1.4 The simulation software
2.1.3 Computational experiments
The timing parameters of every simulation have been set as follows: M = 50 daily sections, each characterize by a length of T = 25, 200 s (corresponding to 7 hours of trading activity). Each simulation is characterized by a particular value of the shape
2.1 Waiting-time distribution of trading activity 21 parameter β of the Weibull distribution modeling order waiting times. 17 different values of β ranging from 0.2 to 1 with step 0.05 have been chosen. The average order waiting time hτoi has been set to 20 s for every simulation. The scale factor η is so
determined by the choice of β and hτoi according to Eq. 2.2. The orders lifespan has
been set to 600 s À hτoi. Sell and buy limit prices are computed following Eq. 2.4
and Eq. 2.5 respectively. The random number ni(th) is a random draw by trader
i from a Gaussian distribution with constant mean µ = 1 and standard deviation σ = 0.005.
The number of agents is set to 10,000. At the beginning of the simulation, the stock price is set at 100.00 units of cash, say dollars and each trader is endowed with an equal amount of cash and of shares of the risky stocks. These amounts are 100,000 dollars and 1,000 shares, respectively.
Computational experiments produce realistic intraday price paths, see Ref. [161] for further details. Log-returns r∆t have been computed in homogeneous time win- dows ∆t, according to the previous-tick interpolation technique [56, pag. 37]. The time window ∆t has been set to 100 s, which is about two times the average value of trade waiting times. Figure 2.1 presents the average values of orders waiting times
hτoi and trade waiting times hτTi, respectively, as a function of the shape parameter
β. hτoi is nearly 20 s, as expected by model construction, whereas hτTi, i.e., an
output of the GASM model, is around 55 s. hτTi appears to be independent from β
when β ≥ 0.3; the pattern of increasing values of hτTi when β < 0.3 may be due to
numerical effects. The choice of ∆t > hτTi has been made in order to avoid spurious
effect in the statistical properties of returns due to long period of trading inactivity, especially in the case of small value of β.
Figure 2.2 presents the values of kurtosis of log-returns as a function of the shape parameter β. The figure shows that the distribution of log-returns is characterized by increasing values of kurtosis as β decreases. In previous works [45, 161], it is already showed that the mechanism of the limit order book was able to recover fat tails in the distribution of log-returns without ad-hoc behavioral assumptions regarding agents. In that cases, β was set to 1, i.e., the order generation process was modeled as a Poisson process with exponentially distributed waiting times. Figure 2.2 generalizes such a result showing that the same conclusion also holds for a more general renewal process, i.e., Weibull distributed waiting times. Furthermore, the tails of the distribution of returns become fatter when β decreases from 1, i.e., the distribution of order waiting times becomes a stretched exponential.
Figure 2.3 and Figure 2.4 show estimates of the survival probability distribution of order waiting times (dots) and of the survival probability distribution of trade waiting times (crosses) in the case β = 0.4 (Fig. 2.3) and β = 1 (Fig. 2.4). As expected by the assumptions of the model, survival probability distributions of order waiting times follow the corresponding Weibull distribution, represented by the continuous line.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 15 20 25 30 35 40 45 50 55 60
β of Weibull distribution for τo
average of
τ
{s} τo
τT
Figure 2.1: Average values of order waiting times and of trade waiting times as a function of the shape parameter β of the Weibull distribution for orders.
2.1 Waiting-time distribution of trading activity 23 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45 50
β of Weibull distribution for τo
kurtosis of r ∆ t ( ∆ t = 100 s)
Figure 2.2: Kurtosis of the distribution of log-returns as a function of the shape parameter β.
0 500 1000 1500 2000 2500 3000 10−5 10−4 10−3 10−2 10−1 100 P > ( τ ) τ {s} τo τT
Figure 2.3: Survival probability distribution of order waiting times (dots) and of trade waiting times (crosses) in the case β = 0.4. The two curves represent the corresponding Weibull fits.
The survival probability distribution of trade waiting times is well fitted by a Weibull distribution only in the case β = 1. Figure 2.3 shows that, in the case β = 0.4, the survival probability distribution of trade waiting times departs from a Weibull distribution determined from data by means of the maximum likelihood principle and represented in the Figure with the dashed curve. In this case, the Kolmogorov- Smirnov test rejects the null hypothesis of Weibull distribution at the significance level of 5 %. Generally speaking, trade waiting times are Weibull distributed only in the case β = 1.
In the case β = 1, the process of trading is a Poisson process as the order arrival process. An identical conclusion follows from theoretical considerations. In fact, consider that every transaction occurs when a new order that arrives in the book finds a matching order in the queue of orders of the opposite type. Therefore, any new order will be satisfied or not in function of the state of the book in that moment. The state of the book varies for each moment and for each simulation path. Given the absence of significant feedbacks in the market, however, it is reasonable to assume that the average state of the book is time invariant. Therefore, each incoming order will be satisfied on average with a constant probability and the trading process can
2.1 Waiting-time distribution of trading activity 25 0 100 200 300 400 500 600 10−5 10−4 10−3 10−2 10−1 100 P > ( τ ) τ {s} τo τT
Figure 2.4: Survival probability distribution of order waiting times (dots) and of trade waiting times (crosses) in the case β = 1 (exponential distribution). The two lines represent the corresponding exponential fits.
be regarded as a random extraction form the Poisson process of orders issuing. The procedure of random extraction from a Poisson process is called “thinning” and it is well known that a thinning from a Poisson process is a new Poisson process; see Ref. [52] for further details.