Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization

Roya Soltani; Seyed Jafar Sadjadi; Mona Rahnama

doi:10.3934/jimo.2017014

Article Contents

2017, Volume 13, Issue 4: 1701-1721. Doi: 10.3934/jimo.2017014

This issue Previous Article Continuity of approximate solution maps to vector equilibrium problems Next Article An extension of hybrid method without extrapolation step to equilibrium problems

Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization

1.
Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2.
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

* Corresponding author: Roya Soltani
* Corresponding author: Roya Soltani

Received: July 31, 2015

Published: November 2017

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Abstract

This study makes use of the artificial intelligence approaches combined with some nonlinear optimization techniques for optimization of a well-known problem in financial engineering called yield curve. Yield curve estimation plays an important role on making strategic investment decisions. In this paper, we use two well-known parsimonious estimation models, Nelson-Siegel and Extended Nelson-Siegel, for the yield curve estimation. The proposed models of this paper are formulated as continuous nonlinear optimization problems. The resulted models are then solved using some nonlinear optimization and meta-heuristic approaches. The optimization techniques include hybrid GPSO parallel trust region-dog leg, Hybrid GPSO parallel trust region-nearly exact, Hybrid GPSO parallel Levenberg-Marquardt and Hybrid genetic electromagnetism like algorithm. The proposed models of this paper are examined using some real-world data from the bank of England and the results are analyzed.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. PSO solution space

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Figure 2. EM solution space

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Figure 3. Parallel trust regions

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Figure 4. Dogleg trajectory

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Figure 5. Interval plot for Extended Nelson-Siegel model

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Figure 6. Interval plot for Nelson-Siegel model

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Figure 7. Comparing the proposed methods regarding 12 sets of data taken from bank of England

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Figure 8. Comparing the proposed methods regarding 12 sets of data taken from bank of England

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Figure 9. Fitted models versus market data set 1

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Figure 10. Fitted Extended Nelson-Siegel models resulted from HGPSOPLM versus market data

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Table 1. Fitness Error

	Methods
Models	Nelson-Siegel			Extended Nelson-Siegel
	Average	Standard deviation	Average Time (in minutes)	Average deviation	Standard	Average Time (in minutes)
HGEM	0.103029	0.020578	1.607608	0.007307	0.002806582	2.761143
HGPSO	0.103976	0.01973	1.05098	0.012025	0.006613485	1.89916
HGPSOPTR_NE	0.102208	0.023577	29.1853	0.004866667	0.002573407	36.364
HGPSOPTR_DL	0.102167	0.023679	31.3681	0.005108333	0.002877644	37.85
HGPSOPLM	0.102533	0.006366	27.4275	0.004783333	0.002648785	35.74

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Table 2. Parameter values for Extended Nelson-Siegel model

	Variables
Methods	$\beta_{0}$		$\beta_{1}$		$\beta_{2}$		$\beta_{3}$		$\tau_{1}$		$\tau_{2}$
	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation
HGEM	9.496	0.384	-8.982	0.378	-6.656	0.142	-12.387	1.031	2.482	0.061	23.948	0.263
HGPSO	9.470	0.350	-8.978	0.381	-6.585	0.035	-11.786	0.471	2.480	0.078	24.239	0.144
HGPSOPTR-NE	9.796	1.007	-9.324	1.035	-6.688	0.549	-13.333	2.883	2.571	0.162	24.619	1.110
HGPSOPTR-DL	9.917	0.849	-9.448	0.866	-6.738	0.531	-13.666	2.442	2.586	0.117	24.462	1.286
HGPSOPLM	9.957	0.850	-9.491	0.873	-6.725	0.574	-13.796	2.469	2.604	0.111	24.610	1.119

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Table 3. Parameter values for Nelson-Siegel model

	Variables
Methods	$\beta_{0}$		$\beta_{1}$		$\beta_{2}$		$\tau_{1}$
	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation	Average	Standard deviation
HGEM	5.2881	0.0004	-3.8043	0.0053	-5.4705	0.0093	1.5192	0.0013
HGPSO	5.2467	0.0919	-4.5282	0.0732	-5.2269	0.5462	1.6515	0.2967
HGPSOPTR-NE	5.0321	0.0581	-4.3450	0.0458	-4.1469	0.0422	2.2402	0.0073
HGPSOPTR-DL	5.0028	0.9806	-4.3148	0.8265	-4.0892	4.9073	2.273	2.6319
HGPSOPLM	5.0832	0.0581	-4.3976	0.0458	-4.2517	0.0422	2.1824	0.0072

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Table 4. Extended Nelson-Siegel Eigenvalues and Gradient norms

Method	Data	Eigenvalues						Gnorm
	Set1	219.717	7.245	2.530	0.041	0	0.0009	0.0004
	Set2	220.752	7.111	2.748	0.042	0.0001	0.001	0.011
	Set3	214.848	6.942	2.733	0.041	0	0.0009	0.006
	Set4	188.925	5.999	3.557	0.046	0.001	0.0006	0.026
	Set5	198.835	6.254	3.134	0.072	-0.005	0.003	0.861
	Set6	184.056	5.863	2.929	0.039	0.0002	0.0006	0.003
HGPSOPTR-NE	Set7	220.411	6.617	3.233	0.043	0	0.002	0.041
	Set8	201.848	6.490	2.754	0.039	0	0.0007	0.002
	Set9	201.782	6.450	2.712	0.047	-0.002	0.002	0.299
	Set10	192.970	6.112	2.971	0.040	0.0007	0.0001	0.002
	Set11	192.574	6.087	3.086	0.043	-0.001	0.001	0.374
	Set12	206.778	6.637	2.896	0.042	0.0001	0.0009	0.004
	Average	203.624	6.484	2.940	0.045	-0.0005	0.0012	0.136
	Set1	215.781	7.093	2.624	0.042	0	0.0009	0.009
	Set2	217.899	7.074	2.697	0.042	0	0.0009	0.0008
	Set3	210.860	6.806	2.826	0.042	0	0.0009	0.0137
	Set4	208.092	6.804	2.728	0.041	0	0.0008	0.009
	Set5	198.835	6.254	3.134	0.072	-0.005	0.003	0.861
	Set6	180.292	5.633	3.236	0.042	0.0004	0.0007	0.009
HGPSOPTR-DL	Set7	220.411	6.617	3.233	0.043	0	0.002	0.041
	Set8	216.129	6.706	3.019	0.041	0.0001	0.002	0.014
	Set9	194.599	6.178	2.895	0.039	0.0001	0.0007	0.007
	Set10	195.845	6.264	2.811	0.039	0	0.0007	0.0121
	Set11	195.095	6.153	2.995	0.040	0	0.0007	0.0002
	Set12	207.949	6.682	2.881	0.042	0	0.0009	0.002
	Average	205.149	6.522	2.923	0.044	-0.0004	0.001	0.0815
	Set1	225.272	7.332	2.588	0.042	0	0.001	0.003
	Set2	217.686	7.065	2.709	0.042	0	0.0009	0.005
	Set3	212.355	6.863	2.781	0.042	0	0.0009	0.001
	Set4	210.691	6.901	2.684	0.042	0	0.0008	0.004
	Set5	198.835	6.254	3.134	0.072	-0.005	0.003	0.861
	Set6	184.161	5.865	2.933	0.039	0.0002	0.0006	0.008
HGPSOPLM	Set7	220.411	6.617	3.233	0.043	0	0.002	0.041
	Set8	201.916	6.494	2.751	0.039	0	0.0007	0.007
	Set9	195.245	6.211	2.868	0.039	0	0.0007	0.003
	Set10	192.97	6.112	2.971	0.04	0.0001	0.0007	0.002
	Set11	192.575	6.088	3.086	0.043	-0.001	0.001	0.374
	Set12	206.734	6.636	2.895	0.042	0	0.0009	0.002
	Average	204.904	6.536	2.886	0.044	-0.0005	0.001	0.109

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Table 5. Nelson-Siegel Eigenvalues and Gradient norms

Method	Data	Eigenvalues				Gnorm
	Set 1	166.4771	6.6645	3.359	0.0499	0.0032
	Set 2	168.9473	6.8236	3.3593	0.0498	0.0069
	Set 3	167.3736	6.8091	3.3541	0.0489	0.0025
	Set 4	166.0604	6.5357	3.3699	0.0494	0.0033
	Set 5	161.8373	5.9944	3.4018	0.0482	0.0056
	Set 6	144.2963	3.0416	0.3247	0.0013	0.005
HGPSOPTR-NE	Set 7	162.1993	5.941	3.4066	0.0456	0.0081
	Set 8	164.1612	6.2443	3.3904	0.0477	0.0105
	Set 9	162.8706	6.1114	3.3942	0.0468	0.0054
	Set 10	162.2295	5.9982	3.4065	0.0473	0.0022
	Set 11	163.7136	6.2145	3.3885	0.0472	0.0009
	Set 12	167.1193	6.6272	3.3702	0.0495	0.0005
	Average	163.1071	6.0838	3.1271	0.0443	0.0045
	Set1	166.4771	6.6645	3.359	0.0499	0.0032
	Set2	168.9473	6.8236	3.3593	0.0498	0.0069
	Set3	167.3736	6.8091	3.3541	0.0489	0.0025
	Set4	166.0604	6.5357	3.3699	0.0494	0.0033
	Set5	161.8373	5.9944	3.4018	0.0482	0.0056
	Set6	145.1737	2.9842	0.0011	0.3113	0.0065
HGPSOPTR-DL	Set7	162.1993	5.941	3.4066	0.0456	0.0081
	Set8	164.1612	6.2443	3.3904	0.0477	0.0105
	Set9	162.8699	6.1117	3.3942	0.0468	0.0035
	Set10	162.2295	5.9982	3.4065	0.0473	0.0022
	Set11	162.7722	5.7516	3.4069	0.02	1.969
	Set12	167.1193	6.6272	3.3702	0.0495	0.0005
	Average	163.1017	6.0405	3.1017	0.0679	0.1685
	Set1	166.4771	6.6645	3.359	0.0499	0.0032
	Set2	168.9473	6.8236	3.3593	0.0498	0.0069
	Set3	168.368	6.8067	3.3543	0.0488	0.0066
	Set4	166.0697	6.5361	3.3699	0.0493	0.0069
	Set5	161.8373	5.9944	3.4018	0.0482	0.0056
	Set6	142.6002	3.1483	0.002	0.3515	0.0065
HGPSOPLM	Set7	162.1993	5.941	3.4066	0.0456	0.0081
	Set8	164.1612	6.2443	3.3904	0.0477	0.0105
	Set9	162.8699	6.1117	3.3942	0.0468	0.0035
	Set10	162.2295	5.9982	3.4065	0.0473	0.0022
	Set11	163.7136	6.2145	3.3885	0.0472	0.0009
	Set12	167.1193	6.6272	3.3702	0.0495	0.0005
	Average	163.0494	6.0925	3.1002	0.0735	0.0051

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Table 6. Fitness Error for the Extended Nelson-Siegel model (Second quarter in 2015)

	Average	Standard deviation	Average Time (in minutes)
HGEM	0.024759	0.009674	3.0325
HGPSO	0.039497	0.007324	1.1797
HGPSOPTR-NE	0.016541	0.015271	35.3814
HGPSOPTR-DL	0.019875	0.01700	39.1531
HGPSOPLM	0.015778	0.014238	36.332

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Table 7. Comparison between LM results with respect to random, yesterday and PSO results as initial conditions

	Random parameters as Initial condition	Yesterday parameters estimation as Initial condition	PSO results as Initial condition
Date	Final best value	Final best value	Final best value
01/05/2008	0.5486	-	0.0368
02/05/2008	0.0540	0.4362	0.0081
06/05/2008	0.7640	0.5221	0.0489
07/05/2008	0.0534	0.4598	0.0084
08/05/2008	0.0536	0.6287	0.0144
09/05/2008	0.0550	0.6578	0.0166
12/05/2008	0.0126	0.7254	0.0672
13/05/2008	0.0705	0.6465	0.0064
14/05/2008	0.0825	0.4961	0.0037
15/05/2008	0.1040	0.4468	0.0346
16/05/2008	0.2365	0.7188	0.1200
19/05/2008	0.3837	0.4385	0.0742
20/05/2008	0.0914	0.4956	0.0023
21/05/2008	0.2751	0.3631	0.1220
22/05/2008	0.2099	0.2960	0.0838
23/05/2008	0.2504	0.4476	0.1214
27/05/2008	0.1863	0.3923	0.0768
28/05/2008	0.2189	0.3314	0.0869
29/05/2008	0.1662	0.1650	0.0880
30/05/2008	0.1637	0.1691	0.0846

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