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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

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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

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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

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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

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Discrete sampling from a continuous process

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Upward range and downward range

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Range and one-sided ranges

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

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

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Distribution assumptions in ACARR

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

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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

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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

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

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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.

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ACARR vs. ACD identical formula 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

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

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

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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

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Figure 11: Q-Q plot of et-DWNR

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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

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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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