A Hybrid Particle Swarm Optimization-Nelder- Mead Algorithm (PSO-NM) for Nelson-Siegel- Svensson Calibration Sofia Ayouche, Rachid Ellaia, Rajae Aboulaich Abstract—Today, insurers may use the yield curve as an indicator evaluation of the profit or the performance of their portfolios; therefore, they modeled it by one class of model that has the ability to fit and forecast the future term structure of interest rates. This class of model is the Nelson-Siegel-Svensson model. Unfortunately, many authors have reported a lot of difficulties when they want to calibrate the model because the optimization problem is not convex and has multiple local optima. In this context, we implement a hybrid Particle Swarm optimization and Nelder Mead algorithm in order to minimize by least squares method, the difference between the zero-coupon curve and the NSS curve. Keywords—Optimization, zero-coupon curve, Nelson-Siegel- Svensson, Particle Swarm Optimization, Nelder-Mead Algorithm. I. I NTRODUCTION A NY investor is exposed to rate risk, and he should anticipate rate movements in the future. Constructing a term structure of interest rates most often refers to the concept of zero-coupon. The zero coupon curves are calculated through the yield curves of the market and are used mainly for evaluating financial contracts. The calibration of the zero coupon curve consists in reconstructing the yield curve using data observed in the market. This reconstruction is necessary due to the fact that there is not enough zero-coupon bonds (strips) listed on the market. In addition, zero coupon bonds often have less liquidity than coupon bonds. In this context, various curve fitting methods have been introduced. The most popular approaches to the term structure modeling are various curve fitting spline methods initiated by McCulloch [12] and who fit a cubic Spline to the discount curve and also Vasicek and Fong [15] who model the discount curve with exponential Spline. Indeed Bliss and Fama [1] developed an iterative method for fitting the forward rate curves, sometimes called unsmoothed Fama-Bliss. Hence, these methods have been criticized for not having economic properties. Therefore, Nelson and Siegel [13] and Svensson [14] proposed parametric curves that are flexible enough to describe the whole family of observed term structure shapes. Despite the absence of the no-arbitrage restriction, the function developed by Nelson Siegel and its augmented version by Svensson matches the yield curve quite well and is widely used by many central banks for yield curve modeling. S. Ayouche, R. Ellaia, and R. Aboulaich are in Laboratory of Study and Research in Applied Mathematics, LERMA, Mohammed V University in Rabat, Mohammadia School of Engineers, BP 765, Ibn Sina avenue, Agdal, Rabat, Morocco (e-mail: sofia.ayouche@gmail.com). In this paper, we restrict ourselves to the Nelson-Siegle-Svensson model in order to reconstruct and forecast the term structure of interest rates, and this can help the investor to better manage a portfolio of products rate. Many authors founded a lot of difficulties in calibrating the model since the function is not convex, especially those who use methods based on derivatives of the objective function. Gilli, Grosse and Schumann [5] analyze the calibration of the model and implement and test an optimization heuristic, Differential Evolution, to obtain parameters. Differential Evolution gives solutions that fit the data very well. Gimeno et al. [6] proposed the use of genetic algorithms as an alternative optimization methodology to the traditional methods. They find better results than traditional methods. We use the hybrid particle swarm optimization and Nelder Mead algorithm to calibrate the model using Moroccan Government bonds. We presented a modified PSO using a direct search complex algorithm to control the PSO heuristic parameters. The paper is structured as follows: Section II discusses in detail the calibration of the NSS model using heuristic optimization methods. Section III concludes and presents new perspectives on the basis of work done. II. CALIBRATING THE NELSON-SIEGLE-SVENSSON MODEL TO CONSTRUCT THE YIELD CURVE A. Description of the NSS Model The Nelson-Siegel [13] and its extension developed by Svensson [14] is a parametric model, which was designed to describe the movement of the entire range of rates and to reconstruct the yield curve. Moreover, it is a dynamic method that uses parameters changing in time. These parameters are estimated to a high level of accuracy and are considered as factors that match the level, slope and the interest rate curve of government bonds. Unlike the model of Nelson-Siegel, the Svensson can fit different shapes of yield curve that can be found on the market, especially curve with a bump or a hollow. See for instance [4]. The resulting Nelson-Siegel approximating forward curve can be assumed to be the solution to a second order differential equation with equal roots for spot rates f t (τ,β)= β 0 + β 1 exp  - t τ 1  + (1) β 2 t τ 1 exp  - t τ 1  + β 3 t τ 2 exp  -t/τ 2  World Academy of Science, Engineering and Technology International Journal of Economics and Management Engineering Vol:10, No:4, 2016 1365 International Scholarly and Scientific Research & Innovation 10(4) 2016 ISNI:0000000091950263 Open Science Index, Economics and Management Engineering Vol:10, No:4, 2016 publications.waset.org/10004553/pdf