Diseño de la BD

Figure 2 compares one typical day of trading in the model (left column) and in the real market (right column). From top to bottom the panels display: the price and value (the latter is depicted only for the model); the price returns; trading volume in terms of value; and the bid-ask spread (at the end of the 10-sec interval). Finally, the bottom panel depicts auto-correlations of raw returns (red), absolute returns (blue), and returns to the power of 2 (orange) for different lags. The gray lines indicate the 1-percent quantile of significance.

53 Of course, trading of the Lloyds plc is not only at this time but can be done for 24 hours at other exchange market or via brokers. However, we refrain to the trading at the LSE to assure that institutional environment is constant.

Parameter Interpretation Dynamic indicator Value 𝛽₀ Basic volatility expectation Measures of price

dynamics 1.376 ∗ 10⁻⁶

𝛽₁ Historic volatility orientation Measures of price

dynamics 0.3

𝑣_{𝐿𝐹,𝑚𝑚} Inventory coefficient of LF-market makers

Average holding

time of HF-traders 1 ∗ 10⁻⁵ 𝑣𝐿𝐹,⅂𝑚𝑚 Inventory coefficient of

aggressive LF-traders 0

𝑣_𝐻𝐹 Inventory coefficient of HF-traders

Average holding

time of HF-traders 5 ∗ 10⁻³

𝛼 Trading power Total trading volume 6 ∗ 10⁶

𝜎0

SD of time gap between

fundamental news. Daily return 30s

𝜎₁ SD of size of fundamental news Daily return 8 ∗ 10⁻⁴

𝜂𝐻𝐹 Latency of HF-traders 20s

𝜂_𝐿𝐹 Latency of LF-traders 2m

𝑚 weight of memory Measures of price

dynamics 0.56

Table 2: Model parameters, interpretation, dynamic indicators the parameters are directly linked to and parameter value.

Agents in the base configuration: 70 LF-market makers, 16 LF-fundamentalists, 154 LF-noise traders.

The figure shows that the model mimics some of the most important stylized facts of the high-frequency dynamics of financial markets. These facts are:

• Absence of autocorrelations in raw returns (Taylor, 2005): The price dynamics does not reveal any predictable pattern so that no significant autocorrelation in returns arise (see bottom panel, red graph).

• Volatility clustering (Taylor, 2005; Pacurar, 2008): Intervals of turbulence alternate with intervals in which prices evolve calmly (see return panels).

• Heavy tails in returns (Taylor, 2005): The proportion of extreme returns is significantly greater than under a normal distribution with the same mean and the same standard deviation. (We will verify this fact by computing tail indices).

• Excessive volatility (surveys by Fama, 1981, 1998, and Cochrane, 1991): Prices are more volatile than the intrinsic value of the asset. (in simulation run displayed, the average 10-second return for prices is 0.065% whereas for value it is 0.025%. The same can be observed for most other simulation days.)

• Bubbles and crashed (Kirilenko et al., 2011; Easley et al., 2010): Prices disconnect from value for significant spans of time (see top left panel). Take the flash crash as a prominent example.

Moreover, we can observe that the model behaves similar to the real market with respect to some other properties. At some times of the day, we observe massive trading volume and extreme bid-ask spreads, in the model as well as in the real market. This may point to the fact that extremely large orders enter the marker or trading is driven by some power law.

Following to the qualitative validation, we test the quantitatively fit of the model. Eight dynamic indicators are used for this purpose. Five of them capture the dynamics of prices: the average return over intervals of 10 seconds as a measure of intraday volatility; the total return over one day; the Hill-tail indices of the 10-second return distribution with tail fraction of 3%

and 5%; and the Hurst coefficient for the 10-second return distribution. Three indicators relate to aspects of market infrastructure and trading: the total number of trades; the total trading volume in £; and the average bid-ask spread. Figure 3 depicts the distribution of the respective indicators for the model (blue columns) and for the real market (red line). The mean of the respective distribution is marked by a dot and the median by a dashed line. The number in the upper right corner represents the percentage to which the model distribution covers the real distribution. It can be used as an indicator for the matching accuracy.⁵⁴

54 We recommend not to over-interpreted this number. On the one hand the number is sensitive to changes in the number of histogram bars. On the other hand, unless the number of observations tends to infinity, divergences between the model and the real distribution might be due to chance. Hence, the indicator would be below 100%

even if both samples would be drawn from the same population.

Figure 2: Simulation run vs. real trading day. Upper left panel: Price (solid) and value (grey). Bottom panels:

Autocorrelation of absolute returns (blue bold), squared returns (orange) and raw returns (red).

Figure 3: Quantitative validation. Blue distribution: 1.000 simulation runs of the model. Red line: 100 trading days of the Lloyds plc equity on LSE from 27.06.2011 to 14.11.2011. Values in upper right corners: percentage to which the simulated distributions cover the real ones.

It can be seen that the fit is quite accurate for the indicators of price dynamics. Only for the 10-second returns, the real market seems to reveal a greater variance than the model, which might be due to the fact that at some days there was profound insecurity about fundamental developments in the economy and the banking business – a factor which the model does not account for. This might also explain the relatively great variance of the distribution of trades, trading volume, and average bid-ask spreads in the real market. If due to an ambiguous economic situation, the expectations of traders are very heterogeneous, there is much potential for trading. This drives up the number of trades and trading volume. At the same time, bid-ask spreads rise as marker maker face high information risks, and thus behave rather observantly.

Overall we believe the model to fit the real market quite well.