# Bond pricing based on nelson siegel model

Licencing Abstract Our paper aims to model the yield curve that corresponds to a graphical representation of the yields offered by the bonds of the same issuer according to their maturity, from the shortest to the longest expiration date in the Tunisian bond market TBM.

To get to our objective, we will compare the Nelson-Siegel modeling strategy, which is most often used for the analysis and the hedging of the interest rate risk of portfolios with known flows in bond pricing based on nelson siegel model, to the Svensson modeling strategy, which is the extension of the Nelson-Siegel model. Our sampling data statistically support the evidence that the more appropriate yield curve for the TBM is that estimated by the Nelson-Siegel model.

Introduction Our paper aims to better understand the behavior of the yield curve in order to be able to create a more efficient and reliable one.

Our yield curve will be essentially based on the Nelson Siegel and the Svensson models since those are widely used amongst the financial institutions and adapted to less liquid and less developed markets similar to the Tunisian bond market TBM. The Nelson-Siegel and Svensson models are zero-coupon parametric models with the advantage of taking into account the different deformations of the yield curves, to allow a dynamic analysis of the market with time-varying parameters that are estimated from market data, and represent the curve by a smooth surface.

In March 2016, TC started publishing a yield curve for treasury bills inspired by the practices of different markets around the world. While working with the available information on TC website, a gap has appeared. The data used to make the yield curve reflect the whole market for intra primary dealer operations and all the operations that are out of the market which leads us to think that TC yield curves may be biased, so it is mandatory to point out a yield curve which is more reliable and efficient.

To do so, we will estimate the yield curve via Nelson-Siegel and Svensson models. The remainder of our paper is organized as follows. Section 2 deals with the literature review. Section 3 points out the yield curve modeling framework.

Section 4 copes with the empirical methodology. In section 5, we run the yield curve estimations. Section 6 exhibits results and findings. Finally, in section 7, we conclude. Literature Review The term interest rate structure or yield curve is the function that, combines on a given date, the level of the associated interest rate for each maturity.

Market curves are constructed directly from the market quotations. It is about the swap curve and yield curve of government bonds. Implied curves are constructed indirectly from the market quotations for instruments such as bonds and swaps zero-coupon ZC yield curve, forward yield curve, instant forward yield curve and finally yield curve at par.

The literature provides two methods to construct a ZC curve; the bond pricing approaches and approaches to yields. An abundant literature exists on the methods of construction of the yield curve.

They can be grouped into two main groups: Other researchers have proposed to group them into three categories: First, based on the spline functions. Second, those that a priori postulate a function class and finally, non-parametric techniques Roncalli 2.

In this paper, we propose to group the methods of construction of the yield curve into 4 categories: The traditional method of constructing a yield curve is to represent the rate of return of a series of bonds by maturity. Regression models respond to this method.

In these models, bond yields are a function of several explanatory variables maturities, coupon rates. The parameters are estimated by minimizing the squared error between theoretical yield derived from the model and the bond yield observed in the bond markets.

In this category, it is important to mention the the work of Mcenally 3Dobbie and Wilkie 4Dobbie and Wilkie 5Bolder and Streliski 6.

- The underlying model is expressed by the following equation 1;
- Given the pragmatic objectives of this research, the analysis focuses on the practical and deals with two key problems;
- To get to our objective, we will compare the Nelson-Siegel modeling strategy, which is most often used for the analysis and the hedging of the interest rate risk of portfolios with known flows in practice, to the Svensson modeling strategy, which is the extension of the Nelson-Siegel model;
- The literature provides two methods to construct a ZC curve; the bond pricing approaches and approaches to yields;
- It highlights how markets and prices coordinate economic activities.

The second category is composed of empirical models. The general idea is to adjust the discount factor by an appropriate mathematical function and then extract the parameters.

These are obtained by minimizing the squared deviation between the theoretical price derived from the model and the price of the bonds observed on the bond markets. Several mathematical forms exist in the literature to adjust the discount factor. This is expressed experimentally by large oscillations of the interpolation polynomial, even if the function is very simple. Hence the idea of interpolating by polynomial functions by bits, of which the degree does not increase with the number of interpolation points.

## Yield Curve Modelling at the Bank of Canada

As one wishes to interpolate by functions more differentiable than X1, we look for the interpolant in the space of the polynomial class functions, of degree k on each interval [ ] called spline functions the terminology was introduced by Schoenberg in 1946 Pansu 8. These methods have been criticized for having undesirable economic properties and are often perceived as "black boxes". The third category is composed of parametric models.

The most used parametric model is the Nelson and Siegel 10 model. Since its appearance, it has been adopted by many experts from the professional world. It is used by many central banks, policy makers and fixed-income portfolio managers. Also, the Nelson-Siegel model has known a great bond pricing based on nelson siegel model in academic research. The Nelson-Siegel model has the advantage of being parsimonious and its parameters have an economic significance.

However, it still presents some disadvantages. Indeed, it does allow to reconstitute all forms of yield curves that can be found on the market. It lacks the bond pricing based on nelson siegel model of adjustment for maturities over 7 years. The Nelson-Siegel model class is in practice most often used for the analysis and the hedging of the interest rate risk of portfolios with known flows.

The Svensson model is the extension of the Nelson-Siegel model to allow an additional curvature Pierre-E. The work of Roncalli 2 showed the existence of two classes of models rates for valuation of financial assets: In the other hand, the theory of general equilibrium studies the allocation of resources in the context of a market economy with perfect competition.

It highlights how markets and prices coordinate economic activities. The general equilibrium models propose theories about the nature of the stochastic process that interest rates follow. The fourth category is composed of the bootstrapping models. The bootstrapping method is a procedure for reconstructing a ZC curve step-by-step, that means maturity segment by maturity segment. The price of the second bond take an equation with two unknowns the first coupon to be discounted at the first year's rate and the second coupon with the second year's rateThe rate for the first year being determined through the first bond, we can find without worry the rate of the second year, and then we can find all the ZC rates.

The big problem with this method is that we have to have the bonds with successive maturities to be able to do the bootstrapping, something that is not always evident in the emerging markets Annaert, Anouk G. More recently, we observe models for the construction of yield curves that takes into account its dynamic character.

- It is used by many central banks, policy makers and fixed-income portfolio managers;
- Finally, although it was decided to employ the Svensson model, there are other functional forms that could be more stable or better describe the underlying data;
- Finally, in section 7, we conclude;
- The Nelson-Siegel and Svensson models are zero-coupon parametric models with the advantage of taking into account the different deformations of the yield curves, to allow a dynamic analysis of the market with time-varying parameters that are estimated from market data, and represent the curve by a smooth surface;
- Section 2 deals with the literature review.

We can enumerate first the principal component analysis models PCA ; when several more than two quantitative variables are studied simultaneously. The difficulty comes from the fact that the individuals studied are no longer represented in a plan, space of dimension 2, but in a space of larger dimension e.

The objective of the Principal Component Analysis PCA is to return to a space of reduced size for example 2 by deforming the least possible reality. It is therefore to obtain the most relevant summary of the initial data. Mathematically, PCA is a simple base change: The Nelson-Siegel dynamic, and the Functional Signal plus Noise FSN are proposed to analyze the dynamics of a wide range of yields or asset prices in which observations are contemporary functionally related.

Bowsher and Roland Meeks 18. Among the authors who used the PCA as a technique for estimating the yield curve zero, we can cite the work of Litterman and Scheinkman 35.

- Other researchers have proposed to group them into three categories;
- Other interesting methods of estimating the yield curve are worth noting;
- In the absence of a developed literature dealing with the practical side of parametric term structure estimation, this paper provides some guidance for those wishing to use parametric models under "real world" constraints.

In the same context, Leoni 21 discusses in detail the applications of the PCA in the determination of interest rates. Diebold and Li 23 introduce a dynamic version of the model of Nelson Siegel, while Bowsher and Meeks 18 applies the "Functional Signal plus Noise FSN "to model and predict the zero-rate curve from treasury.

Other interesting methods of estimating the yield curve are worth noting.

Pennacchi 24Babbs and Nowman 25and Babbs and Nowman 26 use the Kalman filter to estimate gaussian affine models.

Lund 27 uses a non-linear filter to estimate an affine model from bonds with coupons. Finally, Brenner et al. The Yield Curve Modeling Framework 3. The Nelson-Siegel Model 1987 The Nelson-Siegel model class is in practice most often used for the analysis and the hedging of the interest rate risk of portfolios with known flows. In fact the Nelson-Siegel model is distinguished by its dynamic analysis with parameters that change over time and by its simple analysis with few parameters to estimate, consistent with illiquid and less developed markets.

It is used for the construction of a smooth surface curve and also for the modeling of different deformations of the curve level, slope, curvature. The underlying model is expressed by the following equation 1: