Jumps in the Volatility of Financial Markets

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9912

PERRON, Benoît

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Département de sciences économiques

Université de Montréal

Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221

Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 0999.

This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 0999.

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CAHIER 9912

JUMPS IN THE VOLATILITY OF FINANCIAL MARKETS

Benoit PERRON1

1

Centre de recherche et développement en économique (C.R.D.E.) and Département de sciences économiques, Université de Montréal

October 1999

_________________

The author would like to thank Peter Phillips, Oliver Linton, Donald Andrews, seminar participants at Yale, and participants at the 1998 Canadian Econometric Study Group for their helpful comments. Inmaculada Fiteni has gratefully shared some of her programs. The author assumes full responsibility for any remaining errors. Financial assistance from an Alfred P. Sloan Dissertation Fellowship and from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

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RÉSUMÉ

Des études récentes suggèrent que la variance conditionnelle des rendements financiers est sujette à des sauts. Ce papier étend une procédure non paramétrique de détection de sauts développée par Delgado et Hidalgo (1996) à la détection de sauts dans les moments conditionnels supérieurs, en particulier la variance conditionnelle. Les résultats de simulation démontrent que cette procédure estime de façon raisonnable le nombre de sauts ainsi que leurs emplacements. L'application de cette procédure aux rendements journaliers sur l'indice S&P 500 révèle la présence de plusieurs sauts dans la variance conditionnelle.

Mots clés : sauts, variance conditionnelle, noyau, fenêtre unidirectionnelle

ABSTRACT

Recent work suggests that the conditional variance of financial returns may exhibit sudden jumps. This paper extends a non-parametric procedure to detect discontinuities in otherwise continuous functions of a random variable developed by Delgado and Hidalgo (1996) to higher conditional moments, in particular the conditional variance. Simulation results show that the procedure provides reasonable estimates of the number and location of jumps. This procedure detects several jumps in the conditional variance of daily returns on the S&P 500 index.

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(21)

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(22)

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(23)

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(24)

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(25)

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(26)

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2 '

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6

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(27)

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E5& 1 E 5 E2' EAKRn

2s E5

& 2E62nE5& 6n E5&nJR

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E5& 1 E 5~ E2' JR

EA KRn32

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Surri1 Gluhfw iurp Sursrvlwlrq 8 lq Ghojdgr dqg Klgdojr EbbS1

(28)

Wdeoh 41 Sdudphwhu ydoxhv iru vlpxodwlrq h{shulphqw Ydoxhv hvwlpdwhg iurp V)S 833 uhwxuqv/

Mdqxdu| 5qg/ 4<;30Ghfhpehu 64vw/ 4<<9 GJS 4 GJS 5 > Eff 3137<7 313796 {B Ef, <1343< ;14559 {Q Ef, 3 081<4:6 Q 0 31:379 { Ef, ₃ _314;5< 0 316959 k 3 3139;7 q 3 31;;;9 57

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Table 2. Frequency distribution o f estimated number o f jumps ht 9.0109 10 5 True number o f jumps is 0 T 1000, 500 replications p 0 p 1 p 2 p 3 c 0.8 1 1.2 cv c 0.8 1 1.2 cv c 0.8 1 1.2 cv c 0.8 1 1.2 cv M 0 100.0 100.0 100.0 100.0 11.2 39.2 61.8 69.2 0.0 2.6 31.6 32.6 0.0 0.6 16.0 17.4 1 0.0 0.0 0.0 0.0 20.8 23.2 20.4 23.0 1.6 20.4 28.0 38.0 0.0 9.4 32.2 34.8 2 0.0 0.0 0.0 0.0 28.0 23.0 14.8 6.8 16.2 55.4 37.0 25.8 25.0 72.4 50.4 46.4 3 0.0 0.0 0.0 0.0 23.0 12.6 3.0 1.0 66.8 21.6 3.4 3.6 70.4 17.6 1.4 1.4 4 0.0 0.0 0.0 0.0 15.6 2.0 0.0 0.0 15.4 0.0 0.0 0.0 4.6 0.0 0.0 0.0 K 5 0.0 0.0 0.0 0.0 1.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Incomplete 0.0 0.0 0.0 0.0 25.0 15.8 11.0 4.0 81.6 55.5 33.8 22.2 90.6 76.2 51.4 44.4 Mean 0.000 0.000 0.000 0.000 2.152 1.150 0.590 0.396 2.960 1.960 1.122 1.004 2.796 2.070 1.372 1.318

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Table 3. Frequency distribution o f estimated number o f jumps ht Ft 1 ) * 0.0684 yt 1 6 2 0.8886 ht 1 Ft 0 1 1 t T K A1 2 1 t T K A2 True number o f jumps is 2 T 1000, 500 replications p 0 p 1 p 2 p 3 c 0.8 1 1.2 cv c 0.8 1 1.2 cv c 0.8 1 1.2 cv c 0.8 1 1.2 cv M 0 53.0 46.6 27.6 28.6 0.8 1.2 0.2 0.4 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.2 1 13.0 23.8 45.8 43.4 8.4 13.6 20.8 19.0 2.0 12.2 20.4 21.6 1.6 10.2 33.8 32.4 2 5.4 6.4 12.8 12.0 13.2 30.8 52.2 53.8 16.6 53.8 64.0 66.6 33.8 74.6 63.0 62.6 3 6.2 14.8 13.4 8.2 32.2 45.2 26.8 25.8 58.4 33.6 15.6 11.2 59.6 15.2 3.2 4.8 4 10.2 8.2 0.4 4.8 40.4 9.2 0.0 1.0 23.0 0.4 0.0 0.2 5.0 0.0 0.0 0.0 K 5 12.2 0.2 0.0 3.0 5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Incomplete 15.4 16.4 14.2 7.6 63.0 61.6 64.6 66.8 80.2 75.2 76.6 72.2 88.8 83.2 73.6 75.4 Mean 1.442 1.148 1.132 1.262 3.180 2.476 2.056 2.080 3.024 2.222 1.952 1.892 2.680 2.050 1.694 1.720

(31)

Table 4. Results from estimation o f jumps in conditional variance S& P 500 returns, January 2nd, 1980-December 31st, 1996 Bandwidth constant chosen b y cross-validation p 01 2 3 Date Size 10 7 p-value Date Size 10 4 p-value Date Size 10 4 p-value Date Size 10 3 p-value 0.4099 861218 6.878 0.001 0.3829 860707 32.786 0.000 0.4652 871127 129.787 0.000 0.6229 900803 25.152 0.000 0.5399 890307 -0.452 0.013 0.6262 900823 -20.709 0.000 0.6988 911115 -0.038 0.012 0.1800 830121 -3.358 0.421 0.2038 830617 2.712 0.536 0.7720 930216 4.099 0.000 0.1549 820819 0.184 0.358 0.1703 821122 -0.066 0.184 c 1.2 0.9 1.2 1.1

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(33)

(34)

(35)