Annals of Operations Research

, Volume 262, Issue 2, pp 493–518 | Cite as

Equilibrium-based volatility models of the market portfolio rate of return (peacock tails or stotting gazelles)

  • David FeldmanEmail author
  • Xin Xu
S.I.: Financial Economics


We introduce a theoretical and empirical method of studying equilibrium-consistent volatility models. We implement it with the market portfolio’s return, which is central to financial risk management. Within an equilibrium framework, we study two families of such models. One is deterministic volatility, represented by current popular models. The other is in the “constant elasticity of variance” family, in which we propose new models. Theoretically, we show that, together with constant expected returns, the latter family tends to have better ability to forecast. Empirically, our proposed models, while as easy to implement as the popular ones, outperform them in three out-of-sample forecast evaluations of different time periods, by standard predictability criteria. This is true particularly during high-volatility periods, whether the market rises or falls.


Market risk Volatility model Systematic risk Market portfolio Predictive power Equilibrium GARCH RiskMetrics Piecewise constant volatility Constant elasticity of variance 

JEL Classification

G17 G12 C58 


  1. Allen, L., Boudoukh, J., & Saunders, A. (2003). Understanding market, credit, and operational risk: The value at risk approach. Hoboken: Wiley.Google Scholar
  2. Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39, 885–905.CrossRefGoogle Scholar
  3. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71, 579–625.CrossRefGoogle Scholar
  4. BCBS. (2011). Messages from the academic literature on risk measurement for the trading book. Working paper no. 19, Basel Committee on Banking Supervision.Google Scholar
  5. Berndt, E. K., Hall, B. H., Hall, R. E., & Hausman, J. A. (1974). Estimation inference in nonlinear structural models. Annals of Economic and Social Measurement, 4, 653–665.Google Scholar
  6. Bick, A. (1990). On viable diffusion price processes of the market portfolio. The Journal of Finance, 45, 673–689.CrossRefGoogle Scholar
  7. Black, F. (1976). Studies in stock price volatility changes.’ In Proceedings of the 1976 business meeting of the business and economic statistics section (pp. 177–181). American Statistical Association.Google Scholar
  8. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. The Journal of Econometrics, 31, 307–327.CrossRefGoogle Scholar
  9. Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52, 5–59.CrossRefGoogle Scholar
  10. Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. The Journal of Political Economy, 96, 116–131.CrossRefGoogle Scholar
  11. Boudoukh, J., Richardson, M., & Whitelaw, R. (1997). An investigation of a class of volatility forecasting models. Journal of Derivatives, 4, 63–71.CrossRefGoogle Scholar
  12. Christoffersen, P. F., & Diebold, F. X. (2000). How relevant is volatility forecasting for financial risk management? The Review of Economics and Statistics, 82, 12–22.CrossRefGoogle Scholar
  13. Christoffersen, P. F., Hahn, J., & Inoue, A. (2001). Testing and comparing value-at-risk measures. Journal of Empirical Finance, 8, 325–342.CrossRefGoogle Scholar
  14. Clogg, C. C., Petkova, E., & Haritou, A. (1995). Statistical methods for comparing regression coefficients between models. American Journal of Sociology, 100, 1261–1293.CrossRefGoogle Scholar
  15. Corradi, V. (2000). Reconsidering the continuous time limit of the GARCH(1,1) process. Journal of Econometrics, 96, 145–153.CrossRefGoogle Scholar
  16. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985a). An intertemporal general equilibrium model of asset prices. Econometrica, 51, 363–383.CrossRefGoogle Scholar
  17. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985b). A theory of the term structure of interest rates. Econometrica, 53, 385–407.CrossRefGoogle Scholar
  18. Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166.CrossRefGoogle Scholar
  19. Crouhy, M., Galai, D., & Mark, R. (2001). Risk management. New York: McGraw-Hill.Google Scholar
  20. Diebold, F. X., & Mariano, R. S. (2002). Comparing predictive accuracy. Journal of Business and Economic Statistics, 20, 134–144.CrossRefGoogle Scholar
  21. Dothan, M. U., & Feldman, D. (1986). Equilibrium interest rates and multiperiod bonds in a partially observable economy. The Journal of Finance, 41, 369–382.CrossRefGoogle Scholar
  22. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1007.CrossRefGoogle Scholar
  23. Engle, R. F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics, 17, 425–446.CrossRefGoogle Scholar
  24. Feldman, D. (1989). The term structure of interest rates in a partially observable economy. The Journal of Finance, 44, 789–812.CrossRefGoogle Scholar
  25. Feldman, D. (2007). Incomplete information equilibria separation theorems and other myths. Annals of Operations Research, Special Issue on Financial Modeling, 151, 119–149.CrossRefGoogle Scholar
  26. Fornari, F., & Mele, A. (2006). Approximating volatility diffusions with CEV-ARCH models. Journal of Economic Dynamics and Control, 30, 931–966.CrossRefGoogle Scholar
  27. Friedman, M. (1953). Essays in positive economics. Chicago: The University of Chicago Press.Google Scholar
  28. Gujarati, D. N. (2003). Basic econometrics (4th ed.). New York: McGraw-Hill.Google Scholar
  29. Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH(1, 1)? Journal of Applied Econometrics, 20(7), 873–889.CrossRefGoogle Scholar
  30. Heath, D. P., & Platen, E. (2002). Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In J. Yong (Ed.), Recent developments in mathematical finance (pp. 117–126). Singapore: World Scientific.Google Scholar
  31. He, H., & Leland, H. (1993). On equilibrium asset price processes. The Review of Financial Studies, 6, 593–617.CrossRefGoogle Scholar
  32. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343.CrossRefGoogle Scholar
  33. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatility. Journal of Finance, 3, 281–300.CrossRefGoogle Scholar
  34. Jorion, P. (2007). Value at risk: The new benchmark for managing financial risk (3rd ed.). New York: McGraw-Hill.Google Scholar
  35. Liptser, R. S., & Shiryaev, A. N. (2001). Statistics of random processes, I. General theory (2nd ed.). Berlin: Springer.CrossRefGoogle Scholar
  36. Litterman, R., & Winkelmann, K. (1998). Estimating covariance matrices, risk management series. New York: Goldman, Sachs & Co.Google Scholar
  37. Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica, 46, 1429–1445.CrossRefGoogle Scholar
  38. Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373–413.CrossRefGoogle Scholar
  39. Merton, R. (1973). An intertemporal capital asset pricing model. Econometrica, 41, 867–887.CrossRefGoogle Scholar
  40. Mincer, J., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In J. Mincer (Ed.), Economic forecasts and expectations. Cambridge, MA: National Bureau of Economic Research.Google Scholar
  41. Morgan/Reuters, J. P. (1996). RiskMetrics \(^{TM}\)Technical document (4th ed.). New York: Morgan Guaranty Trust Company and Reuters Ltd.Google Scholar
  42. Nelson, D. B. (1990). ARCH models as diffusion approximation. Journal of Econometrics, 45, 7–38.CrossRefGoogle Scholar
  43. Park, B.-J. (2014). Time-varying, heterogeneous risk aversion and dynamics of asset prices among boundedly rational agents. Journal of Banking and Finance, 43, 150–159.CrossRefGoogle Scholar
  44. Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. The Journal of Econometrics, 160, 246–256.CrossRefGoogle Scholar
  45. Phelan, M. J. (1995). Probability and statistics applied to the practice of financial risk management: The case of JP Morgan’s riskmetrics \(^{TM}\). Working paper, Wharton School, the University of Pennsylvania.Google Scholar
  46. Platen, E., & Semmler, W. (2012). A dynamic portfolio approach to asset markets and monetary policy. Stochastic processes, finance and control: A festschrift in honor of Robert J. Elliott, pp. 347–373, Advances in statistics, probability and actuarial sciences, Vol. 1. Singapore: World Scientific.Google Scholar
  47. Platen, E. (2011). A benchmark approach to investing and pricing. In L. MacLean, E. Thorp, & W. T. Ziemba (Eds.), The Kelly capital growth investment criterion. World Scientific handbook in financial economics series (Vol. 3, pp. 409–426). Singapore: World Scientific.CrossRefGoogle Scholar
  48. Schorfheide, F., Song, D., & Yaron, A. (2013). Identifying long-run risks: A Bayesian mixed-frequency approach. Working paper, University of Pennsylvania.Google Scholar
  49. Singleton, K. J. (2006). Empirical dynamic asset pricing: Model specification and econometric assessment. Princeton: Princeton University Press.Google Scholar
  50. Wiener, Z. (1999). Introduction to VaR, risk management and regulation in banking. New York: Kluwer Academic.Google Scholar
  51. Xu, X. (2013). Modeling risk of the multi-period market portfolio: An equilibrium-based approach. PhD thesis, University of New South Wales, Australia. The link for the thesis is:
  52. Zahavi, A., & Zahavi, A. (1997). The handicap principle: A missing piece of Darwin’s puzzle. New York: Oxford University Press.Google Scholar
  53. Zieling, D., Mahayni, A., & Balder, S. (2014). Performance evaluation of optimized portfolio insurance strategies. Journal of Banking and Finance, 43, 212–225.CrossRefGoogle Scholar
  54. Zumbach, G. (2007). The RiskMetrics 2006 methodology. Technical report, RiskMetrics Group.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Banking and Finance, Australian School of BusinessThe University of New South Wales (UNSW)UNSW SydneyAustralia
  2. 2.Fair Isaac (FICO)SydneyAustralia

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