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Highs, lows, and overreaction in intraday price movements Martin Becker Ralph Friedmann oner Stefan Kl ohle Walter Sanddorf-K Saarland University, Saarbr ucken, Germany Abstract We propose measures of upside and

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Highs, lows, and overreaction in intraday price movements

Martin Becker∗ Ralph Friedmann† Stefan Kl¨

ßner‡

Walter Sanddorf-K¨

hle§

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

pposite 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

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1 Introduction

The question whether stock price movements exhibit overreactions is difficult and the answers in the literature are controversial. For example, the study

f 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

f short run, intraday overreactions followed by immediate corrections. We

propose to consider the deviation of daily high and low prices from the starting 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 volatility 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 2T 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 volatility seasonality as well as conditional heteroscedasticity due to intraday high frequency or conventional interday GARCH models. Furthermore, our normalized measures of intraday upside and downside volatility are reduced by accounting for discrete information arrival effects and price jumps. On the

ther 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 empirical findings, based on daily stock price quotes for the shares included in 2

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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 characteristics of these measures are derived as a benchmark under the assumption

f 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

f 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 volatility and moderate autoregression in the drift rate of the price process. Fur- thermore, using an illustrative example from the literature, we provide evidence 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 process, 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 opposite 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

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the intraday price movements of several stock market indices. For comparison, 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

f 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 upside 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 τt until market close at τ c

t . Taking the length of the daily trading time, τ c t −τt, as the time unit we

have τ c

t = τt+1, where τt+1 ≥ τt+1. Using data on the daily open, high, low,

and close log-prices, P o

t = P(τt), P h t , P l t, and P c t = P(τt + 1), respectively,

we suggest measures Vt,max (Vt,min) of intraday upside (downside) volatility defined as Vt,max = 2 (P h

t − P o t )(P h t − P c t ),

(1) Vt,min = 2 (P o

t − P l t)(P c t − P l t).

(2) Both Vt,max and Vt,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 Xt(τ) := P(τt + τ) − P(τt), 0 ≤ τ ≤ 1, (3) with the intraday final returns Xt and intraday maximal (minimal) returns Yt,max (Yt,min) given as Xt := Xt(1) = P c

t − P o t ,

(4) Yt,max := max

0≤τ≤1 Xt(τ) = P h t − P o t ,

(5) Yt,min := min

0≤τ≤1 Xt(τ) = P l t − P o t ,

(6) 4

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the definitions of intraday upside and downside volatility can be rewritten equivalently as Vt,max = 2 Yt,max(Yt,max − Xt), (7) Vt,min = 2 Yt,min(Yt,min − Xt). (8) Under the benchmark assumption that the intraday log-price process follows a Brownian motion with drift rate µt and volatility σt, the suggested measures

f intraday upside and downside volatility have some attractive properties,

which are summarized in the following lemma. Lemma 1: If the intraday log-price process at day t follows a Brownian motion with drift parameter µt and volatility σt, intraday upside (downside) volatility Vt,max (Vt,min) satisfy the following properties: (i) The distribution of Vt,max (Vt,min) is drift independent with E(Vt,max) = E(Vt,min) = σ2

t .

(ii) The distribution of χ2 = 2Vt,max/σ2

(χ2 = 2Vt,min/σ2

t ) is chi-square

with two degrees of freedom. (iii) Vt,max (Vt,min) is stochastically independent of the contemporary intraday final return Xt.1 (iv) The distribution of the ratios obtained by normalizing upside (downside) volatility with the variance estimate of the intraday open-to-close return, Ft,max = Vt,max (Xt − µt)2, and Ft,min = Vt,min (Xt − µt)2, (9) is an F−distribution with two degrees of freedom in the numerator and

ne degree of freedom in the denominator.
Proof. See appendix A.

Notice that the proposed measures of upside (downside) volatility as well as

1Notice that, although each of the random variables Vt,max, Vt,min is independent of

the final return Xt, the vector (Vt,max, Vt,min) is not jointly independent of Xt. Further- more, the joint distribution of (Vt,max, Vt,min) and in particular the correlation between Vt,max, Vt,min depends on the drift rate. This follows from the joint trivariate distribution

f the final return, and the minimal and maximal return, see Billingsley (1968), p. 79.

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the intraday final return volatility used for normalization in (9) refer to the daily trading period from market open to market close. Hence, price changes from the market close price to the next open price (opening jumps), which are important to account for in estimating the common close-to-close return volatility using intraday high and low prices, do not interfere with our analysis.2 the distribution of the volatility ratios is an F−distribution with 2T degrees

f freedom in the numerator and T − 1 degrees of freedom in the denomina-

tor. Before we turn to analyzing the power of the proposed F−test in detecting short run price overreactions, we will show in section 3 that the distribution

f the test statistics Fmax and Fmin is fairly robust against interday depen-

dencies caused by GARCH type variation of the conditional variance σ2

t and

moderate autocorrelation in the mean µt of daily returns. This allows us to apply the test statistic to large samples of daily price data, despite the fact that the ideal assumption of independently, identically distributed intraday price processes is unlikely to hold. Furthermore, section 3 provides an illustrative example which indicates that intraday volatility patterns, such as deterministic U-shaped intraday seasonality and high-frequency GARCH- effects, do not seriously affect the distribution of the test statistic.

3 Robustness of volatility ratios

3.1 Interday variation of volatility

We start our analysis with an illustrative example, simulating a sequence of triples of daily final returns Xt, maximal returns Yt,max and minimal returns Yt,min under the assumption that interday log-prices process follow a Brown- ian motion with constant drift rate µ and volatility parameter σt following an

2See the study of Yang and Zhang (2000), which accounts for opening jumps in an

approach to efficient volatility estimation using daily high and low prices. Generally, the literature on volatility estimation using high and low prices, see Parkinson (1980), Garman and Klass (1980), Wiggins (1991), Rogers and Satchell (1991), relies on the assumption

f a geometric Brownian motion for the price process, which appears to be critical in the

light of empirical evidence, as given, for example, in section 6 of this paper.

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0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

T=50

Value of test statistic Density F2T, T−1 Fmin Fmax 0.6 0.8 1.0 1.2 1.4 2 4 6

T=1000

Value of test statistic Density F2T, T−1 Fmin Fmax

Figure 1: Volatility ratios for an interday EGARCH process asymmetric interday EGARCH process.3 The drift rate µ and the EGARCH parameters were set equal to the estimates obtained from T = 1000 daily returns of the Boeing share, traded at the NYSE, starting the sample in January 2000. Thus the simulated returns are based on µ = 0.065 and ln σ2

t = 0.106 − 0.054εt−1 + 0.205(|εt−1| −

2/π) + 0.939 lnσ2

t−1.

For given µ, σt we generate (Xt, Ymax,t, Ymin,t) ∼ BM(µ, σt), and εt = (Xt − µ)/σt, where BM denotes Brownian motion. Based on the simulation of one million triples (Xt, Yt,max, Yt,min) for Brownian motions with drift rate µ and σt following the specified EGARCH process, figure 1 shows for two sample sizes (T = 50 and T = 1000) that interday variation in the volatility parameter according to the estimated EGARCH process does not seriously distort the distribution of the test statistic under the null of independently and identically distributed intraday price processes.

3In our Monte Carlo simulations of (Xt, Yt,max, Yt,min) for a Brownian motion we make

use of the joint distribution (Xt, Vt,max, Vt,min) rather than simulating a discrete approximation to the realized path of a Brownian motion.

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More generally we consider the effect of varying volatility parameters on the asymptotic distribution of the appropriately transformed test statistic. It is well known that the transformation of an F(ν1, ν2) distributed random variable F into R = ν1F ν2 + ν1F has a Beta distribution over the unit interval with parameters

ν1 2 , ν2 2 , see

Johnson et al. (1995), p.327. Then, for Fmax and Fmin with ν1 = 2T, ν2 = T − 1, for T → ∞ the standardizing linear transformation of Ri = 2T Fi T − 1 + 2T Fi = 2 T

t=1 Vt,i

t=1(Xt − ¯

X)2 + 2

t=1 Vt,i

is asymptotically standard normal. Thus, corresponding to the test statistics Fmax and Fmin we obtain, for i = max, min, the test statistics Zi,T = √ 3T

2Ri − 1

√ 3T

t=1 Vt,i −

t=1(Xt − ¯

X)2

t=1(Xt − ¯

X)2 + 2 T

t=1 Vt,i

(10) which are asymtotically standard normal under the null. The following lemma shows that asymptotically the effect of interday variation in volatility σt solely depends on the coefficient of variation of the time varying σ2

t around the mean variance.

Lemma 2: Under the assumption of a varying volatility σt, with

1 T

t=1(σ2 t − ¯

σ2)2 (¯ σ2)2 =: γ > 0, with ¯ σ2 = 1 T

σ2

the test statistic (10) is asymptotically distributed as N(0, 1 + γ). Proof: See appendix B. As an example, assume the variation in the volatility follows a stationary GARCH(1,1)-process with existing fourth moment, i.e. σ2

t = α0 + α1X2 t−1 + β1σ2 t−1,

(11) with 2α2

1 + (α1 + β1)2 < 1 and

σ2 = E(X2

t ) = E(σ2 t ) =

α0 1 − α1 − β1 , 8

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see Bollerslev (1986). While Xt is conditionally N(0, σ2

t ), Bollerslev has

proved that for the marginal distribution of Xt it holds that E(X4

t )

= 3α2

0(1 + α1 + β1)

(1 − α1 − β1)(1 − 2α2

1 − (α1 + β1)2)

= 3σ4 1 − (α1 + β1)2 1 − 2α2

1 − (α1 + β1)2

and hence Var(X2

t )

= E(X4

t ) − (E(σ2 t ))2 = E(X4 t ) − σ4

= 2σ4

3α2

1 − 2α2

1 − (α1 + β1)2

Thus, using (11) and the stationarity of the process, we obtain for the vari-

ance of σ2

Var(σ2

t ) =

α2

1Var(X2 t )

1 − β2

1 − 2α1β1

= 2α2

1σ4

1 − 2α2

1 − (α1 + β1)2

Inserting parameter values α1 = 0.05, β1 = 0.9, α0 = 1−α1−β1, for instance, we obtain σ2 = 1 and Var(σ2

t ) = 0.05405. With γ = Var(σ2 t )/σ4 = 0.05405,

we obtain Z0 ∼ N(0, 1.05405) under the alternative. This example confirms that the test statistic is fairly robust against typical GARCH type variations in volatility.

3.2 Interday variation of the drift rate

Although the presence of varying drift parameters µ1, . . . , µT in principal causes more serious distortions of the distribution of the volatility ratios Fmax, Fmin than time varying volatility, empirically the drift rate displays

nly small, if any, interday variation over time.

Generally, variation in the drift parameter does not affect the distribution of

t=1 2Vt,i/σ2, which remains to be chi-square with 2T degrees of freedom.

On the other hand, variation of the drift parameter µt around ¯ µ increases

t=1(Xt − ¯

X)2/σ2, with the distribution becoming noncentral χ2 with T − 1 degrees of freedom and noncentrality parameter λ,

t=1(Xt − ¯

X)2 σ2 ∼ χ′2

T−1(λ),

with λ =

(µt − ¯ µ)2 σ2 . 9

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As a consequence, the normalized upside and downside volatility measures Fmax and Fmin decrease, and for their reciprocals we obtain the noncentral F−distribution with T − 1, 2T degrees of freedom and noncentrality parameter λ, see Johnson et al. (1995), p.480, 1 Fi ∼ χ′2

T−1(λ)/(T − 1)

χ2

2T/2T

∼ F ′

T−1,2T(λ),

i = max, min . (12) Using the approximation suggested by Patnaik (1949) (see Johnson et al. (1995), p.492), the noncentral F− distribution can be approximated by P(F ′

T−1,2T(λ) ≤ f) ≈ P

Fν,2T ≤

f 1 +

λ T−1

, with ν = (T − 1 + λ)2

T − 1 + 2λ . (13) Thus, to quantify the shift of the distribution of Fi, i = min, max, to the left side, due to a varying drift rate µt with noncentrality parameter λ, one may compute the tail probability left of the α−quantile of the null distribution by P(Fi ≤ f2T,T−1,α| λ) = P

Fi ≥ 1 f2T,T−1,α | λ

Fν,2T ≥

1 f2T,T−1,α(1 +

λ T−1)

F2T,ν ≤ f2T,T−1,α(1 +

λ T − 1)

For example, assume that the drift rate µt follows an AR(1) process with

autocorrelation ρ. Then we have λ T − 1 = ρ2 1 − ρ2. Taking ρ = 0.1 as parameter value, for instance, the shift of the distribution

f the test statistic to the left side is indicated by an increase of the probability

mass left of the 5%−quantile to 6% for sample size T = 250, and to 7.2% for T = 1000. Thus the distortion of the distribution of the test statistic is quite moderate in empirically relevant cases. Furthermore, for a given sequence µt any distortion of the test distribution can be completely avoided using

t=1(Xt − µt)2 instead of T t=1(Xt − ¯

X)2. 10

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3.3 Intraday volatility patterns

Regarding the effect of well-known typical intraday volatility patterns such as deterministic U-shaped intraday seasonality and high-frequency GARCH effects, we provide evidence that the distribution of the test statistics is not seriously affected by intraday variation in volatility. As an illustrative example we present the results of Monte Carlo simulations of an empirically realistic high frequency intraday AR-EGARCH model for deseasonalized returns Xd

t,i

ver 5 minute intervals, on which we impose a multiplicative deterministic

U-shaped seasonal volatility component. The specification and parameteri- zation of the AR-EGARCH model is taken from the intraday volatility model for the Boeing stock, as estimated by Bauwens and Giot (2001, p.142) from high frequency data for the period from January to May 1997, with 78 intraday 5 minute returns from 9h30 to 16h for every trading day: µt,i = −0.025 + 0.014Xd

t,i−1,

ln σ2

t,i

= 0.011 − 0.012εt,i−1 + 0.163(|εt,i−1| −

2/π) + 0.951 ln σ2

t,i−1.

For given µt,i, σt,i we generate the deseasonalized triples (Xd

t,i, Y d max,t,i, Y d min,t,i) ∼ BM(µt,i, σt,i),

and εt,i = (Xd

t,i − µt,i)/σt,i,

where BM denotes Brownian motion. Thus the deseasonalized intraday log- price process is simulated by a sequence of Brownian motions with drift rates µt,i and volatility parameters σt,i. Finally we impose a multiplicative deterministic U-shaped intraday seasonal component

φ(ti) for the volatility,

see figure 2, similar to the results of Bauwens and Giot, to obtain (Xt,i, Ymax,t,i, Ymin,t,i) =

φ(ti)(Xd

t,i, Y d max,t,i, Y d min,t,i).

For every day t the open-close final return as well as the intraday maximal and minimal return is obtained by appropriately cumulating the 5-minute-

returns. According to our simulation results, the shape of the distribution of

the volatility ratios for different sample sizes from T = 50 to T = 1000 days displays only minor deviations from the null distribution, see figure 2. Notice that distortions caused by bid-ask-spread and tick size, which are common for high frequency returns, should be of minor importance in our case, because intraday maximal and minimal returns refer to cumulated returns rather than to the maximum (minimum) over individual high frequency returns. In the following section we analyze the power of the proposed F−test in detecting overreaction and mean reversion in the intraday price process. 11

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20 40 60 80 5.0 e−06 2.5 e−05

Intraday seasonal volatility

ti φ(ti) 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5

Volatility ratios, T=50

Value of test statistic Density F2T, T−1 Fmin Fmax 0.6 0.8 1.0 1.2 1.4 1.6 1 2 3 4 5

Volatility ratios, T=500

Value of test statistic Density F2T, T−1 Fmin Fmax 0.6 0.8 1.0 1.2 1.4 1.6 2 4 6

Volatility ratios, T=1000

Value of test statistic Density F2T, T−1 Fmin Fmax

Figure 2: Volatility ratios under intraday seasonal volatility and a high frequency AR-EGARCH model 12

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4 Short run overreaction and mean reversion

The persistence of shocks is the major characteristic of any sum process such as a random walk and correspondingly, our benchmark model of Brownian motion, which can be considered as a random walk in continuous time. In this framework, news are transmitted into shocks on the asset price, whose impact does not die out in the future, but remains persistent. In the Brown- ian motion model there is no overreaction in the price response to news, that would require any correction. As an alternative model for the intraday log-price movement, featuring non-persistent price changes, we consider an Ornstein-Uhlenbeck (OU) process, which can be considered as an autoregressive process in continuous time. More precisely, it is a stationary process, given by the following assumption (compare, for example, Gourieroux and Jasiak [2001], pp 249-251). OU-Process: The log-price Pt(τ) over day t is assumed to follow the stochastic differential equation dPt(τ) = (µ − λPt(τ))dτ + ω dWt(τ), (14) with parameters µ ∈ I R, and λ, ω ∈ I R+. Wt(τ) is a standard Brownian motion, and Wt(τ) is independent of Ws(τ ′) for t = s and τ, τ ′ ∈ [0, 1]. Notice that for λ → 0, the process approaches the (non-stationary) Brownian process. The solutions of the stochastic differential equation (14) satisfy, for any τ ′ < τ, Pt(τ) = µ λ

1 − e−λ(τ−τ ′)

+ e−λ(τ−τ ′)Pt(τ ′) + ω

τ ′ e−λ(τ−u)dWt(u).

(15) In particular, if the unit interval is partitioned into N subintervals of length

1 N , and we set Pt,n := Pt( n N ), for n = 0, 1, . . ., N, we obtain the autoregressive

relation Pt,n = δ + ρ

1 N Pt,n−1 + εn,

n = 1, . . . , N (16) where ρ = e−λ, δ = µ λ

1 − ρ

1 N

εn

iid

∼ N

0, σ2

with σ2

ε = ω2

2λ

1 − ρ

2 N

Thus the stationary log-price is Gaussian with E(Pt,n) = µ λ, Var(Pt,n) = ω2 2λ and Corr(Pt,n, Pt,m) = ρ

|n−m| N

. (17) 13

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This result extends to the continuous-time process Pt(τ), with Corr(Pt(τ), Pt(τ ′)) = ρ|τ−τ ′|. Further, as a consequence of (17), the final returns Xt = Pt,N − Pt, 0 are iid normally distributed with E(Xt) = 0, and Var(Xt) = ω2 λ (1 − ρ) =: σ2. For 0 < τ < 1, the intraday cumulated returns Xt(τ) = Pt(τ) − Pt, 0 are Gaussian with E(Xt(τ)) = 0, and Var(Xt(τ)) = ω2 λ (1 − ρτ). Hence, for 0 < τ < 1, the variance of intraday cumulated returns, Var(Xt(τ)), relative to the final return variance, Var(Xt), is larger for an OU-process than for a Brownian motion, because, for τ, ρ ∈ (0, 1), 1 − ρτ 1 − ρ > τ. Due to the mean reverting behavior of the OU process, we expect that the normalized upside (downside) volatility Fmax (Fmin) will be increased, that is, we expect the distribution of the test statistic to be shifted to the right

side. For analyzing the power of the suggested test against the OU alter-

native, we simulate the OU-process by (16), with µ = δ = 0, normalize the variance of the final return by assuming ω2 = λ/(1 − ρ), which implies σ2

ε = 0.5(1 − ρ

2 N )/(1 − ρ), and initialize Pt,0 with a random draw according

to (17). Although the discretization (16) of the OU-process holds exactly, the extremal returns obtained from a realization of (16) are only approximations of the respective realizations of Yt,max and Yt,min of the continuous-time process. For the continuous time process the measures of upside and downside volatility, Vt,max, Vt,min, respectively, are greater or equal than the respective values

btained from simulating the discrete approximation of the process. Thus,

the computed power which is based on the discrete approximation of the process, underestimates the true power. To make this approximation bias small, we have chosen N quite large by setting N = 10000. 14

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Table 1: Power against OU-process (right-sided test, α = 5%)

T λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.4 λ = 0.6 λ = 0.8 λ = 1.0 50 5.35 5.70 7.70 15.15 25.95 37.95 45.40 250 5.35 8.90 16.80 46.55 76.10 91.50 98.45 500 5.75 9.85 27.60 71.65 94.80 99.55 100.00 1000 7.15 12.50 42.10 92.30 99.90 100.00 100.00

Rejection frequencies (Fi > F2T,T −1,0.95, i = max, min) in 2,000 replications in percent.

With the number of replications equal to 2,000, we simulate the process and the resulting volatility ratios Fmax, Fmin for selective values of the parameter λ from λ = 0.05 to λ = 1.0, which corresponds to ρ = 0.951 to ρ = 0.368, and to an autoregressive coefficient in (16) from ρ

1 N = 0.999995 to ρ 1 N = 0.9999.

The results for the power at a significance level of 5% are given for sample sizes from T = 50 to T = 1000 in table 1. For small values of λ, i.e. when the OU-process is close to the Brownian motion, the power remains small even at a sample size of T = 1000. As expected, the power increases with increasing λ, and it is quite high for realistic sample sizes. For example, for λ = 0.6 and sample size T = 500, the power of the test is already higher than 90%; a realization of the daily log price process for λ = 0.6 is given in figure 3. Both the assumption of a Brownian motion as well as the alternative of an OU-process imply a continuous inflow of news or noise. One might wonder how the test statistic would be affected by allowing for discrete information arrival, for nonnormal shocks following leptokurtic distributions, and for price

jumps. In the next section we show that the distribution of the test statistics

is not robust against these quite common violations of the Brownian motion

assumptions. However, conversely to the effect of non-persistent shocks and
verreaction, the volatility ratios generally become smaller by allowing for

these stylized facts of price processes. Thus, the suggested volatility ratios may be used to distinguish between the impact of price overreaction at the

ne hand and, on the other hand, the presence of discrete information arrival,

price jumps, or excess kurtosis in the distribution of shocks. 15

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2000 4000 6000 8000 10000 0.2 0.4 0.6 0.8 1.0 1.2 n Pt, n

Figure 3: Log-price simulation as an OU process with λ = 0.6

5 Discrete information arrival and Merton jump-diffusions

First we analyze the effect of discrete information arrival according to a random walk, where changes of the log-price level occur only N−times per day, instead of a continuous flow of news modeled by a Brownian motion. Subsequently we consider the distribution of the volatility ratio statistics Fmax, Fmin when the price process follows Merton’s jump-diffusion model.

5.1 Discrete information arrival

We consider the effect of discretization of the Brownian motion by a random walk with independently, identically distributed increments. Clearly, by the functional central limit theorem (compare, for example, Billingsley (1968), pp 68 - 69), with N → ∞, and equispaced occurrences of the price changes, the process approaches the Brownian motion, as long as we consider distributions

f the log-price increments with existing variance, such that N−times the

variance is finite. Thus, our focus is on lower values of N; we consider N = 5, 10, 50, 250. For the distributional assumption we consider, apart from the standard normal distribution, standardized t−distributions with 3, 5 and 16

SLIDE 17

10 degrees of freedom. For comparison, we also consider the power against a t−distribution with 2 degress of freedom, i.e., the second moment does not exist, and against the alternative of a stable, non-normal distribution, namely against the standard form of the Cauchy distribution. Before we discuss the power analysis by Monte Carlo simulation, let us briefly look at the cases of only N = 1 or N = 2 persistent price changes per day. For N = 1 we have Vt = 0, for each t, because Yt,i is either equal to zero or equal to Xt, for i = min, max. Thus the F−statistic takes the value zero, and the null hypothesis will be rejected. For N = 2, and subsequent log-price increments εt,1, εt,2, say, it follows that Vt,i, i = max, min, is either zero, with probability 0.75, or Vt,i = 2|εt,1||εt,2|, if εt,1 > 0 and εt,2 < 0, for i = max, and if εt,1 < 0 and εt,2 > 0, for i = min. Thus, for N = 2, it follows that P(Vt,i = 0) = 0.75 and P(Vt,i = 2|εt,1||εt,2|) = 0.25. If we assume independently, normally distributed increments εt,1, εt,2 with mean zero and variance σ2/2, the expectation of Vt,i is E(Vt,i) = 2E(|εt,1||εt,2|) 4 = σ2(E|Z|)2 4 = σ2 2π < σ2, where Z ∼ N(0, 1). Accordingly, the distribution of the volatility ratios will be shifted to the left side. This effect is further increased, when we assume the standardized t−distribution with 3, 5, and 10 degrees of freedom, where E(Vt,i) = 2E(|εt,1||εt,2|) 4 = σ2(ν − 2)(E|tν|)2 4ν = σ2 2π

(ν − 2)(Γ(1

2(ν − 1))2

2(Γ(1

2ν))2

< σ2

2π , see Johnson et al. (1995, p.366). For ν = 3, 5, 10 the multiplier of σ2/(2π) takes the values 0.637, 0.849, 0.940, respectively. With N increasing, we still expect Vt,i to be biased towards zero. In particular, in the case of N normally distributed increments, the simulated daily sequence of log-prices can be considered as a subset of the respective path

f the continuous-time Brownian motion. As a consequence, the maximal

(minimal) cumulated returns from this subset are a.s. less (greater) than the respective extremal returns under the null hypothesis, such that Vt,i will continue to be biased towards zero, although the bias should decrease for increasing N. 17

SLIDE 18

Table 2: Power against N−step random walks (left-sided test, α = 5%)

Normal distribution T N = 5 N = 10 N = 50 N = 250 50 96.40 80.46 32.56 13.93 250 100.00 99.98 78.88 30.40 500 100.00 100.00 95.95 47.10 1000 100.00 100.00 99.94 70.15 t-distribution with ν = 10 degrees of freedom T N = 5 N = 10 N = 50 N = 250 50 96.79 83.01 35.28 14.79 250 100.00 100.00 82.30 32.19 500 100.00 100.00 97.07 51.32 1000 100.00 100.00 99.93 74.26 t-distribution with ν = 5 degrees of freedom T N = 5 N = 10 N = 50 N = 250 50 97.43 86.17 39.66 16.34 250 100.00 100.00 88.13 38.42 500 100.00 100.00 98.94 57.92 1000 100.00 100.00 99.99 82.38 t-distribution with ν = 3 degrees of freedom T N = 5 N = 10 N = 50 N = 250 50 98.30 92.20 55.30 25.60 250 100.00 100.00 97.67 64.42 500 100.00 100.00 99.94 87.22 1000 100.00 100.00 100.00 98.75 t-distribution with ν = 2 degrees of freedom T N = 5 N = 10 N = 50 N = 250 50 98.85 96.87 84.75 67.18 250 99.99 99.99 99.96 99.23 500 100.00 99.99 100.00 99.99 1000 100.00 100.00 100.00 100.00 Cauchy distribution T N = 5 N = 10 N = 50 N = 250 50 99.43 99.25 99.19 99.03 250 99.93 99.94 99.91 99.93 500 99.99 99.96 99.95 99.96 1000 99.99 99.99 99.97 99.95

Rejection frequencies (Fi < F2T,T −1,0.05, i = max, min) in 10,000 replications in percent.

SLIDE 19

For N = 5, 10, 50, 250 information arrivals per day we compute the power of the left-sided F−test, based on the volatility ratios Fmax or Fmin, by Monte Carlo simulations. We set the significance level to 5%, and simulate the log- price process according to the different alternatives with 10,000 replications. In every case, the log-price increments are obtained as random draws from the respective distribution, divided by √

N. For the sample size, we consider

T = 50, 250, 500, 1000. Table 2 presents the respective results of our Monte Carlo simulations. The power is increasing with the sample size, and with the leptokurtosis of the alternative distribution. As expected, the power is inversely related with the number N of price changes, as far as the functional central limit theorem applies, although, the convergence to the Brownian motion and the implied distribution of the extremal returns appears to be remarkably slow. For the t−distribution with 2 degrees of freedom, the power has further increased, as compared with the t−distribution with 3 degrees of freedom. For small sample size, however, the power is still decreasing with a growing number N

f price steps. In the case of the Cauchy-distribution, the power appears to

be close to one, irrespective of N.

5.2 Merton jump-diffusion model

Our analysis of the effect of price jumps on the distribution of volatility ratios is based on the Merton model, see, for example, Cont and Tankov (2004, pp 111-112). In the Merton jump-diffusion model, jumps, which are imposed on a Brownian motion with drift rate µBM and diffusion volatility σBM, occur according to a Poisson process with parameter λ, and the jump size follows a Gaussian distribution N(µJ, σ2

J). The return Xt over the unit period is

distributed as Xt ∼ N(µ, σ2) with µ = µBM + λµJ, σ2 = σ2

BM + λσ2 J + λµ2 J.

(18) Generally, according to our simulation results, imposing price jumps on the Brownian motion leads to a decrease of the volatility ratios. For the Monte Carlo simulation of the Merton model the jump intensity λ is varied over λ = 0.01, 0.1, 1, 5, 10, corresponding to an average occurence

f one jump in one hundred days, one jump in ten days, one jump per day,

five jumps per day, and ten jumps per day, respectively. We denote the proportion of the contribution of the jumps to Var(Xt) by ρ, ρ = λσ2

J + λµ2 J

σ2

BM + λσ2 J + λµ2 J

. (19) 19

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Table 3: Power against Merton jump-diffusion (left-sided test, α = 5%)

µJ = µBM = 0 (T = 250) ρ λ = 0.01 λ = 0.1 λ = 1 λ = 5 λ = 10 0.10 21.79 13.07 8.01 6.23 5.85 0.25 49.32 48.28 20.90 11.69 9.37 0.50 71.06 94.80 75.72 37.64 25.36 µJ = −0.03/λ, µBM = 0.03, Fmax − test (T = 250) ρ λ = 0.01 λ = 0.1 λ = 1 λ = 5 λ = 10 0.10 20.82 12.91 7.73 6.38 5.78 0.25 50.26 48.42 21.74 11.60 9.66 0.50 71.87 94.79 75.75 36.95 25.42 µJ = −0.03/λ, µBM = 0.03, Fmin − test (T = 250) ρ λ = 0.01 λ = 0.1 λ = 1 λ = 5 λ = 10 0.10 20.47 13.04 7.83 6.34 5.81 0.25 50.17 47.81 21.63 11.11 9.87 0.50 71.83 94.75 76.24 36.70 25.28

Rejection frequencies (Fi < F2T,T −1,0.05, i = max, min) in 10,000 replications in percent.

In our simulation study ρ takes the values ρ = 0.1, 0.25, 0.5. We assume µ = 0, hence µJ = −µBM/λ, firstly with µBM = 0 and secondly, as another case, µBM = 0.03. With normalization by σ2 = 1, for a fixed variance proportion ρ, the variance of the jump size varies with the jump intensity, σ2

J = ρ

λ − µ2

J = ρ

λ − µ2

λ2 . (20) The power of the test against Merton jump diffusion is analyzed by Monte Carlo simulation of the test statistics Fmax, Fmin for different sample sizes. Table 3 presents the results for T = 250. In the first case (µJ = µBM = 0) the power, of course, is the same, irrespective of using Fmax or Fmin. It is increasing with growing contribution of the jumps to the return volatility, given by ρ. For fixed ρ, the power is decreasing, when the jump intensity λ is getting large, and thus, according to (20), the variance of the jump size is getting small. Only when the jump intensity is very small (λ = 0.01), we also find that the power is reduced. This reduction in power can be explained by the probability that, for λ = 0.01 and sample size T = 250, no jump, or at least no big jump, occurs. 20

SLIDE 21

In the second case, where we assume that the mean size of jumps is negative, we analyze the power separately for Fmax and Fmin. According to our simulation results, however, the distribution of both of the test statistics is essentially the same. Generally, the pattern of the power results is similar to the first case with zero drift and zero mean of the jump size. Summarizing the results of this section, we find that discrete information arrival as well as jump-diffusion reduces the volatility ratios and thus affects the distribution of the test statistics Fmax and Fmin conversely to overreaction and non-persistent price changes.

6 Empirical findings

Our analysis uses daily price data (open, high, low, close) for the components of the S&P 500, with a closer look on the 30 components which are also included in the Dow Jones Industrial Average (DJIA), furthermore daily quotes for the 30 securities included in the German XETRA DAX. We also consider the respective stock market indices S&P 500, DJIA, XETRA DAX, and, additionally, the Nasdaq Composite Index, Nasdaq 100, Dow Jones Euro STOXX 50, and the Nikkei 225. The presented results refer to a sample of T = 1000 daily price vectors up to November 30, 2005, starting in Decem- ber 2001.4 From using other time periods and sample sizes we find that our conclusions do not crucially depend on the selected time period. Additionally we consider daily price data for a sample of domestic Chinese shares, 27 of them traded at the Shanghai stock exchange (SSE) and 13 traded at the Shenzhen stock exchange (SZE), with sample sizes between T = 805 and T = 1454 up to 2006, February 22. One of the characteristics

f the very young Chinese stock market is market segmentation into domestic

A shares for Chinese citizens and B shares, which are traded in US dollar and designated to attract foreign capital, although there is a progressive opening

f the B share market to Chinese citizens since 2001. We restrict our atten-

tion to domestic A shares. Notice that trading of shares is restricted by a 10 percent daily price change limit and additional regulations to prevent or at least to reduce intraday trading. Generally for every security in our analysis we compute the test statistics Fmin, Fmax, as defined in (??), giving the normalized downside volatility and

4We have excluded 8 out of the set of 500 shares of the S&P 500 portfolio, because the

available sample size is less than T = 1000; the excluded shares are AMP, CIT, FSL-B, GNW, HSP, MHS, PRU, SHLD.

SLIDE 22

upside volatility, respectively. Our focus is on testing for overreaction through a right-sided F−test. From our analysis in the previous section we know that common features of intraday price movements such as discrete information arrival and jumps pull the volatility ratios downside. Despite this fact, the S&P 500 data as well as the XETRA DAX data provides considerable evidence for short run overreaction in the case of bad news. This evidence becomes even stronger, when we apply the F−test separately, on the one hand to a sample of daily price data with negative open-close returns (loss days) and, on the other hand, to a sample of daily price data with positive

pen-close returns (win days).

Regarding the S&P 500 components, for 258 (52.4%) of the companies the normalized downside volatility Fmin is greater than the 5%−critical value F0.95,2T,T−1 = 1.095, and there are still 220 (44.7%) significant at the 1%

level. On the other hand, the normalized upside volatility Fmax does not

indicate comparable non-persistent upside price movements, with only 39 (7.9%) of the test statistics exceeding the 5%−critical value, of which 23 (4.7%) remain to be significant at the 1% level. Regarding the mean of the normalized downside (upside) volatility over the S&P 500 components, we

btain ¯

Fmin = 1.134 ( ¯ Fmax = 0.890), respectively. The distribution of the test statistics Fmin and Fmax is shown in the left graphic of figure 4. Additionally, for the S&P 500 components we consider the empirical distribution of the normalized downside volatility Fmin,loss conditional on loss days, based on a sample of T = 500 daily price data where open-close returns Xt are negative, and, on the other hand, of Fmin,win conditional on win days, based on a sample of T = 500 daily price data with Xt > 0. Here we assume the mean return E(Xt) = 0 to be given, such that the downside volatility ¯ Vmin,loss and ¯ Vmin,win is normalized with the respective semivariance. Remind that under the null hypothesis of a Brownian motion Vt,min is stochastically independent of the intraday return Xt and, assuming drift rate zero, Fmin,loss as well as Fmin,win follow an F distribution with 2T and T degrees of freedom. In contrast to that, the graphic at the right hand side of figure 4 shows that the test statistics Fmin,loss and Fmin,win, which are obtained for the constituents of the S&P 500, are quite different. Conditioning on loss days, the downside volatility indicates significant overreaction at the 5% level for 312 (63.7%) of the S&P 500 components, while conditioning on win days leads to significant downside overreaction only for 93 (19%) of the S&P 500 com-

ponents. Accordingly the mean of the normalized downside volatility over

the S&P 500 components, given by ¯ Fmin = 1.134, rises to ¯ Fmin,loss = 1.247 and falls to ¯ Fmin,win = 1.008 when we condition on loss days and win days, 22

SLIDE 23

0.5 1.0 1.5 2.0 2 4 6

FminandFmax(T=1000)

Valueofteststatistic Density F2T,T−1 Fmin Fmax 0.5 1.0 1.5 2.0 1 2 3 4 5

Fmin,lossandFmin,win(T=500)

Valueofteststatistic Density F2T,T Fmin,loss Fmin,win

Figure 4: Distribution of volatility ratios Fmin, Fmax for S&P 500 shares

respectively. This confirms that non-persistent downside price movements

are mainly due to overreaction to bad news. Notice that, in contrast, conditioning the normalized upside volatility Fmax on win and loss days does not induce any major effects. For those of the S&P 500 components which constitute the DJIA, the test results for Fmin and Fmax are given in table 4. The normalized downside volatility Fmin leads to significant overreaction at the 5% level for 22 (73.3%)

f the shares, while Fmax indicates significant upside overreaction only for

3 (10%) of the Dow Jones shares. Once again, inspection of Fmin conditional

n loss days, for a sample of T = 500 for each of the DJIA shares, and, on

the other hand, conditional on win days, yields that Fmin,loss is significant for 21 (70%) of the shares, while Fmin,win is significant only for 7 (23.3%) of the shares. As an example for a European stock exchange, we consider the 30 components of the German XETRA DAX. The test results are presented in table 5. Here we find an extremely pronounced asymmetry in downside and upside

volatility. The evidence for downside overreaction is even stronger than for

the DJIA or S&P 500 components. For 29 (96.7%) companies Fmin indicates 23

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Table 4: F−test for overreaction (DJIA shares)

Symbol Name Fmin p.Fmin Fmax p.Fmax AA Alcoa 1.2139 0.0002 0.9750 0.6802 AIG American Intern. Group 1.1499 0.0058 0.8630 0.9967 AXP American Express 1.1155 0.0240 1.0444 0.2161 BA Boeing 1.2164 0.0002 1.0215 0.3517 C Citigroup 1.4841 0.0000 0.9896 0.5782 CAT Caterpillar 1.0909 0.0577 0.8300 0.9997 DD Du Pont 1.1183 0.0216 1.0038 0.4749 DIS Walt Disney 1.4749 0.0000 1.0151 0.3948 GE General Electric 1.2911 0.0000 0.9605 0.7714 GM General Motors 1.0205 0.3582 0.7851 1.0000 HD Home Depot 1.2473 0.0000 1.0592 0.1491 HON Honeywell Intern. 1.2187 0.0002 1.0188 0.3694 HPQ Hewlett−Packard 1.1553 0.0046 1.1330 0.0120 IBM IBM 0.9331 0.8988 0.9857 0.6059 INTC Intel 0.8905 0.9837 0.8932 0.9813 JNJ Johnson & Johnson 1.2054 0.0004 1.2513 0.0000 JPM JPMorgan Chase and Co. 1.1412 0.0085 0.9571 0.7908 KO Coca−Cola 1.1636 0.0031 1.0322 0.2841 MCD McDonald’s 1.1384 0.0096 1.0625 0.1365 MMM 3M Company 1.0297 0.2993 0.8676 0.9956 MO Altria Group 1.0241 0.3344 0.8720 0.9942 MRK Merck & Co. 1.0434 0.2214 0.9610 0.7683 MSFT Microsoft Corp. 0.9048 0.9672 0.9638 0.7521 PFE Pfizer 1.5042 0.0000 0.9684 0.7236 PG Procter & Gamble 1.1335 0.0118 0.9202 0.9370 T AT & T 1.2228 0.0001 1.1258 0.0161 UTX United Technologies 1.1192 0.0209 0.8547 0.9981 VZ Verizon Communications 1.1878 0.0010 0.9777 0.6620 WMT Wal−Mart Stores 1.1446 0.0074 1.0412 0.2331 XOM Exxon Mobil Group 1.1758 0.0017 1.0201 0.3610

Sample size T = 1000 until 2005/11/30, Fmin, Fmax : values of the test statistic (normalized downside-, upside-volatility), bold: signif. at 5% (right-sided test), F0.95,2T,T −1 = 1.095, p.Fi = P(F(2T, T − 1) > Fi), i = min, max.

SLIDE 25

Table 5: F−test for overreaction (DAX shares)

Symbol Name Fmin p.Fmin Fmax p.Fmax ADS.DE Adidas Salomon 1.3063 0.0000 0.9945 0.5427 ALT.DE Altana 1.5093 0.0000 1.0666 0.1217 ALV.DE Allianz 1.1654 0.0029 0.8028 1.0000 BAS.DE BASF 1.5189 0.0000 1.0640 0.1308 BAY.DE Bayer 1.2409 0.0001 0.7490 1.0000 BMW.DE BMW 1.4284 0.0000 0.9569 0.7913 CBK.DE Commerzbank 1.3628 0.0000 1.0303 0.2956 CON.DE Continental 1.5592 0.0000 1.0193 0.3664 DB1.DE Deutsche Boerse 1.4972 0.0000 1.0393 0.2434 DBK.DE Deutsche Bank 1.0984 0.0448 0.8907 0.9834 DCX.DE DaimlerChrysler 1.3978 0.0000 1.0023 0.4860 DPW.DE Deutsche Post 1.7050 0.0000 1.1755 0.0018 DTE.DE Deutsche Telekom 0.9084 0.9614 0.8827 0.9892 EOA.DE E.ON 1.3228 0.0000 1.0136 0.4052 FME.DE Fresenius 1.1206 0.0198 0.8484 0.9988 HEN3.DE Henkel 1.3823 0.0000 0.7466 1.0000 HVM.DE HVB 1.3111 0.0000 0.9479 0.8377 IFX.DE Infineon 1.3185 0.0000 0.9458 0.8476 LHA.DE Lufthansa 1.6595 0.0000 1.1563 0.0044 LIN.DE Linde 1.2444 0.0000 1.0296 0.2999 MAN.DE MAN 1.2774 0.0000 0.8762 0.9926 MEO.DE Metro 1.4772 0.0000 0.9679 0.7266 MUV2.DE Muenchner Rueck 1.1894 0.0009 0.8860 0.9871 RWE.DE RWE 1.4174 0.0000 1.0436 0.2203 SAP.DE SAP 1.1294 0.0139 0.8685 0.9953 SCH.DE Schering 1.4348 0.0000 1.0310 0.2914 SIE.DE Siemens 1.2883 0.0000 0.9348 0.8927 TKA.DE ThyssenKrupp 1.6461 0.0000 1.0762 0.0919 TUI1.DE TUI 1.4440 0.0000 1.1505 0.0057 VOW.DE Volkswagen 1.3030 0.0000 0.9024 0.9706

SLIDE 26

significant overreaction at the 5% level, and still 26 (86.7%) are significant at the 1% level. Only for Deutsche Telekom the normalized downside volatility is quite low. For the mean of the normalized downside volatility over the XE- TRA DAX components, we obtain ¯ Fmin = 1.355, rising to ¯ Fmin,loss = 1.539 and falling to ¯ Fmin,win = 1.179, when we condition on loss days and win days, respectively. On the other hand, for only three companies (Deutsche Lufthansa, Deutsche Post and TUI) we find significant upside overreaction. For 27 (90.0%) of the shares, the normalized upside volatility Fmax is quite low and there is no significant upside overreaction. The mean of the normalized upside volatility over the XETRA DAX components is obtained as ¯ Fmax = 0.982, and it stays below one when we condition either on win days

r on loss days.

The empirical results are different, when we consider the intraday price movements on the more regulated stock markets at the Shanghai and Shenzhen stock exchange. For a sample of 40 shares the values of the normalized downside and upside volatility are presented in table 6. Although downside

verreaction is significant at the 5% level still for 15 (37.5%) of the shares,

the mean value of the normalized downside volatility over the 40 shares, ¯ Fmin = 1.082, is lower than for the S&P 500 or XETRA DAX shares, and it does not rise when we condition either on loss or win days. On the other hand, upside overreaction is more frequently significant than in the previous cases, for 9 (22.5%) of the shares. Conditioning on win days, upside overreaction is significant for 26 (65.0%) of the shares, while conditioning on loss days there is no significant upside overreaction at all. Thus the daily open, high, low, and close prices provide some evidence for overreaction to good

news. Accordingly, conditioning on win days, the mean normalized upside

volatility over the 40 A shares in our sample, ¯ Fmax,win = 1.126, although moderate, is quite high as compared to the respective value for the S&P 500 and for the XETRA DAX shares, which stays below one in both cases. Finally, for the portfolios given by the stock market indices NASDAQ Com- posite, S&P 500, Nikkei 225, NASDAQ 100, DJ Euro STOXX 50, and XE- TRA DAX we find no evidence for overreaction, neither upside or downside, see table 7. Normalized upside and downside volatility is quite low for these indices, and the downside overreaction that we find in many individual shares appears to be compensated by converse price movements of other components

f the respective portfolio. Only for the DJIA the results are quite different.

Here the test statistics Fmin and Fmax indicate highly significant overreaction to both sides. 26

SLIDE 27

Table 6: F−test for overreaction (Chinese A shares)

Symbol Name T Fmin p.Fmin Fmax p.Fmax 600002.SS Qilu Petrochemical (A) 1014 1.0209 0.3541 1.1398 0.0086 600005.SS Wuhan Iron & Steel (A) 975 1.0261 0.3238 1.1228 0.0193 600073.SS SH Maling Aquar (A) 1000 0.9336 0.8970 1.1350 0.0110 600086.SS Hubei Duojia (A) 1004 1.4810 0.0000 1.0463 0.2060 600088.SS China Television (A) 1018 0.9276 0.9185 1.0905 0.0569 600186.SS Lianhua Gourmet (A) 1021 1.3695 0.0000 0.8372 0.9995 600205.SS SD Aluminium (A) 999 0.8960 0.9784 0.9096 0.9593 600601.SS Founder Sci&Tec (A) 1020 1.2210 0.0001 1.1327 0.0115 600631.SS Shanghai Bailian (A) 1008 0.8672 0.9958 1.1423 0.0079 600652.SS Shanghai ACE (A) 1020 1.3449 0.0000 1.1318 0.0119 600654.SS SH Fello (A) 998 1.0294 0.3014 1.0422 0.2278 600685.SS GZ Shipyard (A) 1020 0.9831 0.6256 1.0002 0.5013 600698.SS Jinan Qingqi Mot (A) 932 0.9686 0.7153 0.9074 0.9579 600715.SS Songliao Auto (A) 1004 1.0338 0.2738 0.9465 0.8446 600748.SS SH Ind Deve (A) 946 1.0892 0.0664 1.0093 0.4370 600770.SS Zongyi (A) 1002 0.9660 0.7384 1.2749 0.0000 600788.SS Xian Diamond Indl (A) 805 0.9003 0.9587 0.8441 0.9975 600797.SS Insigma Tech (A) 1000 1.2894 0.0000 1.0603 0.1449 600823.SS Shanghai Shi Mao (A) 1019 0.9125 0.9555 0.9977 0.5192 600831.SS Shaanxi Broad (A) 993 1.3364 0.0000 1.0826 0.0764 600832.SS Oriental Pearl (A) 1001 0.8143 0.9999 1.1656 0.0028 600876.SS Luoyang Glass (A) 1017 1.1767 0.0015 1.0462 0.2053 600879.SS Long Mar Vehicle (A) 995 1.3941 0.0000 1.0452 0.2130 000008.SZ Yorkpoint S & T (A) 1420 1.1427 0.0020 1.0843 0.0403 000066.SZ China Greatwall (A) 1433 0.7892 1.0000 1.0072 0.4395 000402.SZ Fin Street Hldg (A) 1434 1.2617 0.0000 0.7467 1.0000 000503.SZ Searainbow Hldg (A) 1434 1.1833 0.0001 1.0490 0.1496 000517.SZ Success Info Ind (A) 1405 1.0593 0.1082 0.9988 0.5122 000535.SZ KMK Company (A) 1247 1.0727 0.0781 1.0535 0.1462 000546.SZ Jinlin Indl Group (A) 1198 1.1951 0.0002 1.0623 0.1156 000550.SZ Jiang Ling Motor (A) 1404 0.9350 0.9284 1.0502 0.1465 000557.SZ Guangxia Yinchuan (A) 1275 1.0436 0.1916 0.8963 0.9885 000585.SZ NE Elec Devlpmnt (A) 1328 1.1402 0.0031 0.9729 0.7203 000720.SZ LN Taishan Cab (A) 1451 0.9490 0.8770 1.0026 0.4794 000787.SZ Powerise Info (A) 1450 0.9338 0.9352 0.9351 0.9312 000795.SZ TY Corundum (A) 1442 1.5352 0.0000 0.9995 0.5061 000799.SZ Jiuguijiu (A) 1454 1.064 0.0876 1.0474 0.1559 000912.SZ SC Lutianhua (A) 1432 1.1900 0.0001 0.9836 0.6434 000917.SZ HN TV & Brdcast (A) 1406 0.9455 0.8890 1.0605 0.1034 000920.SZ South Huiton (A) 1448 0.9487 0.8782 1.0750 0.0574

Sample size T until 2006/02/22, Fmin, Fmax : values of the test statistic (normalized downside-, upside-volatility), bold: signif. at 5% (right-sided test), p.Fi = P(F(2T, T − 1) > Fi), i = min, max.

SLIDE 28

Table 7: F−test for overreaction (Indices)

Symbol Name Fmin p.Fmin Fmax p.Fmax GDAXI XETRA DAX 0.9286 0.9137 0.6291 1.0000 DJI DJ Ind Ave. 1.9394 0.0000 1.7163 0.0000 STOXX50E DJ Euro STOXX 50 0.7733 1.0000 0.6463 1.0000 NDX NASDAQ 100 0.7395 1.0000 0.6924 1.0000 N225 Nikkei 225 0.7876 1.0000 0.7450 1.0000 GSPC S&P 500 0.7334 1.0000 0.5308 1.0000 IXIC NASDAQ Compos. 0.6822 1.0000 0.5611 1.0000

7 Summary

In this paper we provide empirical evidence for short run, intraday overreaction to bad news, based on daily open, high, low, and close quotes for the S&P 500 shares as well as for the 30 XETRA DAX shares. In contrast, we find some evidence for intraday upside overreaction in a sample of Chinese domestic shares. Our analysis of overreaction is based on the measurement

f 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 intraday log-price process these measures of intraday downside and upside volatility fulfil some basic relationships with the intraday return volatility. In particular, the different measures have the same expectation, and the volatility ratios of upside (downside) volatility to the variance of intraday returns are distributed with an F−distribution. The distribution of the volatility ratios is shown to be fairly robust against interday variation of the volatility

parameter. Further, the effects of interday variations of the drift rate on the

exact distribution of the F−statistic are analyzed. The suggested volatility ratios can discriminate between two opposite vi-

lations of the Brownian motion framework. Volatility ratios of normalized

upside (downside) volatility are increased by jump-diffusions and discrete information arrival, and on the other hand, they are decreased by short term mean reversion, which is captured with the model of a (stationary) Ornstein Uhlenbeck process. Thus the empirical evidence for violations of the Brownian motion towards the overreaction hypothesis cannot be explained by well known features of intraday volatility patterns, such as high 28

SLIDE 29

frequency GARCH-processes and intraday seasonality in volatility, nor by jump-diffusions or discrete information arrival.

Appendix

A Proof of Lemma 1

To simplify notation, we drop the time index t, when we derive the joint distribution of (X, Vt,i), i = max, min. In the first step we state the joint distributions of (X, Ymax) and (X, Ymin). Subsequently, we derive the joint densities of (X, Vmax) and (X, Vmin), using (7), (8) for Vmax, Vmin. For a Brownian motion X(τ) with drift µ ∈ I R and volatility σ ∈ I R+, that is X(τ) ∼ µ τ + σ W(τ), 0 ≤ τ ≤ 1, where W(τ) is a standard Brownian motion with (a) W(0) = 0 a.s. (b) W(τ2) − W(τ1) ∼ N(0, τ2 − τ1) for 0 ≤ τ1 < τ2 (c) W(τ4) − W(τ3) is independent of W(τ2) − W(τ1) for 0 ≤ τ1 < τ2 ≤ τ3 < τ4, the joint distribution of X = X(1) and Ymax = max0≤τ≤1 X(τ) is given by FX,Ymax(x, y) = Φ

x − µ

− exp

2 µ y

σ2

x − 2y − µ

, y ≥ 0, y ≥ x,

see, for example, Korn and Korn (2001), Lemma 4.5., p. 168. Furthermore, using that Ymin = min0≤τ≤1 X(τ) = − max0≤τ≤1(−X(τ)), the joint distribution function of X, Ymin for y ≤ 0, y ≤ x is easily derived as FX,Ymin(x, y) = Φ

y − µ

+ exp

2 µ y

σ2 Φ

y + µ

− Φ

2y − x + µ

Then the joint bivariate density functions of (X, Yi), i = max, min, which are obtained from the respective cumulative distribution functions, can be 29

SLIDE 30

written as pX,Ymax(x, y) = pX(x) 2(2y − x) σ2 exp

−2y(y − x)

σ2

for y > 0, y > x,

pX,Ymin(x, y) = pX(x) 2(x − 2y) σ2 exp

−2y(y − x)

σ2

for y < 0, y < x,

where pX(x) denotes the normal density function of X ∼N(µ, σ2). For the transformation of (X, Yi) into (X, Vi), i = max, min, with Vi = 2Yi(Yi − X), i = max, min, and with the inverse transformations given by Ymax = X 2 +

+ Vmax 2 , Ymin = X 2 −

+ Vmin 2 , the Jacobian is

= |2(2y − x)| > 0.

Thus the joint densities of (X, Vi), i = max, min, are given as pX,Vi(x, v) = 1 |2(2y − x)| pX,Yi(x, y) = pX(x) 1 σ2 exp

− v

σ2

for v > 0.

Obviously, the joint bivariate density function is the product of the marginal densities of X and of Vi, i =min, max. Hence, Vt,max (Vt,min) is stochastically independent of the contemporary intraday final return Xt, as stated in (iii). The marginal distribution of Vmax (Vmin) is an exponential distribution with rate 1/σ2, hence it is drift independent with expectation σ2, as stated in (i). As an immediate consequence, for i = max, min, the distribution of χ2 = 2Vt,i/σ2

t is chi-square with two degrees of freedom, as stated in (ii).

Finally, proposition (iv) follows, because (ii) and (iii) imply that the ratio Ft,i = Vt,i (Xt − µt)2 = (2Vt,i/σ2

t )/2

(Xt − µt)2/σ2

= χ2

2/2

χ2

, i = max, min, is distributed with an F−distribution with two degrees of freedom in the numerator and one degree of freedom in the denominator. ⋄ 30

SLIDE 31

B Proof of Lemma 2

Considering the asymptotic distribution of the test statistics (10) the sample mean ¯ X can be replaced by the drift parameter µ. Then, the test statistic can be rewritten as Zi,T = √ 3T

1 T

t=1

σ2

t − (Xt−µ)2

σ2

¯ σ2 1 T

t=1

(Xt−µ)2

σ2

+ 2 Vt

σ2

¯ σ2

(21) For the denominator the expectation as well as the probability limit is obtained as 3, hence, Zi,T

σ2

− (Xt − µ)2 σ2

σ2

¯ σ2

For any T ∈ I N we obtain E

σ2

− (Xt − µ)2 σ2

σ2

¯ σ2

0, Var

σ2

− (Xt − µ)2 σ2

σ2

¯ σ2

3 T 2

σ2

¯ σ2

= 3 T (γ + 1), hence, Zi,T

∼ N(0, 1 + γ).

⋄

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