterative step , the modified financial constraints 3.1 are bootstrapped to obtain the piecewise () () () constant interpolation function 2.2a, 2.2b with values 1 , 2 , . . . , . As before, we use the quadratic, area-preserving spline ³ ´ () for −1 (1 − ) (4.1a) () = −1 (1 − ) + − 3 −1 + − 2
for = 1 2 , and flat extrapolation beyond the endpoints, (4.1b) (4.1c)
for 0 , for .
() = 0 () = 3
f(t)
2.0
PWC SQ 1.5
APQSpline
1.0
0.5
0.0 0
6
12
18
24
30
36
42
months Fig. 3.1. The forward curve using pw constant interpolation (blue), ordinary smart quadratic interpolation (red), and area preserving quadratic spline interpolation (black). The SQ interpolation frequently exhibits double humps and other artifacts; these are eliminated by using the area preserving quadratic spline.instead.
¡ ¢ ¡ ¢ Now, however, we choose ≡ ( ) so that 0 − = 0 + at the domain boundaries . This requires choosing ³ ´ ³ ´ ∆+1 ∆ 3 () 1 3 () 1 − − (4.2a) = + −1 +1 2 ∆+1 + ∆ 2 ∆+1 + ∆ 2 +1 2 for = 1 2 − 1, where (4.2b)
∆ = − −1
In addition, we set 0 (0 ) = 0 ( ) = 0 to be consistent with flat extrapolation. This requires (4.3a) (4.3b)
()
0 ≡ (0 ) = 32 1 − 12 1
() ≡ ( ) = 32 − 12 −1
Together, eqs. 4.2a-4.3b form a tri-diagonal system, (4.4a)
()
00 0 + 01 1 = 32 1 ()
(4.4b) (4.4c)
−1 −1 + + +1 +1 =
3 2
()
∆+1 + ∆ +1 ∆+1 + ∆
() −1 −1 + = 32
4
for = 1 2 − 1
whose non-zero matrix elements are 00 = 1
(4.5a)
1 2
01 = 12
∆+1 ∆+1 + ∆
(4.5b)
−1 =
(4.5c)
−1 = 12
= 1
+1 =
1 2
∆ ∆+1 + ∆
= 1
is a diagonally dominant, nonnegative-element, tri-diagonal, regular matrix. This tridiagonal system is easily solved for 0 1 . See ref. [3]. With these values in hand, 4.1a-4.1c provide the instantaneous forward curve (). As in the smart quadratic method, the new discount factors very nearly satisfy the modified financial constraints in 3.1. As before, we calculate the error in satisfying the original financial constraints, X
(4.6a)
0 −
=1
0
(0 )0
()
− =
for = 1
add this to the right hand side, (+1)
(4.6b)
()
=
()
−
and iterate. We believe that the area preserving quadratic spline is superior to the smart quadratic method: It is the smarter quadratic. The resulting curves () are 1 , where the smart quadratic curves are only 0 , and the curves appear free of obvious artifacts. See figure 3.1. In particular, the continuous derivative at ¡ the¢ boundaries ensures that double maxima and minima cannot occur, since double extrema require − ¡ ¢ and + to have different signs. The forward curves also depend smoothly on the data, and thus yield stable risks. Specifically, the off-diagonal elements of the matrix sum to 12 for all rows, so = 12 and = 32 are two of the eigenvalues of ; Perron-Frobenius theory[2] ensures that |1 − | 12 for all other eigenvalues; so the eigenvalues of the inverse matrix −1 must have || 2. Thus, risks obtained from the instantaneous forward curve are well conditioned, since changes in the vector f = (0 1 ) cannot be more than twice as large as the changes in the right hand side, (4.7)
3 2
Ã
() 1
()
()
() () ∆ −1 + ∆−2 ∆2 1 + ∆1 2 () ∆1 + ∆2 ∆−2 + ∆
!
REFERENCES [1] Hagan, P. and West, G. (2006) Interpolation methods for curve construction, Appl Math Fin, 13: 2, 89-129 [2] Henryk M. (1988) Nonnegative matrices, Wiley:New York [3] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2007) Numerical recipes 3rd Edition, Cambridge: New York
5
