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Interest Rate Modeling                        
Page last updated  13/01/2009

bullet Nelson Siegel Yield Curve Model
bullet Nelson Siegel Yield Curve Model with Svensson 1994 Extension
bulletOne-Factor Interest Rate Models (Vasicek. Cox, Ingersoll & Ross)
bulletInterest Rate Trinomial Tree - Hull & White Method


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Nelson-Siegel Yield Curve Model  »download        NEW: Nelson & Siegel with Svensson 1994 extension


The idea of the Nelson-Siegel (N&S) approach is to fit the empirical form of the yield curve with a pre-specified
functional form of the spot rates which is a function of the time to maturity of the bonds (see below).

The Excel model does not just illustrate the model but, as an illustrative example, fits a term structure of spot rates
to the small universe of NZ Government bonds. This can in turn be used to detect over-, respectively underpriced bonds.
(See other examples of bond relative value models on this website)
The Excel workbook uses SOLVER, the built in optimization module. There are some installation requirements
given in the file to run SOLVER and also macros using SOLVER. The N&S parameters are found by minimizing the sum of
squared differences between model and market prices. A more sophisticated version weighs these differences with
the inverse of the duration as proposed in Bliss (1998).

Illustration of MS Excel model
Text Box: Illustration of MS Excel model

References:
Nelson, C. R. & Siegel, A. F. (1987). Parsimonious modeling of yield curves, Journal of Business 60(4): 473—489.
Bliss, R. R. (1997). Testing Term Structure Estimation Methods. Advances in Futures and Options Research(9), 197-231.
Formulas from Van Landschoot, Astrid. The Term Structure of Credit Spreads on Euro Corporate Bonds. Working Paper, CentER,
    Tilburg University. April 2003. Downloaded from http://ideas.repec.org/p/dgr/kubcen/200346.html February 2004.
Note that the formula in this paper has a small typo and just shows the basic Nelson & Siegel where t2 = t1.

 

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Nelson-Siegel Yield Curve Model with Svensson 1994 Extension  »download    


Svensson (1994) extend Nelson-Siegel (1987) model described above with b3 parameter term.

References:
Svensson, L. (1994). Estimating and interpreting forward interest rates: Sweden 1992-4.
    Discussion paper, Centre for Economic Policy Research(1051).
 Anderson, N., Breedon, F., Deacon, M., Derry, A., & Murphy, G. (1996). Estimating and interpreting the yield curve.
    Chichester: John Wiley Series in Financial Economics and Quantitative Analysis. Chapter 2.4.6, pgs. 36-41.

 

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One-Factor Interest Rate Models  »download

Error in calculation of A (missing brackets) in CIR model fixed 26/10/05. Hint from Rafael Nicolas Fermin from the Richard Ivey School of Business, University of Western Ontario

This model introduces to the well-known stochastic one-factor interest rate models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985)
which we call "CIR" in the following. There are three worksheets in this Excel file, "Simulation One-Factor Models" shows discrete versions of
the two models so one gets a feel for their mean-reverting nature of the stochastic processes.
Each time the spreadsheet is recalculated (press F9), a new time series series is generated. The two sheets "Term structure Vasicek/CIR
model" show the the term structure implied by the stochastic process for the spot rates for each model.
 
Note that Vasicek interest rate may actually become negative unlike CIR where the random term becomes increasingly smaller as the rate approaches zero (multiplication with square root of r):
      Vasicek discrete version                                                             CIR discrete version
               
with
r spot rate
s instantaneous standard deviation of short rate
a pull-back factor
b long-term equilibrium of short-term rates
Dt a small time increment

The pictures below show how rates will be distributed in the very long-run, i.e. their probability distribution after the effect of the initial spot rate r at t=0 has vanished. Both distributions have a mean of  b (the long-term equilibrium). Yet under Vasicek they are normally distributed and so there is a probability for them being negative. For the CIR model this density function has the property of a Gamma distribution, not allowing for negative interest rates.

Model parameters for above distributions: a = 0.15, b = 6%, s =0.02 (Vasicek)/0.05(CIR)

References: 
Vasicek, O. 1977 "An Equilibrium Characterization of the term structure." Journal of Financial Economics 5: 177-188.
Cox, Ingersoll, and Ross. "A Theory of the Term Structure of Interest Rates". Econometrica, 53 (1985). 385-407.
Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. P. 567f

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Interest Rate Trinomial Tree - Hull & White Method  »download 
Update 17 August 2005: removed an error in normal model in the calculation of alpha thanks to a hint from
Ricard Caballero Montesso,  a specialist in financial mathematics and in particular interest rate models.

This model presents the general tree-building procedure as proposed by Hull and White for a more general group of one-factor models.
It handles spot interest rate processes that follow of the following general form: 

      With constants a (pullback factor of mean reversion process) and s the volatility parameter of Wiener process dz
for f(r) = r and f(r) = ln(r) At time t the process reverts to a level q(t)/a. q(t) is set to fit the current term structure of interest.

It is a property of these models that they can be fitted to any term structure of interest.

Normal Model
Lognormal Model


Below is a screenshot of the Excel/VBA model. Note the threshold factor 0.184 which was proposed by Hull & White to avoid unnecessary sampling of unlikely interest rate scenarios. It basically determines the maximum number of interest rate steps in the grid. Obviously the model as it stands now has limited practical use but it can now easily be extended to price interest rate sensitive instruments like bonds and interest rate options. This is done by rolling the payoff of the instrument back trough the tree, discounting at the interest rates determined at each node. To understand the basic idea, refer to Cox, Ross & Rubinstein Binomial Tree in this website for an example of equity option valuation in a binomial tree.

 

Reference:
Adapted from Hull, John C., Options, Futures & Other Derivatives. Fourth edition (2000). Prentice-Hall. p. 580. 
Literature:
Hull, J. White, A. "Numerical Procedures for Implementing Term Structure Models I: Single factor models", Journal of Derivatives 2, 1 (1994), 7-16.
Hull, J. White, A. "Using Hull-White Interest Rate Trees", Journal of Derivatives, (Spring 1996), 26-36.f

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Copyright © January, 2009 Kurt Hess, University of Waikato
Last modified: 13-Jan-2009