Article Contents
Article Contents

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

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

• 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.

Mathematics Subject Classification: Primary:91G30;Secondary:60G22.

 Citation:

• Figure 1.  Log-returns index of ITLUP (top) and UBI (bottom)

Figure 2.  Autocorrelation Function (left) and Partial Autocorrelation Function (right) in ITLUP returns. Top to bottom: log returns index, absolute returns, square returns

Figure 3.  Autocorrelation Function (left) and Partial Autocorrelation Function (right) in UBI returns. Top to bottom: log-returns index, absolute returns, square returns

Figure 4.  Time-lagged relations between absolute and squared returns in the indices

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

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
•  [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. [2] E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, (2006). [3] W. Arthur, Complexity Economics: A Different Framework for Economic Thought, Complexity and the Economy, Oxford University Press, (2014). [4] L. Bacheliere, Théorie de la Spéculation, Phd. Thesis, University of Paris, (1900). [5] J. Beran, Statistics for Long Memory Processes, Chapman and Hall, New York, (1994). [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. [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. [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. [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. [10] J. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, (2000). [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. [12] D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, (2006). [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. [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. [15] R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.  doi: 10.1080/713665670. [16] R. Cont, Long range dependence in financial markets, Fractals in Engineering, Springer, (2005), 159–180. doi: 10.1007/1-84628-048-6_11. [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. [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. [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. [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. [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. [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. [23] A. Harvey, Long memory in stochastic volatility, Forecasting Volatility in Financial Markets, (1998), 307–320. doi: 10.1016/B978-075066942-9.50018-2. [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. [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. [26] H. Hurst, The long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 1116 (1951), 770-808. [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. [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. [29] J. Kalemkerian, Parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes, preprint, arXiv: 2004.10369 (2020). [30] A. Kolmogorov, Wienersche Spiralen und Einige Andere Interessante Kurven im Hilbertschen Raum., C. R. Acad. Sci. URSS, (1940), 115–118. [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. [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. [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. [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. [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. [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. [37] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer, (2005). [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. [39] W. Palma, Long Memory Time Series-Theory and Methods, John Wiley, Hoboken, NJ, (2007). doi: 10.1002/9780470131466. [40] W. Petty, Political Arithmetick, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, (1690). [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. [42] P. Theodossiou, Financial data and the skewed generalized T distribution, Management Science, 44 (1998), 1650-1661. [43] G. Uhlenbeck and L. Ornstein, On the theory of Brownian Motion, Physical Review, 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823. [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.

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