Equivalence between forward rate interpolations and discount factor interpolations for the yield curve construction
AArticle
Equivalence between forward rate interpolations anddiscount factor interpolations for the yield curveconstruction
Jherek Healy * Correspondence: [email protected]: December 2019
Abstract:
The traditional way of building a yield curve is to choose an interpolation on discount factors,implied by the market tradable instruments. Since then, constructions based on specific interpolations ofthe forward rates have become the trend. We show here that some popular interpolation methods onthe forward rates correspond exactly to classical interpolation methods on discount factors. This paperalso aims at clarifying the differences between interpolations in terms of discount factors, instantaneousforward rates, discrete forward rates, and constant period forward rates.
Keywords: monotone interpolation; cubic spline; yield curve; finance
1. Introduction
The traditional way of building a yield curve is to choose an interpolation on discount factors, impliedby the market tradable instruments [1,2]. Since then, with the introduction of the monotone convexinterpolation by Hagan and West [3], constructions based on specific interpolations of the discrete forwardrates have become the trend, mostly because the discrete forward rates are more directly related to themarket observables. We show here that some popular interpolation methods on the consecutive discreteforward rates, or on instantaneous forward rates, correspond actually to classic interpolation methods ondiscount factors.The equivalence was suggested in [4]. The aim of this paper is to clarify this equivalence further, andto show in particular that the smart quadratic interpolation, often used in banks, and described in [5] isequivalent to a Hermite spline interpolation on the logarithm of discount factors, and the area-preservingquadratic spline interpolation of Hagan [5] is equivalent to a cubic spline interpolation of class C on thelogarithm of discount factors.Since the 2008 financial crisis, there is not a single curve used for discounting or projecting ratesanymore, but multiple, jointly calibrated, curves. Distinct curves are used for discounting or for projectingrates of specific tenors and currency. In this context, cubic spline interpolations may be applied directly todiscrete forward rates of constant period as in [6]. We show that additional knots must be introduced toderive the equivalent cubic spline interpolation in terms of the logarithm of pseudo-discount factors.To conclude, we present a few alternative piecewise cubic interpolation schemes with natural shapepreserving properties. a r X i v : . [ q -f i n . P R ] M a y of 13
2. Yield curve construction
We start by introducing the concepts of discount factors, discrete forward rate and instantaneousforward rate, key for the curve construction. Let P ( t ) denote the price of a zero coupon bond deliveringfor certain $1 at maturity t . The continuously compounded yield y is given by P ( t ) = e − y ( t ) t . (1)The instantaneous forward rate f is defined by P ( t ) = e − (cid:82) t f ( u ) du . (2)The forward rate f di from t i − to t i , which corresponds to the rate fixed at time 0 to borrow from t i − and t i is defined by P ( t i − ) e − f di ( t i − t i − ) = P ( t i ) ,or equivalently, f di = − ln P ( t i ) − ln P ( t i − ) t i − t i − . (3)From equations 2 and 3, we deduce that discrete forward rate corresponds to the area of theinstantaneous forward rate between t i − and t i . We have f di = t i − t i − (cid:90) t i t i − f ( u ) du . (4)Let z be the logarithm of the discount factors: z ( t ) = ln P ( t ) = − y ( t ) t . (5)The relation between the logarithm of the discount factor and the instantaneous forward rate is then f ( t ) = − ∂ ln P ∂ t ( t ) = − ∂ z ∂ t ( t ) , (6)and the relationship with the discrete forward rate is f di = − z ( t i ) − z ( t i − ) t i − t i − . (7)In the above equations, we followed Hagan and West [3] and defined f di as a continuouslycompounded rate. Yet, it still represents effectively a discrete forward rate between two dates, in contrastwith the instantaneous forward rate defined by Equation (2). In particular, the continuously compoundedrate f di is equivalent to, and may be trivially converted into a specific single period discrete rate. We willdiscuss the latter in more details, in Section 6.The yield curve is the curve followed by y ( t ) at any time t , or equivalently the curve which definesthe (pseudo-)zero coupon bond value P ( t ) . The market quotes a limited number of securities, whose pricesdepend on the (pseudo-)zero coupon bond values at a discrete set of dates ( t j ) j = M . A proper yieldcurve must be able to reprice the market securities. A complete description of yield curve construction isgiven in [1,2]. Let V i be the price of N securities. Typically, for a Libor curve, those securities are Libor of 13 deposits for the first few months, Eurodollar futures for up to 3 or 4 years and par swaps for the rest of thecurve. We assume that the securities can be written as a linear combination of discount bond prices: V i = M ∑ j = c i , j P ( t j ) , i =
1, ..., N (8)with t , ..., t M a finite set of dates, in practice corresponding to the cash flow dates of the N benchmarksecurities. In the context of multiple curves construction, the coefficients c i , j may depend onpseudo-discount factors obtained from the other curves. In the case of the OIS curve, the price of somesecurities is not necessarily a linear combination of a discrete set of discount factors, but only a non-linearfunction of this set. We will detail this in Section 6.3.If we want to calibrate a set of discount factors to the market instruments, we only need to solve alinear system in the discount factors ( P ( t j )) j = M . In reality, the interpolation plays however a role, sincethe number of securities N is typically smaller than the number of discount factors M and the systemis under-determined. The interpolation allows then to restrict the possible shapes allowed to a spaceof shapes implied by N parameters. The typical yield curve calibration algorithm consists in solvingfor the N parameters of a given interpolation function so that Equation (8) holds. This is not a linearproblem anymore, as the interpolation function is not a linear in the discount factors and a non-linearsolver (Levenberg-Marquardt in our numerical examples) must be used. The linearity property of Equation(8) is effectively not used in the calibration.
3. Hagan and West interpolation
Let us recall Hagan and West monotone convex spline construction [3].Firstly, a suitable set of forward rates is computed according to the following procedure. Let f di be theinput discrete forward rate at node i , the rate at point t i is defined for i =
1, 2, ..., n − f i = f ( t i ) = t i − t i − t i + − t i − f di + + t i + − t i t i + − t i − f di , (9) f = f ( t ) = f d − ( f − f d ) , (10) f n = f ( t n ) = f dn − ( f n − − f dn ) . (11)Secondly, for i =
1, 2, ..., n , let g i ( x ) = f ( t i − + ( t i − t i − ) x ) − f di . The quadratic g is defined for x ∈ [
0, 1 ] by g i ( x ) = g i ( )( − x + x ) + g i ( )( − x + x ) .The instantaneous forward rate is computed from g i through f ( t ) = g i (cid:16) t − t i − t i − t i − (cid:17) + f di , for t ∈ [ t i − , t i ) .This definition creates a continuous interpolation which preserves the consecutive discrete forward ratesas we have t i − t i − (cid:82) t i t i − f ( t ) dt = f di by construction.Secondly, the function g is modified so that the positivity and monotonicity of the instantaneousforward rates is preserved. Since the advent of negative rates, and because those modifications may createunstable hedges in some circumstances [4], practitioners often discard the modifications to g . A recentpaper by Hagan [5] also presents this same interpolation (without the modifications for monotonicityand positivity) with the name "smart quadratic" and suggests it is close to what Bloomberg was usinginternally at the time. of 13
4. Cubic Hermite spline interpolation
Given the data z ( t ) , z ( t ) , ..., z ( t n ) with t < t < ... < t n , following [7, p. 39–32], a piecewise cubicinterpolant p is of the following form for i ∈
0, ..., n −
1, for t ∈ [ t i , t i + ] , p ( t ) = p i ( t ) = c i ,0 + c i ,1 ( t − t i ) + c i ,2 ( t − t i ) + c i ,3 ( t − t i ) , (12)with coefficients c i , j ∈ R . The conditions for an interpolation of class C are: p i ( t i ) = z ( t i ) , p (cid:48) i ( t i ) = s i , i =
0, ..., n (13) p i ( t i + ) = z ( t i + ) , p (cid:48) i ( t i + ) = s i + , i =
0, ..., n − s i are free parameters. Let d i = z ( t i + ) − z ( t i ) t i + − t i . (15)The interpolation conditions give: c i ,0 ( t i ) = z ( t i ) , (16) c i ,1 = s i , (17) c i = d i − s i + − s i t i + − t i , (18) c i = − d i − s i + − s i ( t i + − t i ) . (19)For local interpolation schemes, the s i are chosen so that the i th cubic polynomial depends onlyon information from, or near the interval [ t i , t i + ] . While the s i directly correspond to the slope of theinterpolation at t i , the variables d i are just used as a convenient notation.Now if we let z ( t i ) correspond to the logarithm of the discount factor at date t i , per Equation (7), wehave then the identity d i = − f di + . (20)In a Bessel spline, for i =
1, ..., n −
1, the s i are chosen to be the slope of the parabola interpolatingthree consecutive data-points. Its order of accuracy is O ( δ t ) . s i = ( t i − t i − ) d i + ( t i + − t i ) d i − t i + − t i − . (21)The remaining s and s n are determined by an appropriate choice of boundary conditions. From equations21 and 9, along with the identity d i = − f di + , we obtain the identity s i = − f i for i =
1, ..., n − p (cid:48)(cid:48) ( t ) = = p (cid:48)(cid:48) ( t n ) .According to Equation (18), for the left boundary t , this corresponds to3 d − s − s t − t = of 13 or equivalently s = d − ( s − d ) . (23)Similarly for the right boundary t n , according to equations 18 and 19,we have2 3 d n − − s n − s n − t n − t n − − d n − − s n − s n − t n − t n − = s n = d n − − ( s n − − d n − ) . (25)The boundaries thus corresponds to the boundaries set in the forward rate interpolation defined byequations 10 and 11 and we thus also have s = − f and s n = − f n .The derivative of the cubic Hermite spline − p is a continuous quadratic spline, which interpolates f i at t i for i =
0, ..., n . It must thus be the same quadratic spline as Hagan and West. This may also be verifiedby an explicit calculation of p (cid:48) .The cubic Hermite spline on the logarithm of discount factors will also preserve the area from t i to t i + since, by construction, we impose p ( t i ) = z ( t i ) and we have( z ( t i ) − z ( t i − ) = − (cid:82) t i t i − f ( u ) du according toequations 4 and 7.The flat extrapolation of the forward rates for t < t or t n < t corresponds to a C linear extrapolationin the logarithm of discount factors.
5. Hagan smoother area preserving interpolation
In the area preserving quadratic spline interpolation of Hagan [5], the following spline is consideredfor the instantaneous forward: f ( t ) = f i − ( − x i ( t )) + f i x i ( t ) − ( f i − + f i + f di ) x i ( t )( − x i ( t )) for t ∈ [ t i − , t i ) (26)with x i ( t ) = t − t i − t i − t i − . The f i are not defined by Equations (9) anymore but are chosen so that f (cid:48) ( t − i ) = f (cid:48) ( t + i ) and the boundaries are defined by f (cid:48) ( t ) = f (cid:48) ( t n ) =
0. This leads to a tridiagonal system on the f i .Let us now explore its equivalence to a cubic spline on the logarithms of discount factors. The firstderivative of the Hermite cubic spline defined by Equation (12) is p (cid:48) ( t ) = c i − + c i − ( t − t i − ) + c i − ( t − t i − ) for t ∈ [ t i − , t i ) (27)with c i ,1 = s i and c i ,2 , c i ,3 defined by equations 18, 19.When the Hermite cubic spline is applied to the logarithm of discount factors z ( t i ) , we have s i = − f i and by rewriting the Equation (26) in the same form, it can easily be verified that p (cid:48) ( t ) = − f ( t ) . Thecondition f (cid:48) ( t − i ) = f (cid:48) ( t + i ) correspond to p (cid:48)(cid:48) ( t − i ) = p (cid:48)(cid:48) ( t + i ) and the boundary conditions correspond to thenatural boundary conditions. Such a spline is thus the standard cubic spline of class C as described in[7, p. 43]. The derivative of the cubic spline − p is a C quadratic spline, which interpolates f i at t i for i =
0, ..., n . It must thus be the same area preserving quadratic spline as Hagan.It is well known, that on some monotone or convex input data, a standard cubic spline interpolationmay oscillate [8,9]. A typical example is data from a Heaviside like function. If the input data were directlyzero rates, this would happen on many real world examples. In our case, the input data are the logarithmof discount factors, that is, zero rates multiplied by their corresponding maturity. This multiplication will,in effect, considerably smooth out the input data. In order to reproduce a clear oscillation, much largerzero rates variations are needed, so large, that they are unlikely to be realistic at all. of 13
6. Cubic spline interpolation on the discrete forward rates
In a multi-curve environment, each curve is associated to a specific index, and a specific tenor. Fora given index and tenor, instead of interpolating the logarithm of discount factors, or the instantaneousforward rates, Henrard [6] proposes to interpolate the discrete forward rates with start dates u and maturitydates v defined in terms of pseudo-discount factors P byˆ f d ( u , v ) = δ (cid:18) P ( u ) P ( v ) − (cid:19) (28)so that the price of an Ibor coupon with start date u and maturity v is P D ( v ) δ ˆ f d ( u , v ) , with P D being thediscount factor associated to the relevant discount curve, and δ the accrual factor for the period. In amulti-curve environment, P D is different from P .In order to be consistent with our previous notation, we transform the one-period discrete rate ˆ f d into a continuously compounded rate f d through the relation e f d ( u , v )( v − u ) = ˆ f d ( u , v ) δ + f d , Equation (28) becomes f d ( u , v ) = − ln P ( v ) − ln P ( u ) v − u . (30)In particular, we have f d ( t i − , t i ) = f di where f di is defined by Equation (4). Let p be a C cubic spline interpolation on the logarithm of pseudo-discount factors at the dates t i asin Section 5. From Equation 4, we have f d ( t , t + ∆ ) = p ( t + ∆ ) − p ( t ) ∆ . (31)In the discrete forward curve construction, the discrete forward rates used as input of the interpolationare all of the same tenor, and thus ∆ is kept constant for all t . A priori, the set of observations times t i ishowever not evenly distributed.If we start from pseudo-discount factors observation times ( t i ) and want to construct an equivalentdiscrete forward interpolation, we also need the pseudo-discount factors at times ( t i + ∆ ) . Let ( τ j ) j = m bethe sorted set of dates such that ( τ j ) j = m = ( t i ) i = n ∪ ( t i + ∆ ) i = n and let us define the interpolationfunction ¯ f d by ¯ f d ( t ) = f ( t , t + ∆ ) . (32)We know that p is a C cubic piecewise polynomial with knots ( t i ) i = n , and thus ¯ f d is also a C cubicpiecewise polynomial with knots ( τ j ) j = m . Furthermore, we have by construction¯ f d ( t i ) = p ( t i + ∆ ) − p ( t i ) ∆ . (33) We neglect any mismatch related to week-ends or holidays. of 13
We have thus shown that a C cubic spline interpolation on the logarithm of the pseudo-discount factorsat knots ( t i ) i = n implies the same discrete forward rates at all times t as a C cubic spline interpolationon the continuously compounded discrete forward rates at knots ( τ j ) j = m , assuming a constant periodlength ∆ for the continuous compounding. The equivalent interpolation in terms of discrete forward ratesintroduces additional discontinuities in the third derivative of ¯ f d , located at the knots ( t i + ∆ ) .On the interval ( t i , t i + − ∆ ) , the same piecewise polynomial p i will be used to compute the differenceand Equation (31) simplifies to ¯ f d ( t ) = p (cid:48) i ( t ) + ∆ p (cid:48)(cid:48) i ( t ) + ∆ p (cid:48)(cid:48)(cid:48) i ( t ) . (34)If p i is cubic in t , the discrete forward rate will thus be quadratic in t . In contrast, on the interval ( t i + − ∆ , t i + ) , two different piecewise polynomials will be used, and the discrete forward rate will, apriori, not be quadratic, but stay cubic.The choice of interpolating on pseudo-discount factors with a spline thus implies a less smooth splineinterpolation in terms of discrete forward rates when compared to a direct spline interpolation on thediscrete forward rates. Hence a direct interpolation in terms of discrete forward rates may be preferable, ifall we need to compute the yield curve instrument prices are only discrete forward rates.As evidenced in [10], there is however an additional subtlety when interpolating directly on thediscrete forward rates, related to the ambiguity of the start date, in relation with a given end date. Indeed,because of holidays and week-ends, in a standard modified following business day convention, a givenend date (for example a Monday) will correspond to multiple start dates, assuming a constant period forall the forward rates (for example three months). Using the end date as variable to interpolate on wouldcreate an ambiguity. A simple way to resolve this issue is to define the interpolation on the start datesinstead as in Equation (32) . If we start from discrete forward observation dates ( t i ) , and assume a C cubic piecewise polynomialrepresentation on the knots ( t i ) for ¯ f d , the values at t i − k ∆ for 0 ≤ k ≤ floor (cid:16) t i ∆ (cid:17) are also needed toestablish the equivalent C cubic piecewise interpolation in terms of pseudo discount factors, as, bysumming Equation (33), we obtain the relation p ( t ) − p ( t − K ∆ ) = ∆ K ∑ k = ¯ f d ( t − k ∆ ) , (35)with K = floor (cid:0) t ∆ (cid:1) . The value at p ( t − K ∆ ) may be determined by the choice of a specific extrapolationof ¯ f d for t < ∆ . For example, for a constant extrapolation in ¯ f d , p ( t − K ∆ ) = ( t − K ∆ ) f d ( ∆ ) with theconvention p ( ) = p will thus be a C piecewisepolynomial on ( τ j ) j = m = ∪ i = n ∪ ≤ k ≤ floor (cid:16) ti ∆ (cid:17) ( t i − k ∆ ) . Discontinuities in the third derivative of ¯ f d at t i will translate to discontinuities in p (cid:48)(cid:48)(cid:48) at t i − k ∆ for 0 ≤ k ≤ floor (cid:16) t i ∆ (cid:17) . The USD Overnight Index Swap (OIS) curve is typically built from the Fed fund spot rates, Fed fundfutures, and Fed fund OIS swaps quotes (see Table A1 for an example). All those instruments involveonly the discrete one day forward rates. The OIS curve thus allows to capture this one day forward rate,across maturities. If we were to interpolate directly this discrete forward rate, the pricing of a 50y OIS of 13 swap would involve to interpolate the forward rate value every business day during 50 years, as a couponof the OIS swap is calculated by C = N l ∏ j = ( − r ( t j ) δ j ) − Nr par δ ,where N is the notional amount, r ( t ) is the effective OIS rate for the date t , δ j is the accrual period at t j ,which is one day (except around non-business days where the accrual period is adjusted). Finally, r par represents the interest rate of the fixed leg, δ is the coupon period (typically one year) and is defined in theActual/360 daycount convention. We assume that a coupon period consists of l accrual periods. In termsof one day discrete forward rates ˆ f d , the corresponding price for this coupon is C = NP ( T ) (cid:34) l ∏ j = ( + ˆ f d ( t j ) δ j ) − r par δ (cid:35) , (36)where T is the coupon payment date. Using Equation (28), the coupon price may be also expressed interms of pseudo discount factors: C = NP ( T ) (cid:34) l ∏ j = P ( t j ) P ( t j + ) − r par δ (cid:35) .If we neglect the payment lag, the above equation simplifies to C = N [ P ( t ) − r par δ P ( T )] . (37)The pricing of an OIS swap is much more efficient and practical with pseudo-discount factors, as thosedirectly record the accrued rate. It will thus be preferable to interpolate on the pseudo-discount factors,even if the resulting interpolation is less smooth than an interpolation on the discrete one-day forwardrates.Fed fund futures do not involve the compounded rate, but the arithmetic average rate over the futuresperiod. The price F of one contract is F = − R with R being the arithmetic average of daily effectivefunds rates during the contract months: R = ∑ lj = r ( t j ) δ j ∑ lj = δ j . (38)In terms of discount factors the average rate is R = ∑ lj = P ( t j ) P ( t j + ) − ∑ lj = δ j . (39)The equation does not simplify, and all discount factors from the start date to the end date of the futurecontract are needed. It is not problematic since the Fed fund future’s period is one month, and thus less In the settlement, R is rounded to the nearest one-tenth of one basis point. of 13 Time f o r w a r d r a t e (a) 1d forward rate −0.010.000.010.02 0 1 2 3 Time (b) Second derivative
Figure 1.
Fed fund curve using the data from Table A1, zoomed between zero and three years. than 30 discount factors are needed. Note that, contrary to the usual curve instruments , the price of a Fedfund future is not a linear combination of discount factors. As explained in section 2, the usual algorithmto calibrate the yield curve may however still be applied.In Figure 1, we plot the one day forward rate, implied by a cubic spline interpolation of class C onthe pseudo-discount factors. The second derivative of the one day forward rate is flat, and appears to jumpat the knots. The jump represents the change of interpolation in the interval ( t i , t i − ∆ ) from a quadratic toa cubic (see Equation 34): the second derivative is actually continuous, but changes significantly in a shortperiod of time ∆ .
7. Alternative interpolations
What Hagan calls an area preserving spline is also known as a histospline in the literature [11,12].More generally, an interpolation f chosen as to preserve the area (cid:82) t i t i − f ( u ) du = ( t i − t i − ) h i for a givenset of values h i and knots t i is often called an histopolation . Typically, the h i correspond to the values of adiscrete histogram, often a discrete probability density.We can deduce that there is also an equivalence between the quadratic C histospline representationon h i [7, p. 79–81], and the classic cubic spline representation on the values z i = z i − + h i ( t i − t i − ) at knot t i . If the ( h i ) represent a discrete probability density, the ( z i ) correspond then to the discrete cumulativedistribution. The equivalence holds beyond polynomial splines, as shown in [13] for tension splines. This is also the case for the Fed fund basis swap, which also involves an arithmetic average.0 of 13
The harmonic spline of Fritsch and Butland [14] can be applied directly to the forwards by replacingthe forward rates f i of Equation (9) with the following:1 f i = t i − t i − + ( t i + − t i ) ( t i + − t i − ) f di + ( t i − t i − ) + t i + − t i ( t i + − t i − ) f di + . (40)In [4], the above was used only for f di f di + <
0, otherwise f i was set to zero. With the advent of negativerates, it does not necessarily make sense to set f i = Huynh [9] proposes a limiter approach to ensure the monotonicity of a cubic interpolation. Amongthe many limiters proposed, it was found in [4] that the rational limiter was attractive. In terms of forwardrates, this translates to replacing f i of Equation (9) with the following: f i = f di + f di ( f di + + f di )( f di + ) + f di + f di + ( f di ) (41)Again, when the above is used for f di f di + < f i = ( f di + ) f di + f di + ( f di ) ( f di + ) + ( f di ) . (42) As a remedy to the cubic spline oscillations, Lavery [15] proposes to use the optimal spline under the L -norm, that is the cubic piecewise-polynomial p of class C , which minimizes (cid:82) t n t | p (cid:48)(cid:48) ( u ) | du . In contrast,the classic C cubic spline minimizes (cid:82) t n t | p (cid:48)(cid:48) ( u ) | du . By discretizing the integral, the Lavery spline isthe solution of a linear programming problem. While it is more involved numerically, many numericalsoftware libraries offer fast algorithms to solve this kind of problem, for example, GPLK or CBC.Hagan and West [3] provide an interesting innocuous example, corresponding to their Figure 4,where a quartic spline shows major oscillations. Figure 2 shows some undesirable small oscillations when t ∈ [
2, 4 ] for the Bessel, Cubic or Limited splines. The Lavery spline does not oscillate. While we don’tdisplay it here, the smart quadratic interpolation would lead to exactly the same curve as the Bessel spline.
8. Conclusion
The smart quadratic interpolation on the forward rates corresponds exactly to the Bessel-Hermitecubic spline interpolation on the logarithm of discount factors with natural boundary conditions.The area preserving C quadratic interpolation on the instantaneous forward rates correspondsexactly to the C cubic spline interpolation on the logarithm of discount factors with natural boundaryconditions. T Z e r o r a t e Spline
BesselCubicLaveryLimiter
Figure 2.
Zero curve for the data of [3, Figure 4]
We have also shown that a cubic spline interpolation on the logarithm of discount factors translatesto a cubic spline interpolation on the discrete forward rates with a constant accrual period, but usingadditional knots defined in relation with the accrual period. The resulting interpolation will thus be lesssmooth than a direct cubic spline interpolation in terms of discrete forward rates on the forward periodsstart dates. This justifies the industry move towards interpolations in terms of discrete forward rates.Finally, we note that, although the interpolation methods of Hagan and West [3], Hagan [5] arealso defined in terms of discrete forward rates, the latter must be defined on consecutive periods. As aconsequence, the corresponding forward rates periods are, in most cases, not constant. The methods arethus not better adapted to the interpolation of a set of discrete forward rates of constant period, whicharises in the multiple curve framework, than a classical pseudo-discount factors interpolation.As the Hagan interpolation methods directly model the instantaneous forward rate, they becomemore relevant for the case of the OIS curve construction, where the one-day forward rate is very close tothe instantaneous forward rate.While the use of cubic splines is relatively standard for the yield curve construction, if the goal isto produce a very smooth yield curve, it may be more appropriate to consider smoothing splines, andparticularly penalized B-splines (also known as P-splines) or alternatively, some radial basis functioninterpolation. We leave this for further research.
Funding:
This research received no external funding.
Conflicts of Interest:
The authors declare no conflict of interest.1. Ametrano, F.; Bianchetti, M. Bootstrapping the illiquidity .2. Andersen, L.; Piterbarg, V. Interest Rate Modeling–Volume I: Foundations and Vanilla Models.
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