Information Theoretic Ranking of Extreme Value Returns

Abstract

This study highlights the importance of the information contained extreme value ratios (or returns) in the volatility estimation of financial assets. Most popular extreme value estimators like Parkinson (Journal of Business, 61–65, 1980), Garman Klass (Journal of business, 67–78, 1980), Rogers Satchell (The Annals of Applied Probability, 504–512, 1991) and Yang Zhang (The Journal of Business, 73 (3), 477–492, 2000) use a subset of all available extreme value ratios but not the full set. We examine if there are other extreme value ratios which contain more information than the most widely used ratios. This study shows empirically how much information is contained in various extreme value ratios of financial assets, using both real and simulated data. Using information theory, we find out their variability in relation to a uniform distribution in each quarter. We then rank them using the Kullback–Leibler metric (in accordance with a scoring methodology we developed in this study) to ascertain which set of ratios are more variable than others and thus may provide better estimation in computing volatility. We also calculate the rank of the matrix to identify the set of linearly independent ratios, for ascertaining the number of ratios that would be enough to generate a class of volatility estimators. The empirical results demonstrate that the need for incorporating other ratios in volatility estimation. We also observe that each dataset has other more informative ratios which are uniquely attributed to that dataset.

This is a preview of subscription content, access via your institution.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

  1. 1.

    We could not include the ranking scores of all the ratios due to brevity. However it can be made available on request.

References

  1. Alizadeh, S., M. Brandt, and F.X. Diebold. 2002. Range-Based Estimation of Stochastic Volatility Models. Journal of Finance 57: 1047–1092.

    Article  Google Scholar 

  2. Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedas- ticity. Journal of Econometrics 31 (3): 307–327.

    Article  Google Scholar 

  3. Beckers, S. 1983. Variances of security price returns based on high, low, and closing prices. Journal of Business 56 (1): 97–112.

    Article  Google Scholar 

  4. Ding, Z., and C. Granger. 1996. Varieties of long memory models. Journal of Econometrics 73 (1): 61–67.

    Article  Google Scholar 

  5. Engle, R. 1982. Autoregressive Conditional Heteroscedasticity with Esti- mates of the Variance of United Kingdom Inflation. Econometrica 50 (4): 987–1007.

    Article  Google Scholar 

  6. Garman, M.B., and M.J. Klass. 1980. On the estimation of security price volatilities from historical data. Journal of business 53 (1): 67–78.

    Article  Google Scholar 

  7. Horst, E.T., A. Rodriguez, H. Gzyl, and G. Molina. 2012. Stochastic volatil- ity models including open, close, high and low prices. Quantitative Finance 12 (2): 199–212.

    Article  Google Scholar 

  8. Kullback, S., and R.A. Leibler. 1951. On information and sufficiency. The Annals of Mathematical Statistics 22 (1): 79–86.

    Article  Google Scholar 

  9. Mandelbrot, B. 1963. The Variation of Certain Speculative Prices. The Journal of Business 36 (4): 394–419.

    Article  Google Scholar 

  10. Markowitz, H. 1952. Portfolio Selection. The Journal of Finance 7 (1): 77–81.

    Google Scholar 

  11. Meilijson, I. (2008). The garman-klass volatility estimator revisited. arXiv preprint arXiv:0807.3492.

  12. Ong, M.A. 2015. An information theoretic analysis of stock returns, volatility and trading volumes. Applied Economics 47 (36): 3891–3906.

    Article  Google Scholar 

  13. Ozturk, H., U. Erol, and A. Yuksel. 2016. Extreme value volatility estimators and realized volatility of istanbul stock exchange: Evidence from emerging market. International Journal of Economics and Finance 8 (8): 71.

    Article  Google Scholar 

  14. Parkinson, M. 1980. The extreme value method for estimating the variance of the rate of return. Journal of business 53 (1): 61–65.

    Article  Google Scholar 

  15. Philippatos, G.C., and C.J. Wilson. 1972. Entropy, market risk, and the selection of efficient portfolios. Applied Economics 4 (3): 209–220.

    Article  Google Scholar 

  16. Rogers, L.C.G., and S.E. Satchell. 1991. Estimating variance from high, low and closing prices. The Annals of Applied Probability 1 (4): 504–512.

    Article  Google Scholar 

  17. Ross, S.A. (1981). The Recovery Theorem. NBER Working Paper No. 17323.

  18. Samuelson, P.A. 1970. The Fundamental Approximation Theorem of Portfolio Analysis in terms of Means, Variances and Higher Moments. The Review of Economic Studies 37 (4): 537–542.

    Article  Google Scholar 

  19. Shu, J., and J.E. Zhang. 2006. Testing range estimators of historical volatil- ity. Journal of Futures Markets 26 (3): 297–313.

    Article  Google Scholar 

  20. Taylor, S.J. 1986. Modeling financial time series. New York: Wiley.

    Google Scholar 

  21. Todorova, N., and S. Husmann. 2012. A comparative study of range-based stock return volatility estimators for the german market. Journal of Futures Markets 32 (6): 560–586.

    Article  Google Scholar 

  22. Yang, D., and Q. Zhang. 2000. Drift-independent volatility estimation based on high, low, open, and close prices. The Journal of Business 73 (3): 477–492.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Parthajit Kayal.

Appendices

Appendix

Appendix A: Score Tables for Real Data

See appendix Tables 12, 13, 14, 15, 16.

Table 12 Scores for Crude Oil Simple Ratios
Table 13 Scores for 20 most informative Crude Oil derived ratios
Table 14 Scores for Gold Futures Simple Ratios
Table 15 Scores for 20 most informative Gold Futures derived ratios
Table 16 Scores for Crude Oil Simple Ratios

Appendix B: Score Tables for Simulated Data

See appendix Tables 16, 17, 18, 19.

Table 17 Scores for 20 most informative Crude Oil derived ratios
Table 18 Scores for Gold Futures Simple Ratios
Table 19 Scores for 20 most informative Gold Futures derived ratios

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kayal, P., Dutta, S., Khandelwal, V. et al. Information Theoretic Ranking of Extreme Value Returns. J. Quant. Econ. 19, 1–21 (2021). https://doi.org/10.1007/s40953-020-00214-y

Download citation

Keywords

  • Extreme value estimators
  • Information theory
  • Volatility

JEL Classification

  • G10
  • G140
  • G150
  • G170