Highs, lows, and overreaction in intraday price movements Martin Becker ∗ Ralph Friedmann † oßner ‡ Stefan Kl¨ ohle § Walter Sanddorf-K¨ Saarland University, Saarbr¨ ucken, Germany Abstract We propose measures of upside and downside volatility which mea- sure the deviation of daily high and low prices from the respective open and close prices. Under the benchmark assumption of a Brownian motion for the log-price process we derive some relationships between upside/downside volatility and intraday return volatility. We show that the proposed measures of upside and downside volatility react sensitively to non-persistent, overreacting price changes and, in the opposite way, to price jumps and discrete information arrival. An em- pirical application to the S&P 500-stock shares and to the German XETRA-DAX-stock shares provides strong support for overreactions to bad news. In contrast, for a sample of domestic Chinese A-shares, we find some evidence for overreaction to good news. JEL-classifications: C22, C52, G10 Keywords: Intraday volatility, High-Low-Prices, Overreaction ∗ email: martin.becker@mx.uni-saarland.de † Corresponding author. Tel.: +49 681 302 2111 Fax.: +49 681 302 3551 Address: Saarland University, Im Stadtwald, Building C3.1, Room 207, 66123 Saarbr¨ ucken, email: friedmann@mx.uni-saarland.de ‡ email: S.Kloessner@mx.uni-saarland.de § email: wsk@mx.uni-saarland.de 1
1 Introduction The question whether stock price movements exhibit overreactions is difficult and the answers in the literature are controversial. For example, the study of DeBondt and Thaler (1985) was followed by an intensive debate. One reason for the controversial treatment of this question lies in the fact that the adequate reaction to news can hardly be observed, unless the market reaction is considered to be adequate by assumption. Another reason for the intensive controversy about whether stock market prices overreact lies in the regulatory implications which follow from how this question is answered. In this paper we focus on a statistical approach for detecting the existence of short run, intraday overreactions followed by immediate corrections. We propose to consider the deviation of daily high and low prices from the start- ing and end point of the intraday price movement. Clearly, the extremal intraday deviations from open and close price depend on the volatility of high frequency returns. Thus we propose measures of upside and downside volatility, which are normalized by the common estimator of intraday return volatility. The resulting normalized measures of upside and downside volatil- ity are called volatility ratios . Under the assumption of a Brownian motion for the log-price process, which is considered as a benchmark, we derive some basic properties of the proposed volatility ratios. In particular, if the volatil- ity ratios are based on a sample of T daily quotes of open, high, low, and close prices, they are shown to follow an F − distribution with 2 T and T − 1 degrees of freedom. Although geometric Brownian motion is still a standard assumption for spec- ulative price processes, empirical research provides evidence that the Brown- ian motion model is only a poor approximation to intraday and interday stock price movements and volatility. Therefore, we analyze whether and how the volatility ratios are affected by several well-known stylized facts for the behavior of stock prices and asset returns which are in conflict with the Brownian motion assumption. According to our results, the proposed volatility ratios appear to be quite robust against U-shaped intraday volatil- ity seasonality as well as conditional heteroscedasticity due to intraday high frequency or conventional interday GARCH models. Furthermore, our nor- malized measures of intraday upside and downside volatility are reduced by accounting for discrete information arrival effects and price jumps. On the other hand, under the alternative of an Ornstein Uhlenbeck log-price process, modelling non-persistent price changes and mean reversion, intraday upside and downside volatility are increased. Comparing these results with our em- pirical findings, based on daily stock price quotes for the shares included in 2
the S&P 500 index and for the constituents of the German XETRA DAX, we claim to provide strong evidence for short run overreaction to bad news, which is particularly strong in the German stock market. In contrast, from a sample of domestic Chinese shares, which are subject to a daily price change limit of ten percent, we find some evidence for intraday overreaction to good news rather than to bad news. The paper is organized as follows. Section 2 presents formal definitions for the suggested measures of upside and downside volatility. Some basic charac- teristics of these measures are derived as a benchmark under the assumption of a Brownian motion for the log-price process. The relations which are de- rived in this section are also used in the sequel for the random generation of triples of intraday cumulated final returns, maximal returns, and minimal returns, without resorting to the simulation of discrete approximations of a Brownian motion. In section 3 we consider the robustness of the distribution of the proposed volatility ratios in the presence of interday GARCH-type variation of volatil- ity and moderate autoregression in the drift rate of the price process. Fur- thermore, using an illustrative example from the literature, we provide evi- dence for the robustness of the normalized measures of upside and downside volatility with respect to intraday seasonality in volatility and high frequency GARCH dependencies in volatility. Non-persistent price changes are modelled with an Ornstein Uhlenbeck pro- cess, which exhibits mean reversion in the intraday log-price movements. In section 4 we show that allowing for overreaction by this model assumption leads to an increase of upside (downside) volatility relative to intraday return volatility. In section 5 we show that the suggested F − statistic is affected in the oppo- site way by discrete information arrivals, instead of a pure Brownian motion diffusion model. First we consider a discrete N − step random walk, allowing for leptokurtic increments, for the intraday log-price process. Alternatively, we assume the intraday log-price process to follow Merton’s jump-diffusion model. Both types of distortion of the Brownian motion assumption turn out to reduce upside and downside volatility relative to the intraday final return volatility. Our empirical findings in section 6 are based on the analysis of daily open, high, low, and close price data for the components of the S&P 500, includ- ing the constituents of the Dow Jones Industrial Average, and on the 30 shares included in the German XETRA DAX. Further we have a look on 3
the intraday price movements of several stock market indices. For compar- ison, we also consider the intraday stock price behaviour for a sample of 40 domestic Chinese A shares, which are traded on the Shanghai or Shenzhen stock exchange. Generally, for the majority of individual shares included in the S&P 500 and in the XETRA DAX we find highly significant increases of normalized downside volatility as compared with the benchmark. This is considered as strong evidence for overreaction with respect to bad news. For the domestic Chinese stocks, which are traded under specific regulations, we find that short run downside overreaction is comparably weak, while up- side overreaction is particularly strong when we condition on win days. We conclude with a summary in section 7. 2 Measures for intraday upside and downside volatility For any trading day t = 1 , 2 , 3 , . . . we consider the movement of the log-price P ( τ ) of a security from the opening of the market at time τ t until market close at τ c t . Taking the length of the daily trading time, τ c t − τ t , as the time unit we have τ c t = τ t +1, where τ t +1 ≥ τ t +1. Using data on the daily open, high, low, and close log-prices, P o t = P ( τ t ) , P h t , P l t , and P c t = P ( τ t + 1), respectively, we suggest measures V t,max ( V t,min ) of intraday upside (downside) volatility defined as 2 ( P h t − P o t )( P h t − P c V t, max = t ) , (1) 2 ( P o t − P l t )( P c t − P l V t, min = t ) . (2) Both V t, max and V t, min are nonnegative and can be considered as measuring the distance of the daily extremal prices from open and close price. If the intraday return process is denoted with X t ( τ ) := P ( τ t + τ ) − P ( τ t ) , 0 ≤ τ ≤ 1 , (3) with the intraday final returns X t and intraday maximal (minimal) returns Y t, max ( Y t, min ) given as X t (1) = P c t − P o X t := t , (4) 0 ≤ τ ≤ 1 X t ( τ ) = P h t − P o Y t, max := max t , (5) 0 ≤ τ ≤ 1 X t ( τ ) = P l t − P o Y t, min := min t , (6) 4
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