Volatility Forecasting Using Support Vector Regression and a Hybrid Genetic Algorithm
Volatility forecasting is an important process required to measure variability in equity prices, risk management, and several other financial activities. Generalized autoregressive conditional heteroscedastic methods <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(\textit{GARCH})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="italic">GARCH</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> have been used to forecast volatility with reasonable success due unreal assumptions about volatility underlying process. Recently, a supervised learning machine called support vector regression <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(SVR)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mi>V</mi> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> has been employed to forecast financial volatility. Nevertheless, the quality and stability of the model obtained through <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$SVR$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <mi>R</mi> </mrow> </math> </EquationSource> </InlineEquation> training process depend strongly on the selection of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$SVR$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <mi>R</mi> </mrow> </math> </EquationSource> </InlineEquation> parameters. Typically, these are tuned by a grid search method <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$(SVR_{GS})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>; however, this tuning procedure is prone to get trapped on local optima, requires a priori information, and it does not concurrently tune the kernels and its parameters. This paper presents a new method called <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$SVR_{GBC}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> for the financial volatility forecasting problem which selects simultaneously the proper kernel and its parameter values. <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$SVR_{GBC}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> is a hybrid genetic algorithm which uses several genetic operators to enhance the exploration of solutions space: it introduces a new genetic operator called Boltzmann selection, and the use of several random number generators. Experimental data correspond to two ASEAN and two latinoamerican market indexes. <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$SVR_{GBC}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> results are compared against <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\textit{GARCH}\left( 1,1\right) \hbox { and }SVR_{GS}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">GARCH</mi> <mfenced close=")" open="(" separators=""> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mfenced> <mspace width="0.333333em"/> <mtext>and</mtext> <mspace width="0.333333em"/> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>S</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> method. It uses the mean absolute percentage error and directional accuracy functions for measuring quality results. Experimentation shows that, in general, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$SVR_{GBC}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation> overcomes quality of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\textit{GARCH}\left( 1,1\right) \hbox { and }SVR_{GS}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">GARCH</mi> <mfenced close=")" open="(" separators=""> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mfenced> <mspace width="0.333333em"/> <mtext>and</mtext> <mspace width="0.333333em"/> <mi>S</mi> <mi>V</mi> <msub> <mi>R</mi> <mrow> <mi>G</mi> <mi>S</mi> </mrow> </msub> </mrow> </math> </EquationSource> </InlineEquation>. 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