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Modeling the Asymmetry of Stock Movements Using Price Ranges

Modeling the Asymmetry of Stock Movements Using Price Ranges. Ray Y. Chou Academia Sinica “ The 2002 NTU International Finance Conference” Taipei. May 24-25, 2002. Motivation. Provide separate dynamic models for the upward-range and the downward-range to allow for asymmetries.

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Modeling the Asymmetry of Stock Movements Using Price Ranges

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  1. Modeling the Asymmetry of Stock Movements Using Price Ranges Ray Y. Chou Academia Sinica “ The 2002 NTU International Finance Conference” Taipei. May 24-25, 2002

  2. Motivation • Provide separate dynamic models for the upward-range and the downward-range to allow for asymmetries. • Factors driving the upward movements and the downward movements maybe different. • Upward range applications: market rallies, call options, historical new highs, limit order to sell • Downward range applications: Value-at-Risk, put options, limit order to buy

  3. Main Results • ACARR is similar to CARR and ACD but with a different limiting distribution and with new interpretations and implications. • Properties: QMLE, Distribution • Empirical results using daily S&P500 index show asymmetry in dynamics, leverage effect, periodic patterns and interactions of upward and downward movements. • Volatility forecast accuracy: ACARR>CARR>GARCH

  4. Range as a measure of the “realized volatility” • Simpler and more natural than the sum-squared-returns (measuring the integrated volatility) of Anderson et.al.(2000) • Parkinson (1980) and others have established the efficiency gain of range over standard method in estimating volatilities • Chou (2001) proposed CARR, a dynamic model for range with satisfactory performance

  5. Discrete sampling from a continuous process

  6. Upward range and downward range

  7. Range and one-sided ranges

  8. The Conditional Autoregressive Range Expectation (CARR) model in Chou (2001)

  9. The Asymmetric Conditional Autoregressive Range Expectation -ACARR(p,q) model

  10. Distribution assumptions in ACARR

  11. ACARRX(p,q) – ACARR(p,q) with exogenous variables

  12. Explanatory variables in the ACARRX(p,q) model • Lagged returns – leverage effect • Periodic (weekday) pattern • Transaction volumes • Interaction tems – lagged DWNR in expected UPR and lagged UPR in expected DWNR

  13. Properties of ACARR • Same as ACD of Engle and Russell (1998) but with a known limiting distribution for the error term • A conditional mean model • An asymmetric model for volatilities

  14. Sources of asymmetry for an ACARRX(1,1) model • a – short term shock impact • b – long term persistence of shocks • a+b – speed of mean-reverting • g ‘s – effects of leverage, periodic pattern, interaction terms, among others

  15. A special case of ACARR: Exponential ACARR(1,1) or EACARR(1,1) • It’s useful to consider the exponential case for f(.), the distribution of the normalized range or the disturbance. • Like GARCH models, a simple (p=1, q=1) specification works for many empirical examples.

  16. ACARR Range data, positive valued, with fixed sample interval QMLE with EACARR Known limiting distribution A new volatility model ACD Duration data, positive valued, with non-fixed sample interval QMLE with EACD Unknown limiting distribution Hazard rate interpretation ACARR vs. ACDidentical formula

  17. The QMLE property • Assuming any general density function f(.) for the disturbance term et, the parameters in ACARR can be estimated consistently by estimating an exponential-ACARR model. • Proof: see Engle and Russell (1998), p.1135

  18. The QMLE Estimation • Consistent standard errors are obtained by employing the robust covariance method in Bollerslev and Wooldridge (1987). • See Engle and Russell (1998).

  19. Empirical example: S&P500 daily index • Sample period: 1962/01/03 – 2000/08/25 • Data source: Yahoo.com • Models used: EACARR(1,1), EACARRX(p,q) • Both daily and weekly observations are used for estimation • Forecast comparison of CARR and ACARR

  20. Figure 11: Q-Q plot of et-DWNR

  21. Extensions • Robust ACARR – Interquartile range • Multivariate ACARR • Nonparametric or semiparametric ACARR • Other data sets and simulations • Long memory ACARR’s – IACARR, FIACARR,… • ACARR and option price models

  22. Conclusion • ACARR is effective in modeling upward and downward market movements. • Asymmetry found: dynamics, leverage effect, periodic patterns, interaction terms • CARR provides more accurate volatility forecasts than GARCH (Chou (2001)) and ACARR gives further improvements.

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