American Institute of Mathematical Sciences

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July  2020, 7(3): 225-237. doi: 10.3934/jdg.2020016

Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices

Juan Kalemkerian 1, and  Andrés Sosa 2,

1. 

Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225, CP 11400, Montevideo, Uruguay
2. 

Instituto de Estadística, Facultad de Ciencias Económicas y de Administración, Universidad de la República, Eduardo Acevedo 1139, CP 11200, Montevideo, Uruguay

We are grateful to the Guest Editors, Viktoriya Semeshenko, Gabriel Brida and Andrea Roventini, and the two anonymous referees for helpful comments and suggestions.

Received  February 2020 Revised  April 2020 Published  July 2020

One consequence of the fact that a large number of agents with different behaviors operate in financial systems is the emergence of certain statistical properties in some time series. Some of these properties contradict the hypotheses that are established in the traditional models of efficient market and portfolio optimization. Among them is the long-range dependence that is the objective of this work. The approach is proposed by fractional calculus, as a generalization of the classic approach to financial markets through semi-martingales. This paper study the existence of this property in variables dependent on the term structure curves of Uruguayan sovereign debt after the 2002 economic crisis.

Keywords: Long-range dependence, fractional calculus, financial markets, volatility, sovereign debt.
Mathematics Subject Classification: Primary:91G30;Secondary:60G22.
Citation: Juan Kalemkerian, Andrés Sosa. Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices. Journal of Dynamics & Games, 2020, 7 (3) : 225-237. doi: 10.3934/jdg.2020016
References:
[1]

M. Alatriste-Contreras, J. Brida and M. Puchet, Structural change and economic dynamics: Rethinking from the complexity approach, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019). doi: 10.3934/jdg.2019007.  Google Scholar

[2]

E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, (2006).  Google Scholar

[3]

W. Arthur, Complexity Economics: A Different Framework for Economic Thought, Complexity and the Economy, Oxford University Press, (2014). Google Scholar

[4]

L. Bacheliere, Théorie de la Spéculation, Phd. Thesis, University of Paris, (1900).  Google Scholar

[5]

J. Beran, Statistics for Long Memory Processes, Chapman and Hall, New York, (1994).  Google Scholar

[6]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, (2006). doi: 10.1007/978-1-84628-797-8.  Google Scholar

[7]

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[8]

T. Bollerslev, Generalized autoregressive conditional heterocedasticity, History of Economic Thought Books, McMaster University Archive for Journal of Econometrics, 31 (1986), 307-327.  doi: 10.1016/0304-4076(86)90063-1.  Google Scholar

[9]

T. Bollerslev and H. Mikkelsen, Modeling and pricing long memory in stock market volatility, Journal of Econometric, 73 (1996), 151-184.  doi: 10.1016/0304-4076(95)01736-4.  Google Scholar

[10]

J. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, (2000).  Google Scholar

[11]

F. Breidt, N. Crato and P. de Lima, The detection and estimation of long memory in stochastic volatility, Journal of Econometrics, 83 (1998), 325-348.  doi: 10.1016/S0304-4076(97)00072-9.  Google Scholar

[12]

D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, (2006).  Google Scholar

[13]

P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck Processes, Electronic Journal of Probability, 8 (2003), 1-14.  doi: 10.1214/EJP.v8-125.  Google Scholar

[14]

A. Chronopoulou and F. Viens, Estimation and pricing under long-memory stochastic volatility, Annals of Finance, 8 (2012), 379-403.  doi: 10.1007/s10436-010-0156-4.  Google Scholar

[15]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.  doi: 10.1080/713665670.  Google Scholar

[16]

R. Cont, Long range dependence in financial markets, Fractals in Engineering, Springer, (2005), 159–180. doi: 10.1007/1-84628-048-6_11.  Google Scholar

[17]

F. Engle, Autoregressive conditional heterocedasticity whit estimates of the variance of United Kingdom inflation, History of Economic Thought Books, McMaster University Archive for Econometrica, 50 (1982), 987-1008.  doi: 10.2307/1912773.  Google Scholar

[18]

E. Ghysels, A. Harvey and E. Renault, Stochastic volatility, Handbook of Statistics, 14 (1996), 119-191.  doi: 10.1016/S0169-7161(96)14007-4.  Google Scholar

[19]

A. Gloter and M. Hoffmann, Stochastic volatility and fractional Brownian motion, Stochastic Processes and their Applications, 113 (2004), 143-172.  doi: 10.1016/j.spa.2004.03.008.  Google Scholar

[20]

C. Granger and R. Joyeux, An introduction to long memory time series models and fractional differencing, Journal of Time Series Analysis, 1 (1980), 15-29.  doi: 10.1111/j.1467-9892.1980.tb00297.x.  Google Scholar

[21]

W. Granger, S. Spear and Z. Ding, Statistics and Finance: An Interface. Stylized Facts on the Temporal and Distributional Properties of Absolute Returns: An Update, Imperial College Press, London, (2000), 97–120. Google Scholar

[22]

T. Graves, R. Gramacy, N. Watkins and C. Franzke, A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980, Entropy, 19 (2017). doi: 10.3390/e19090437.  Google Scholar

[23]

A. Harvey, Long memory in stochastic volatility, Forecasting Volatility in Financial Markets, (1998), 307–320. doi: 10.1016/B978-075066942-9.50018-2.  Google Scholar

[24]

J. Hasslett and E. Raftery, Space-time modelling with long memory dependence: Assessing Irelands win power resource, Journal of Applied Statistics, 38 (1989), 1-50.  doi: 10.2307/2347679.  Google Scholar

[25]

S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, Society for Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[26]

H. Hurst, The long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 1116 (1951), 770-808.   Google Scholar

[27]

B. Jacobsen, Long term dependence in stock returns, Journal of Empirical Finance, 3 (1996), 393-417.  doi: 10.1016/S0927-5398(96)00009-6.  Google Scholar

[28]

J. Kalemkerian and J. R. León, Fractional iterated Ornstein-Uhlenbeck processes, Latin American Journal of Probability and Mathematical Statistics, 16 (2019), 1105-1128.  doi: 10.30757/ALEA.v16-41.  Google Scholar

[29]

J. Kalemkerian, Parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes, preprint, arXiv: 2004.10369 (2020). Google Scholar

[30]

A. Kolmogorov, Wienersche Spiralen und Einige Andere Interessante Kurven im Hilbertschen Raum., C. R. Acad. Sci. URSS, (1940), 115–118.  Google Scholar

[31]

S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019). doi: 10.3934/jdg.2019011.  Google Scholar

[32]

L. Lima and Z. Xiao, Is there long memory in financial time series?, Applied Financial Economics, 20 (2013), 487-500.  doi: 10.1080/09603100903459733.  Google Scholar

[33]

T. Lux, Long-term stochastic dependence in financial prices: Evidence from the German stock market, Applied Economics Letters, 3 (1996), 701-706.  doi: 10.1080/135048596355691.  Google Scholar

[34]

B. Mandelbrot, Une Classe Processus Stochastiques Homothétiques a soi: Application a la loi Climatologique H. E. Hurst, C.R. Acad. Sci. Paris, (1965), 3274–3277.  Google Scholar

[35]

B. Mandelbrot and J. Wallis, Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resour. Res., 5 (1969), 967-988.   Google Scholar

[36]

R. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 22 (1974), 449-470.  doi: 10.1142/9789814759588_0003.  Google Scholar

[37]

M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer, (2005).  Google Scholar

[38]

H. Niu and J. Wang, Volatility clustering and long memory of financial time series and financial price model, Digital Signal Processing, 23 (2013), 489-498.  doi: 10.1016/j.dsp.2012.11.004.  Google Scholar

[39]

W. Palma, Long Memory Time Series-Theory and Methods, John Wiley, Hoboken, NJ, (2007). doi: 10.1002/9780470131466.  Google Scholar

[40]

W. Petty, Political Arithmetick, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, (1690). Google Scholar

[41]

M. Taqqu, M. Teverovsky and W. Willinger, Estimators for long range dependence: An empirical study, Fractals, 3 (1995), 785-788.  doi: 10.1142/S0218348X95000692.  Google Scholar

[42]

P. Theodossiou, Financial data and the skewed generalized T distribution, Management Science, 44 (1998), 1650-1661.   Google Scholar

[43]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian Motion, Physical Review, 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.  Google Scholar

[44]

R. Weron, Estimating long-range dependence: Finite sample properties and confidence intervals, Physica A: Statistical Mechanics and its Applications, 312 (2002), 285-299.  doi: 10.1016/S0378-4371(02)00961-5.  Google Scholar

show all references

References:
[1]

M. Alatriste-Contreras, J. Brida and M. Puchet, Structural change and economic dynamics: Rethinking from the complexity approach, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019). doi: 10.3934/jdg.2019007.  Google Scholar

[2]

E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, (2006).  Google Scholar

[3]

W. Arthur, Complexity Economics: A Different Framework for Economic Thought, Complexity and the Economy, Oxford University Press, (2014). Google Scholar

[4]

L. Bacheliere, Théorie de la Spéculation, Phd. Thesis, University of Paris, (1900).  Google Scholar

[5]

J. Beran, Statistics for Long Memory Processes, Chapman and Hall, New York, (1994).  Google Scholar

[6]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, (2006). doi: 10.1007/978-1-84628-797-8.  Google Scholar

[7]

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[8]

T. Bollerslev, Generalized autoregressive conditional heterocedasticity, History of Economic Thought Books, McMaster University Archive for Journal of Econometrics, 31 (1986), 307-327.  doi: 10.1016/0304-4076(86)90063-1.  Google Scholar

[9]

T. Bollerslev and H. Mikkelsen, Modeling and pricing long memory in stock market volatility, Journal of Econometric, 73 (1996), 151-184.  doi: 10.1016/0304-4076(95)01736-4.  Google Scholar

[10]

J. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, (2000).  Google Scholar

[11]

F. Breidt, N. Crato and P. de Lima, The detection and estimation of long memory in stochastic volatility, Journal of Econometrics, 83 (1998), 325-348.  doi: 10.1016/S0304-4076(97)00072-9.  Google Scholar

[12]

D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, (2006).  Google Scholar

[13]

P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck Processes, Electronic Journal of Probability, 8 (2003), 1-14.  doi: 10.1214/EJP.v8-125.  Google Scholar

[14]

A. Chronopoulou and F. Viens, Estimation and pricing under long-memory stochastic volatility, Annals of Finance, 8 (2012), 379-403.  doi: 10.1007/s10436-010-0156-4.  Google Scholar

[15]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.  doi: 10.1080/713665670.  Google Scholar

[16]

R. Cont, Long range dependence in financial markets, Fractals in Engineering, Springer, (2005), 159–180. doi: 10.1007/1-84628-048-6_11.  Google Scholar

[17]

F. Engle, Autoregressive conditional heterocedasticity whit estimates of the variance of United Kingdom inflation, History of Economic Thought Books, McMaster University Archive for Econometrica, 50 (1982), 987-1008.  doi: 10.2307/1912773.  Google Scholar

[18]

E. Ghysels, A. Harvey and E. Renault, Stochastic volatility, Handbook of Statistics, 14 (1996), 119-191.  doi: 10.1016/S0169-7161(96)14007-4.  Google Scholar

[19]

A. Gloter and M. Hoffmann, Stochastic volatility and fractional Brownian motion, Stochastic Processes and their Applications, 113 (2004), 143-172.  doi: 10.1016/j.spa.2004.03.008.  Google Scholar

[20]

C. Granger and R. Joyeux, An introduction to long memory time series models and fractional differencing, Journal of Time Series Analysis, 1 (1980), 15-29.  doi: 10.1111/j.1467-9892.1980.tb00297.x.  Google Scholar

[21]

W. Granger, S. Spear and Z. Ding, Statistics and Finance: An Interface. Stylized Facts on the Temporal and Distributional Properties of Absolute Returns: An Update, Imperial College Press, London, (2000), 97–120. Google Scholar

[22]

T. Graves, R. Gramacy, N. Watkins and C. Franzke, A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980, Entropy, 19 (2017). doi: 10.3390/e19090437.  Google Scholar

[23]

A. Harvey, Long memory in stochastic volatility, Forecasting Volatility in Financial Markets, (1998), 307–320. doi: 10.1016/B978-075066942-9.50018-2.  Google Scholar

[24]

J. Hasslett and E. Raftery, Space-time modelling with long memory dependence: Assessing Irelands win power resource, Journal of Applied Statistics, 38 (1989), 1-50.  doi: 10.2307/2347679.  Google Scholar

[25]

S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, Society for Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[26]

H. Hurst, The long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 1116 (1951), 770-808.   Google Scholar

[27]

B. Jacobsen, Long term dependence in stock returns, Journal of Empirical Finance, 3 (1996), 393-417.  doi: 10.1016/S0927-5398(96)00009-6.  Google Scholar

[28]

J. Kalemkerian and J. R. León, Fractional iterated Ornstein-Uhlenbeck processes, Latin American Journal of Probability and Mathematical Statistics, 16 (2019), 1105-1128.  doi: 10.30757/ALEA.v16-41.  Google Scholar

[29]

J. Kalemkerian, Parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes, preprint, arXiv: 2004.10369 (2020). Google Scholar

[30]

A. Kolmogorov, Wienersche Spiralen und Einige Andere Interessante Kurven im Hilbertschen Raum., C. R. Acad. Sci. URSS, (1940), 115–118.  Google Scholar

[31]

S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019). doi: 10.3934/jdg.2019011.  Google Scholar

[32]

L. Lima and Z. Xiao, Is there long memory in financial time series?, Applied Financial Economics, 20 (2013), 487-500.  doi: 10.1080/09603100903459733.  Google Scholar

[33]

T. Lux, Long-term stochastic dependence in financial prices: Evidence from the German stock market, Applied Economics Letters, 3 (1996), 701-706.  doi: 10.1080/135048596355691.  Google Scholar

[34]

B. Mandelbrot, Une Classe Processus Stochastiques Homothétiques a soi: Application a la loi Climatologique H. E. Hurst, C.R. Acad. Sci. Paris, (1965), 3274–3277.  Google Scholar

[35]

B. Mandelbrot and J. Wallis, Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resour. Res., 5 (1969), 967-988.   Google Scholar

[36]

R. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 22 (1974), 449-470.  doi: 10.1142/9789814759588_0003.  Google Scholar

[37]

M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer, (2005).  Google Scholar

[38]

H. Niu and J. Wang, Volatility clustering and long memory of financial time series and financial price model, Digital Signal Processing, 23 (2013), 489-498.  doi: 10.1016/j.dsp.2012.11.004.  Google Scholar

[39]

W. Palma, Long Memory Time Series-Theory and Methods, John Wiley, Hoboken, NJ, (2007). doi: 10.1002/9780470131466.  Google Scholar

[40]

W. Petty, Political Arithmetick, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, (1690). Google Scholar

[41]

M. Taqqu, M. Teverovsky and W. Willinger, Estimators for long range dependence: An empirical study, Fractals, 3 (1995), 785-788.  doi: 10.1142/S0218348X95000692.  Google Scholar

[42]

P. Theodossiou, Financial data and the skewed generalized T distribution, Management Science, 44 (1998), 1650-1661.   Google Scholar

[43]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian Motion, Physical Review, 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.  Google Scholar

[44]

R. Weron, Estimating long-range dependence: Finite sample properties and confidence intervals, Physica A: Statistical Mechanics and its Applications, 312 (2002), 285-299.  doi: 10.1016/S0378-4371(02)00961-5.  Google Scholar

Figure 1.  Log-returns index of ITLUP (top) and UBI (bottom)
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Figure 2.  Autocorrelation Function (left) and Partial Autocorrelation Function (right) in ITLUP returns. Top to bottom: log returns index, absolute returns, square returns
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Figure 3.  Autocorrelation Function (left) and Partial Autocorrelation Function (right) in UBI returns. Top to bottom: log-returns index, absolute returns, square returns
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Figure 4.  Time-lagged relations between absolute and squared returns in the indices
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Table 1.  ITLUP index
Method Estimation Parameter (square) Parameter (absolute)
Simple $ R/S $ 0.7308 0.7489
Corrected R over S 0.8407 0.8633
Empirical Hurst exponent 0.7703 0.7876
Corrected empirical Hurst 0.7158 0.7342
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Method Estimation Parameter (square) Parameter (absolute)
Simple $ R/S $ 0.7308 0.7489
Corrected R over S 0.8407 0.8633
Empirical Hurst exponent 0.7703 0.7876
Corrected empirical Hurst 0.7158 0.7342
Table 2.  UBI index
Method Estimation Parameter (square) Parameter (absolute)
Simple $ R/S $ 0.6987 0.7679
Corrected R over S 0.7887 0.8892
Empirical Hurst exponent 0.7481 0.8304
Corrected empirical Hurst 0.6953 0.7776
Table Options

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Method Estimation Parameter (square) Parameter (absolute)
Simple $ R/S $ 0.6987 0.7679
Corrected R over S 0.7887 0.8892
Empirical Hurst exponent 0.7481 0.8304
Corrected empirical Hurst 0.6953 0.7776
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