On yield curves of the European central bank.pdf In June 1996, the Bank for International Settlements (BIS, Basel) agreed that the central banks of Europe should submit their data to the BIS for the calculation of zero-coupon yield curves and model parameter estimates. It was found that most European banks use the Nelson - Siegel approach (Italy and Finland) or its modification of Svensson (Belgium, Germany, Spain, Norway, France, Switzerland and Sweden) to model yield curves [1]. Unfortunately, the Nelson - Siegel - Svensson (NSS) approach does not give recommendations on how to determine the parameters of the model (only to evaluate them) and does not explain in any way whether such a model is arbitrage-free. Therefore, it makes sense to consider this problem. 1. Dynamics of parameters change in yield (spot) rate and forward curve Statistics of the Directorate General of the European Central Bank (ECB) publishes the euro zone's yield curves every TARGET working day at 12 noon on Central European time [2]. In the ECB technical note [3] formulas are given for calculating three types of yield curves: the yield curve, the forward curve and the nominal yield curve. Formulas for the first two of them have the form: - yield (spot) rate (1 - e-X/X1 ^ „(1 - e-X/X1 -X/ХЛ „( 1 - e-X/X2 -X/ Л (1) + P --e X / X2 J / / X / X1 J I X / X1 --e-x/X1 7 ( x) = Po +P1 - forward curve f (x) = Po + P1 e-X/X + p2 -e-X/X + P3 - e-X/X2. (2) X1 X2 Here x is a term to maturity, рг-, хг- are the parameters that are estimated on the market data. In Fig. 1 for example, the yield curve and the forward curve from the ECB website on June 1, 2017 presented. On the ECB website the curves are shown for term to maturities x from 0 to 30 years. In contrast to this, in order to show these curves "whole" for the entire range of values of x from 0 to да, on Fig. 1 we used the mapping of the positive semiaxis x e (0, да) of terms to maturity to an unit interval of new variable u e (0, 1) by means of nonlinear transformation u = 1 - e-px. The parameter p was chosen in such a way that the interval of maturities x e (0, 30) corresponds to the values of u e (0, 0.9) , i. e. p = ln10/30 = 0.07675. It is relevant to note here that the curves always start from one common pointy(0 |P) = f (0|P) = r(t), current value of short-term interest rate, and for x ^ да always tend to one common limit value у(да |P) = f (да |P) = p0. Because the yield curve is to be time-varying (as it obviously is) and following [4], [5] let us here recognize that the Nelson-Siegel-Svensson (NSS) parameters p0, pb p2, p3 must be time-varying too. Thus in the NSS yield models the yield curveу(т) = у(т | p0, Pi, p2, p3) and the forward curve J(t) = fir | p0, pb Y,F 1.5 1.0 Fig. 1. The yield curve (the lower curve) and the forward curve (the upper curve) from the ECB website on June 1, 2017. Parameter values: p0 = 1.7810, p1 = - 2.5350, p2 = 23.2948, p3 = - 27.6452, т = 1.5822, т2 = 1.7081 u p2, p3) are determined by following relations 1 - e-JT y(T\p) = p0(T) + p1(t)1-e- + p2(t) ут Л -бт 1 - e -бт --e бт Л 1- е-ут (3) -ут + p3(t) --e ут f (xip> = (т Po(т))' + p1(t) ехр (- ут) + p2(0 ут ехр (- ут) + p3(t) бт ехр (-бт). (4) For convenience, some rename have been made here: у = 1/т1, б = 1/т2. The parameters p0, p1, p2, p3 now depend on the current time t. The dependence of the parameter p0 on the current time is somewhat different: p0(T - t) = p0(x), where T is the maturity date. The need for this will be clarified later. Prime means derivative with respect to the term to maturity т = T - t. In this case the state of the market can be described by a set of parameters {p1(t), p2(t), p3(t)} and each of the parameters considered as a component of the state vector p(t). Whether such yield curves satisfy the no arbitrage conditions? In order to clarify this it at first it is need to determine how the dynamics of the state variables p(t) are described. At present the diffusion stochastic processes are most often used to describe the dynamics of these variables, the mathematical model of which is the stochastic differential equation dp(t) = |(p)dt + c(p)dW(t), where |(p) is a state drift vector, c(p) is a state vector volatility matrix, W(t) is a vector of Wiener processes. In order to the arbitrage opportunities to be absent, it is necessary that the price of the zero-coupon bond P(t, T, p) = exp[- тy (т, p)] on a fixed date t satisfies the so-called term structure equation [6]: £P

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