CONCLUSIONES Y RECOMENDACIONES

In our study, the physical time scale projects the transaction data to every time interval of one second. If there is an imbalance of buy and sell market orders in a time interval, the trade sign of this interval is either +1 or _{−1. If there is no trade or there is a balance of} buy and sell market orders in a time interval, it is necessary or even inevitable to introduce the zero trade sign for this time interval.

For each trading day, on average, more than half of the total physical time features zero trade signs, listed in Table 2.4. For those non-trading time intervals, the price is not affected by trades so that the price response is zero at that time. After averaging the response over entire time, the effect of absence of trades is involved, diluting the impact of the trade itself. For instance, the impacts from the last trade one minute ago and from the last trade five hours ago, respectively, on the price at this moment are very different. For the latter, the current price may be influenced more significantly by the news during that five hours than by the last trade five hours ago. The time with or without trading is a non-trivial feature in the transaction data and contains much information on the trading activity. Especially, for whatever economic or other reasons, the time without trading reflects the disagreement of traders on the price. In view of this, the inclusion of zero trade signs in the cross-response function is reasonable.

Alternatively, we rule out all the zero trade signs from the cross-response function. This means the impact of trades is unaltered by the time without trading. The alternative choice implies the sign for one trade is fixed and independent of the time period τ0 without

trading. However, the empirical result reveals a possible reversion of the trade sign after τ0,

as shown in Fig. 3.3 for five successive trading days of AAPL in 2008. The ps(τ0) and pd(τ0)

in Fig. 3.3 are the probabilities of finding the same and different trade signs, respectively, before and after the time period τ0 without trading, where we have ps(τ0) + pd(τ0) = 1.

When this period τ0 enlarges, the probability of finding the same sign falls down slowly,

Chapter 3. Cross-responses in correlated financial markets: individual stocks τ₀/s 10-1 100 101 p ( τ0 ) 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 ps(τ0) pd(τ0)

Figure 3.3: Probabilities ps(τ0) and pd(τ0) for the change of trade signs versus the time τ0without trading. The intraday data used stems from five successive trading days of AAPL in 2008. The strong fluctuations at large τ0 are due to the limited statistics.

while the probability of finding the different signs raises up gradually. Thus, for a long time without trading, it is unlikely to keep the trade sign unchanged. Likewise, it is not possible to maintain the same impact of trades after a long time period without trading. From this perspective, the exclusion of zero trade signs may introduce a bias. Although we incline towards the case including zero trade signs, we display the empirical results of both cases for comparisons.

When there is a balance of buy and sell market orders in time interval t, the signs of buy and sell market orders cancel each other out. However, whether or not the behaviour of the trading itself causes the cross-response is unknown. To quantify the effect of zero trade signs, we use R_ij(inc. 0)(τ ) and R(exc. 0)_ij (τ ) to represent the cross-response in- and excluding ε(t) = 0, respectively. Thus, the response R(only 0)_ij (τ ) stemming from ε(t) = 0 can be quantified by the difference of the two kinds of cross-responses,

R(only 0)_ij (τ ) = R(inc. 0)_ij (τ )− R(exc. 0)ij (τ ) . (3.5)

Here, we do not distinguish the two sources of ε(t) = 0, i.e., a lack of trading and a balance of buy and sell market orders. Analogously, the sign cross-correlator for ε(t) = 0 can be measured by

Θ(only 0)_ij (τ ) = Θ(inc. 0)_ij (τ )_{− Θ}(exc. 0)_ij (τ ) . (3.6) As shown in Fig. 3.4, both the cross-response and the sign cross-correlator for ε(t) = 0 exhibit negative values, which implies the inclusion of zero trade signs will weaken the impact of trades. In contrast, the exclusion of the zero trade signs enlarges this impact. In this sense, the results with the inclusion of ε(t) = 0 gives a conservative estimation for the impact of trades and the price cross-response. In addition, the non-linear response with a reversed trend in the case of ε(t) = 0 further corroborates our analysis on the influence of the time without trading.

3.3. Cross-responses for pairs of stocks τ/s 100 101 102 103 Rij ( τ ) ×10-5 -3 -2 -1 0 1 2 3 4 5 6 7 τ/s 100 101 102 103 Θ ij ( τ ) -0.04 -0.02 0 0.02 0.04 0.06 0.08 for εj(t) = 0 included for εj(t) = 0 excluded for εj(t) = 0

Figure 3.4: The three cross-responses (left) and sign cross-correlators (right) of stock pair (MSFT, AAPL) versus the physical time lag τ : in- and excluding ε(t) = 0 as well as only for ε(t) = 0. The shaded regions indicate the standard error bars.