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Bond_Curve_Fit_Method__Method

Method refers to List of non linear bond curve fitting methods used to construct a fitted bond discount curve supported by QuantLib.
Available Method types:
CubicB Splines
Corresponds to the CubicBSplinesFitting method in QuantLib.
The resulting discount factor P(T) for maturity T is similar to the one used in
Bond Curve Fit Method::Method::Exponential Splines with the exponentials replaced with spline basis functions.
The number of basis functions equals the number of knots minus 4.
The number of parameters equals the number of basis functions if there is no constrain at 0. Otherwise it is less by 1.
See: McCulloch, J. 1971, "Measuring the Term Structure of Interest Rates." Journal of Business, 44: 19-31
McCulloch, J. 1975, "The tax adjusted yield curve."Journal of Finance, XXX811-30
Warning: The results are extremely sensitive to the number and location of the knot points, and there is no optimal way of selecting them.
James, J. and N. Webber, "Interest Rate Modelling" John Wiley, 2000, pp. 440
Exponential Splines
Corresponds to the ExponentialSplinesFitting method in QuantLib.
The resulting discount factor P(T) for maturity T depends on whether a constrain at 0 is imposed or not.
Without the constrain, it depends on 10 parameters and has the form:
P(T) = α(1)exp(-κt)+α(2)exp(-2κt)+...+α(9)exp(-9κt)
with parameter ordering: α(1), α(2), ..., α(9), κ
With the constrain, it depends on 9 parameters and has the form:
P(T) = βexp(-κt)+α(1)exp(-2κt)+α(2)exp(-3κt)+...+α(8)exp(-9κt)
where β = 1-α(1)-α(2)-...-α(8)
and parameter ordering: α(1), α(2), ..., α(8), κ
See: Li, B., E. DeWetering, G. Lucas, R. Brenner and A. Shapiro (2001): "Merrill Lynch Exponential Spline Model." Merrill Lynch Working Paper
Warning: Convergence may be slow.
Nelson Siegel
Corresponds to the NelsonSiegelFitting method in QuantLib.
The resulting zero rate r(T) for maturity T depends on 4 parameters and has the form:
r(T) = α+(β+γ)[1-exp(-κT)]/(κT)-γ*exp(-κT)
The QuantLib implementation actually replaces the product κT with (κ+ε)*(T+ε), where ε a very small positive number, so that setting κ or T to 0 will not result in infinity.
The parameter ordering is: α, β, γ, κ
See: Nelson, C. and A. Siegel (1985): "Parsimonious modeling of yield curves for US Treasury bills." NBER Working Paper Series, no 1594.
Simple Polynomial
Corresponds to the SimplePolynomialFitting method in QuantLib.
The resulting discount factor P(T) for maturity T depends on whether a constrain at 0 is imposed or not.
Without the constrain, it depends on N+1 parameters, where N is the degree of the polynomial and has the simple polynomial form:
P(T) = α(0)+α(1)T+α(2)T^2+...α(N)T^N ]
with parameter ordering: α(0), α(1), α(2), ..., α(N)
With the constrain, it depends on N parameters as follows:
P(T) = 1+α(1)T+α(2)T^2+...α(N)T^N ]
with parameter ordering: α(1), α(2), ..., α(N)
This is a simple/crude, but fast and robust, means of fitting a yield curve
Corresponds to the SpreadFittingMethod method in QuantLib.
Fits a spread curve on top of a discount function according to given parametric method.
The resulting discount factor P(T) for maturity T is calculated as follows:
P(T) = ƒ(α,T)D(T)/D(T0) ]
where ƒ(α,T) is the discount factor as calculated by the referenced fitting method and D(T) is the discount factor emerged from the given discount curve.
T0 is the reference date used in the construction of the fitted curve.
The denominator D(T0) will be other than 1 only if it T0 differs from the reference date of the discount curve.
The parameter ordering is that of the referenced parametric method.
Svensson
Corresponds to the SvenssonFitting method in QuantLib.
The resulting zero rate r(T) for maturity T depends on 6 parameters and has the form:
r(T) = α+(β+γ)[1-exp(-κT)]/(κT)-γ*exp(-κT)+δ[(1-exp(-λT))/(λT)-exp(-λT)]
The QuantLib implementation actually replaces the productσ κT, λT with (κ+ε)*(T+ε), (λ+ε)*(T+ε), where ε a very small positive number, so that setting κ, λ or T to 0 will not result in infinity.
The parameter ordering is: α, β, γ, δ, κ, λ
See: Svensson, L. (1994). Estimating and interpreting forward interest rates: Sweden 1992-4. Discussion paper, Centre for Economic Policy Research(1051).