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Various stochastic volatility models have been designed to model the variance of the asset price. Among these various models, the Heston model, as one-factor stochastic volatility mode, is the most popular and easiest to implement. Unfortunately, recent findings indicate that existing Heston modelis not able to characterize all aspects of asset returns, such as persistence, mean reverting, and clustering. Therefore, a modified Heston model is proposed in this paper. Compared with the original Heston model, the mean-reverting Cox Ingersoll and Ross process is modified to include a cosine term with the intention of capturing the periodicity of volatility. The phenomenon that high-volatile period is interchanged with low-volatile periods can thus be better described by adding such a period term to the volatility process. In addition, the geometric Brownian motion is replaced by a random walk in the presence of a cubic nonlinearity proposed by Bonanno et al. By doing so, a financial market with two different dynamical regimes (normal activity and extreme days) can be modeled. Closed-form solution for the modified Heston model is not derived in this paper. Instead, Monte-Carlo simulation is used to generate the probability density function of log-return which is then compared with the historical probability density function of stock return. Parameters are adjusted and estimated so that the square errors can be minimized. Daily returns of all the component stocks of Dow-Jones industrial index for the period from 3 September 2007 to 31 December 2008 are used to test the proposed model, and the experimental results demonstrate that the proposed model works very well. The mean escape time and mean periodic escape rate of the proposed modified Heston model with periodic stochastic volatility are studied in this paper with two different dynamical regimes like financial markets in normal activity and extreme days. Also the theoretical results of mean escape time and mean periodic escape rate can be calculated by numerical simulation. The experimental results demonstrate that 1) larger value of rate of return, smaller long run average of variance and smaller magnitude of periodic volatility will reduce the mean periodic escape rate, and thus stabilize the market; 2) by analyzing the mean escape time, an optimal value can be identified for the magnitude of periodic volatility which will maximize the mean escape time and again stabilize the market. In addition, an optimal rate of relaxation to long-time variance, smaller frequency of the periodic volatility, larger rate of return, and stronger correlation between noises will furtherreduce the mean escape time and enhance the market stability.
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Keywords:
- periodic volatility /
- mean escape time /
- escape rate /
- econophysics
[1] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[2] Mei D C, Du L C, Wang C J 2009 J. Stat. Phys. 137 625
[3] Du L C, Mei D C 2012 Eur. Phys. J. B 85 1
[4] Jia Y, Zheng X, Hu X, Li J 2001 Phys. Rev. E 63 293
[5] Zhang X Y, Xu W, Zhou B C 2011 Acta Phys. Sin. 60 060514 (in Chinese)[张晓燕, 徐伟, 周丙常 2011 60 060514]
[6] Cao L, Wu D J 2007 Europhys. Lett. 61 593
[7] Shen C S, Zhang J Q, Chen H S 2007 Acta Phys. Sin. 56 6315 (in Chinese)[申传胜, 张季谦, 陈含爽 2007 56 6315]
[8] Spagnolo B, Valenti D, Fiasconaro A 2004 Math. Biosci. Eng. 1 185
[9] Valenti D, Fiasconaro A, Spagnolo B 2004 Physica A 331 477
[10] Bonanno G, Spagnolo B, Valenti D 2008 Int. J. Bifurcat. Chaos 18 2775
[11] Bonanno G, Spagnolo B 2005 Fluct. Noise Lett. 5 325
[12] Jia Z L 2008 Physica A 387 6247
[13] Yoshimoto M, Shirahama H, Kurosawa S 2008 J. Chem. Phys. 129 014508
[14] Zeng C H, Zhang C, Zeng J K, Liu R F, Wang H 2015 J. Stat. Mech. 2015 08027
[15] Krawiecki A, Holyst J A 2003 Physica A 317 597
[16] Babinec P 2002 Chaos 13 1767
[17] Li J C, Mei D C 2013 Phys. Rev. E 88 012811
[18] Bonanno G, Valenti D, Spagnolo B 2007 Phys. Rev. E 75 016106
[19] Bonanno G, Valenti D, Spagnolo B 2006 Eur. Phys. J. B 53 405
[20] Valenti D, Spagnolo B, Bonanno G 2007 Physica A 382 311
[21] Masoliver J, Perell J 2009 Phys. Rev. E 80 016108
[22] Masoliver J, Perell J 2008 Phys. Rev. E 78 056104
[23] Zhou W X 2007 Introduction to Econophysics (Shanghai:Shanghai University of Finance & Economics Press) pp1-14 (in Chinese)[周炜星 2007 金融物理学导论 (上海:上海财经大学出版社) 第1–14 页]
[24] Yalama A, Celik S 2013 Econ. Model. 30 67
[25] Baaquie B E 1997 J. Phys. I 7 1733
[26] Angelovska J 2010 VaR based on SMA, EWMA and GARCH(1, 1) Volatility Models (Germany:VDM Verlag Dr. Müller) pp1-5
[27] Andersen T G, Tim B, Diebold F X, Paul L 2001 Econometrica 71 579
[28] Bouchaud J P, Potters M 2000 Mpra. Paper 285 18
[29] Gencay R, Dacorogna M, Muller U A, Pictet O, Olsen R 2001 An Introduction to High-Frequency Finance (America:Academic Press) pp1-10
[30] Bollerslev T 1986 J. Econom. 31 307
[31] Ding Z, Granger C W J, Engle R F 1993 J. Empir. Financ. 1 83
[32] Bansal R, Kiku D, Shaliastovich I, Yaron A 2014 J. Financ. 69 2471
[33] Jebabli I, Arouri M, Teulon F 2014 Energ. Econ. 45 66
[34] Heston S L 1993 Rev. Financ. Stud. 6 327
[35] Forde M, Jacquier A, Lee R 2012 SIAM J. Financ. Math. 3 690
[36] Drǎgulescu A A, Yakovenko V M 2002 Quant. Financ. 2 443
[37] Poon S H, Granger C W J 2003 J. Econ. Literature 41 478
[38] Fouque J P, Papanicolaou G, Sircar R, Solna K 2004 Financ. Stoch. 8 451
[39] Cox J C, Ingersoll Jr J E, Ross S A 1985 Econometrica 385
[40] Lux T, Marchesi M 2000 IJTAF 03 675
[41] Gopikrishnan P M, Martin Amaral, Nunes L, Stanley H E 1998 Eur. Phys. J. B 3 139
[42] Lillo F, Mantegna R N 2000 Phys. Rev. E 62 6126
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[1] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[2] Mei D C, Du L C, Wang C J 2009 J. Stat. Phys. 137 625
[3] Du L C, Mei D C 2012 Eur. Phys. J. B 85 1
[4] Jia Y, Zheng X, Hu X, Li J 2001 Phys. Rev. E 63 293
[5] Zhang X Y, Xu W, Zhou B C 2011 Acta Phys. Sin. 60 060514 (in Chinese)[张晓燕, 徐伟, 周丙常 2011 60 060514]
[6] Cao L, Wu D J 2007 Europhys. Lett. 61 593
[7] Shen C S, Zhang J Q, Chen H S 2007 Acta Phys. Sin. 56 6315 (in Chinese)[申传胜, 张季谦, 陈含爽 2007 56 6315]
[8] Spagnolo B, Valenti D, Fiasconaro A 2004 Math. Biosci. Eng. 1 185
[9] Valenti D, Fiasconaro A, Spagnolo B 2004 Physica A 331 477
[10] Bonanno G, Spagnolo B, Valenti D 2008 Int. J. Bifurcat. Chaos 18 2775
[11] Bonanno G, Spagnolo B 2005 Fluct. Noise Lett. 5 325
[12] Jia Z L 2008 Physica A 387 6247
[13] Yoshimoto M, Shirahama H, Kurosawa S 2008 J. Chem. Phys. 129 014508
[14] Zeng C H, Zhang C, Zeng J K, Liu R F, Wang H 2015 J. Stat. Mech. 2015 08027
[15] Krawiecki A, Holyst J A 2003 Physica A 317 597
[16] Babinec P 2002 Chaos 13 1767
[17] Li J C, Mei D C 2013 Phys. Rev. E 88 012811
[18] Bonanno G, Valenti D, Spagnolo B 2007 Phys. Rev. E 75 016106
[19] Bonanno G, Valenti D, Spagnolo B 2006 Eur. Phys. J. B 53 405
[20] Valenti D, Spagnolo B, Bonanno G 2007 Physica A 382 311
[21] Masoliver J, Perell J 2009 Phys. Rev. E 80 016108
[22] Masoliver J, Perell J 2008 Phys. Rev. E 78 056104
[23] Zhou W X 2007 Introduction to Econophysics (Shanghai:Shanghai University of Finance & Economics Press) pp1-14 (in Chinese)[周炜星 2007 金融物理学导论 (上海:上海财经大学出版社) 第1–14 页]
[24] Yalama A, Celik S 2013 Econ. Model. 30 67
[25] Baaquie B E 1997 J. Phys. I 7 1733
[26] Angelovska J 2010 VaR based on SMA, EWMA and GARCH(1, 1) Volatility Models (Germany:VDM Verlag Dr. Müller) pp1-5
[27] Andersen T G, Tim B, Diebold F X, Paul L 2001 Econometrica 71 579
[28] Bouchaud J P, Potters M 2000 Mpra. Paper 285 18
[29] Gencay R, Dacorogna M, Muller U A, Pictet O, Olsen R 2001 An Introduction to High-Frequency Finance (America:Academic Press) pp1-10
[30] Bollerslev T 1986 J. Econom. 31 307
[31] Ding Z, Granger C W J, Engle R F 1993 J. Empir. Financ. 1 83
[32] Bansal R, Kiku D, Shaliastovich I, Yaron A 2014 J. Financ. 69 2471
[33] Jebabli I, Arouri M, Teulon F 2014 Energ. Econ. 45 66
[34] Heston S L 1993 Rev. Financ. Stud. 6 327
[35] Forde M, Jacquier A, Lee R 2012 SIAM J. Financ. Math. 3 690
[36] Drǎgulescu A A, Yakovenko V M 2002 Quant. Financ. 2 443
[37] Poon S H, Granger C W J 2003 J. Econ. Literature 41 478
[38] Fouque J P, Papanicolaou G, Sircar R, Solna K 2004 Financ. Stoch. 8 451
[39] Cox J C, Ingersoll Jr J E, Ross S A 1985 Econometrica 385
[40] Lux T, Marchesi M 2000 IJTAF 03 675
[41] Gopikrishnan P M, Martin Amaral, Nunes L, Stanley H E 1998 Eur. Phys. J. B 3 139
[42] Lillo F, Mantegna R N 2000 Phys. Rev. E 62 6126
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