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Smooth Interpolation of Zero Curves (Pdf) Smooth Interpolation of Zero Curves Ken Adams Smoothness is a desirable characteristic of interpolated zero curves; not only is it intuitively appealing, but there is some evidence that it provides more accurate pricing of securities. This paper outlines the mathematics necessary to understand the smooth interpolation of zero curves, and describes two useful methods: cubic-spline interpolation—which guarantees the smoothest interpolation of continuously compounded zero rates—and smoothest forward- rate interpolation—which guarantees the smoothest interpolation of the continuously compounded instantaneous forward rates. Since the theory of spline interpolation is explained in many textbooks on numerical methods, this paper focuses on a careful explanation of smoothest forward-rate interpolation. Risk and other market professionals often show a smooth interpolation. There is a simple keen interest in the smooth interpolation of mathematical definition of smoothness, namely, a interest rates. Though smooth interpolation is smooth function is one that has a continuous intuitively appealing, there is little published differential. Thus, any zero curve that can be research on its benefits. Adams and van represented by a function with a continuous first Deventer’s (1994) investigation into whether derivative is necessarily smooth. However, smooth interpolation affected the accuracy of interpolated zero curves are not necessarily pricing swaps lends some credence to the smooth; for example, curves that are created by intuitive belief that smooth interpolation gives the well-known technique of linear interpolation more accurate results than linear interpolation. of a set of yields are not smooth, since the first The authors took swap rates for maturities of derivative is discontinuous. This article extends one, two, three, five, seven and 10 years together the simple mathematical definition of with the six-month zero rate, removed the seven- smoothness, and then describes the implications year swap rate from the data and created the of the extended definition for the smooth implied zero-rate curve from the remaining data. interpolation of zero curves. The resulting zero-rate curve was used to calculate the missing seven-year swap rate which The mathematics of zero curves was then compared to the actual seven-year swap The mathematics of zero curves is derived from rate. The authors found that swap rates the prices of discount bonds; a discount bond calculated with smoothly interpolated zero-rate being a security that pays, with certainty, a unit curves were closer to the actual seven-year swap amount at maturity. The following axioms define rate than swap rates calculated with curves that the discount bond market: were linearly interpolated. • The market trades continuously over its This paper aims to describe the mathematics and trading horizon: it extends from the current finance theory necessary for an understanding of time to some distant future time such that ALGO RESEARCH QUARTERLY 11 VOL. 4, NOS.1/2 MARCH/JUNE 2001 Interpolation of zero curves the maturities of all the instruments to be maturity of coupon-bearing bonds and yield valued fall between now and the trading curves derived from the continuously horizon. compounded yields to maturity of zero coupon (discount) bonds. This article is concerned only • The market is frictionless: no transaction with the latter, which are called zero curves. In costs or taxes are incurred in trading, there addition, the term zero yield is used to refer to are no restrictions on trade (legal or the continuously compounded yield to maturity otherwise) such as margin requirements on of a zero coupon bond. Note that all rates are short sales, and the goods in the market are continuously compounded. It is an easy matter to infinitely divisible. convert to and from periodic compounding, and the use of continuous compounding enables the • The market is competitive: every trader can expression of the mathematics of zero curves in a buy and sell as many bonds as desired particularly simple and elegant form, which without changing the market price. greatly simplifies the discussion. • The market is efficient: information is Prices and yields available to all traders simultaneously, and every trader makes use of all the available Consider a bond which is sold now, at time t, and ≤ < information. is due to mature at time x, where txx∞ . The • The market is complete: any desired cash trading horizon, x∞ , is much greater than zero flow can be obtained from a suitable self- and is longer than the maturity of any bond. financing strategy based on a portfolio of Suppose the price of the bond is denoted by discount bonds. P()tx, . Since the bond pays a unit amount at < • There are no arbitrage opportunities: the maturity, we must have P()xx, = 1 . When tx price of a portfolio is the sum of its the bond sells at a discount and Ptx(), < 1 . Thus, constituent parts. in general, we have Ptx(), ≤ 1 . • All traders in the market act to maximize Now, define the zero yield in terms of the bond their profits: they are rational and prefer price. The zero yield, as seen at time t, of a bond more to less. that matures at time x, t ≤ xx< ∞ , is denoted by ytx(), and is defined, for tx< , by the relationship These axioms are necessary for the development of the mathematics of zero curves. However, they Ptx(, ) = exp()–()xt– ytx(), (1) may not apply to real markets. For example, one conclusion that can be drawn from the axioms is Equation 1 states that the price of the bond at that the market prices equal the intrinsic value of time t is equal to its discounted value. Note that the bonds; that is, there is no “noise” in the this relationship does not define ytt(), since market prices. Further, the market completeness (), axiom implies that all points on the zero curve PTT = 1 for all T; so, for a discount bond are known. In fact, the zero curve is not defined maturing at tT= , both sides of the equation are by an infinite set of values, but rather by a equal to unity, irrespective of the value of ytt(), . discrete, finite set. The zero yield in terms of the price of a bond is obtained by rearranging Equation 1: Discount bonds are often called zero coupon bonds, in contrast to coupon bonds that make –ln()Ptx(), ytx(,) = -------------------------- (2) more than one cash payment to their owner. It is xt– usual to distinguish between yield curves derived from the periodically compounded yields to provided tx< . ALGO RESEARCH QUARTERLY 12 MARCH/JUNE 2001 Interpolation of zero curves Forward rates where Suppose that at time t we enter into a forward ∂ y ()tx, = ytx(), x ∂ contract to deliver at time x1 a bond that will x mature at time x . Let the forward price of the 2 To derive an equation for the instantaneous , bond be denoted by Pt() x1,x2 . At the same time, forward rates in terms of the bond prices, Equation 2 can be rearranged to obtain a bond that matures at time x1 is purchased; the (), –ln()Ptx(), = ()xt– yt(), x (8) price of this bond is Ptx1 . Further, again at time t, a bond that matures at time x2 is bought; Differentiating Equation 8 with respect to x gives (), the price of this bond is Ptx2 . Note that the , –Px()tx ------------------- = yt(), x + ()xt– y ()tx, (9) complete-market axiom guarantees that these Pt(), x x bonds exist. In addition, the axiom specifying that there are no arbitrage opportunities implies Finally, by direct comparison of Equation 7 and that the price of the bond maturing at time x2 Equation 9 must be equal to the product of the price of the –Px()tx, ftx(), = ------------------ (10) bond maturing at time x1 and the forward price: Ptx(), It is now possible to define yt(), t , which, as noted, Ptx(), 2 = Ptx(), 1 Ptx(), 1,x2 (3) is not defined by Equation 2. First, note that Let the implied forward rate, as seen at time t, for Equation 7 implies ftt(), = ytt(), . Then, noting (),, , the period x1 to x2 be ftx1 x2 , defined by: that Pt() t = 1 , Equation 10 can be used to obtain ()()() Ptx(), 1,x2 = exp – x2 – x1 ftx, 1,x2 (4) –Px()tx, ytt(), ==lim------------------ lim – Px()tx, Note the similarity between the definition of the xt→ Ptx(), xt→ implied forward rate (as defined in Equation 4) Defining the zero curve and the zero rates (as defined in Equation 1). On substituting Equation 1 and Equation 4 into Assume that the prices of all bonds in the market Equation 3 we obtain are known; the implication being that the value (), ()≤≤ of ytx for txx∞ is known. Then, the exp()–()x – t ytx(), = exp()–() x – t ytx(), 2 2 1 i (5) current zero curve (i.e., the one seen at time t) × ()() exp – x2 – x1 ftx(), 1,x2 comprises the zero yields, as seen at time t, of the zero coupon bonds, which mature between t and Rearranging Equation 5 gives x∞ , inclusive; that is, the current zero curve () () defined by ytx(), for ()txx≤≤ . x2 – t ytx(,2 )– x1 – t ytx(,1 ) ∞ ftx(),,x = ------------------------------------------------------------------- (6) 1 2 x – x 2 1 Though the zero curve is defined in terms of the zero yields, it can be defined in terms of the The forward rate ftx(),,x , defined in 1 2 instantaneous forward rates. In this case, the zero Equation 6, is the period forward rate. However, curve is defined by the instantaneous forward the instantaneous forward rate is of much greater rates ftx(, ) for txx≤≤∞ . The zero yield in terms of importance in the theory of the term structure. the instantaneous forward rate is obtained by The instantaneous forward rate for time x, as integrating Equation 7: seen at time t, is denoted by ftx(, ) and is the continuously compounded rate defined by x 1 ytx(), = --------- òft(),u du (11) , () x – t ftx(), ==limftxx(), , + h y() t x + xt– yx()tx, (7) h0→ t ALGO RESEARCH QUARTERLY 13 MARCH/JUNE 2001 Interpolation of zero curves This completes the essential mathematical methods, here, we consider those methods that theory of zero curves.
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