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OHara identifies three main market-microstructure agent types: market-makers, uninformed noise traders and informed traders. The first two agent-types are clearly identifiable in our framework. Our three remaining types of agent are different types of informed agent. While the market microstructure literature does not distinguish between different types of informed agent, behavioural finance researchers make precisely this distinction e. Using a multi-month return horizon, Jegadeesh and Titman showed that exploiting observed momentum i.
De Bondt and Thaler found the opposite effect at a different time horizon. They showed how persistent reversal negative serial correlation observed in multi-year stock returns can be profitably exploited by a similar, but opposite, buy-losers and sell-winners trading rule strategy. A re-examination of the market microstructure literature bearing these ideas in mind is revealing. Almost all market microstructure models about informed trading, dating back to Bagehot , assume that private information is exogenously derived.
This is consistent with our liquidity consumer agent type and also with the view of information being based on fundamental information about intrinsic value but it is at odds with our momentum and mean reversion traders. However, an empirical market microstructure paper by Evans and Lyons opens the door to the idea that private information could be based on endogenous technical i.
Evans and Lyons show that price behaviour in the foreign exchange markets is a function of cumulative order flow. Order flow is the difference between buyer-initiated trading volume and seller-initiated trading volume. It can be thought of as a measure of net buying selling pressure. Crucially, order flow does not require any fundamental model to be specified.
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Endogenous technical price behaviour is sufficient to generate it. The preceding enables us to conclude that while our 5 types of market participant initially seem at odds with the standard market microstructure model, closer scrutiny reveals that all 5 of our agent types have very firm roots in the market microstructure literature. In this section we begin by performing a global sensitivity analysis to explore the influence of the parameters on market dynamics and ensure the robustness of the model.
Subsequently, we explore the existence of the following stylised facts in depth-of-book data from the Chi-X exchange compared with our model: fat tailed distribution of returns, volatility clustering, autocorrelation of returns, long memory in order flow, concave price impact function and the existence of extreme price events. In this section, we asses the sensitivity of the agent-based model described above. To do so, we employ an established approach to global sensitivity analysis known as variance-based global sensitivity Sobol In variance-based global sensitivity analysis, the inputs to an agent-based model are treated as random variables with probability density functions representing their associated uncertainty.
Consequently, the total variance is calculated as follows:.
Partial variances are then defined as:. Once the above is computed, the total sensitivity indicies can be calculated as:. In this paper, twenty three input parameters and four output parameters are considered. The following output parameters are monitored: the Hurst exponent H of volatility [as calculated using the DFA method described by Peng et al. In the following, ten thousand samples from within the parameter space were generated with the input parameters distributed uniformly in the ranges displayed in Table 1. For each sample of the parameters space, the model is run for , trading periods to approximately simulate a trading day on a high-frequency timescale.
The global variance sensitivity, as defined in Eq. To find the set of parameters that produces outputs most similar to those reported in the literature and to further explore the influence of input parameters we perform a large scale grid search of the input space. This yields the optimal set of parameters displayed in Table 2. Figure 2 displays a side-by-side comparison of how the kurtosis of the mid-price return series varies with lag length for our model and an average of the top 5 most actively traded stocks on the Chi-X exchange in a period of days of trading from 12th February to 3rd July In our LOB model, only substantial cancellations, orders that fall inside the spread, and large orders that cross the spread are able to alter the mid price.
This generates many periods with returns of 0 which significantly reduces the variance estimate and generates a leptokurtic distribution in the short run, as can be seen in Fig. Kurtosis is found to be relatively high for short timescales but falls to match levels of the normal distribution at longer timescales. This not only closely matches the pattern of decay seen in the empirical data displayed in Fig. To test for volatility clustering, we compute the Hurst exponent of volatility using the DFA method described by Peng et al.
Once again, in the shortest time lags volatility clustering seems to be present at short timescales in all the simulations but rapidly disappears for longer lags in agreement with Lillo and Farmer Table 3 reports descriptive statistics for the first lag autocorrelation of the returns series for our agent based model and for the Chi-X data. In both instances, there is a very weak but significant autocorrelation in both the mid-price and trade price returns.
This has been empirically observed in other studies see Sect.
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The result is similar for the trade price autocorrelation but as a trade price will always occur at the best bid or ask price a slight oscillation is to be expected and is observed. As presented in Table 4 , we find the mean first lag autocorrelation term of the order-sign series for our model to be 0. Most studies find the order sign autocorrelation to be between 0.
Comparing Kurtosis. Figure 4 a illustrates the price impact in the model as a function of order size on a log-log scale. The concavity of the function is clear. The shape of this curve is very similar t that of the empirical data from Chi-X shown in Fig. Both of these estimates of the exponent of the impact function agree with the findings of Lillo et al.
When the market order volume is reduced, the volume at the opposing best price reduces compared to the rest of the book. This allows smaller trades to eat further into the liquidity stretching the right-most side of the curve. This parameter appears to have very little influence on the shape of the price impact function. However, it does appear to have an effect on the size of the impact.
Log—log price impact. Figure 6 shows the effects on the price impact function of adjusting the relative probabilities of events from the high frequency traders. It is clear that strong concavity is retained across all parameter combinations but some subtle artefacts can be seen.
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Firstly, increasing the probability of both types of high frequency traders equally seems to have very little effect on the shape of the impact function. This is likely due to the strategies of the high frequency traders restraining one another. Although the momentum traders are more active—jumping on price movements and consuming liquidity at the top of the book—they are counterbalanced by the increased activity of the mean reversion traders who replenish top-of-book liquidity when substantial price movements occur.
In the regime where the probability of momentum traders acting is high but the probability for mean reversion traders is low the dotted line we see an increase in price impact across the entire range of order sizes. In this scenario, when large price movements occur, the activity of the liquidity consuming trend followers outweighs that of the liquidity providing mean reverters, leading to less volume being available in the book and thus a greater impact for incoming orders.
The price impact function with different liquidity consumer parameterisations. We follow the definition of Johnson et al. Figure 7 shows a plot the mid-price time-series provides with an illustrative example of a flash occurring in the simulation.
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Table 5 shows statistics for the number of events for each day in the Chi-X data and per simulated day in our ABM. On average, in our model, there are 0. Upon inspection, we can see that such events occur when an agent makes a particularly large order that eats through the best price and sometimes further price levels. This causes the momentum traders to submit particularly large orders on the same side, setting off a positive feedback chain that pushes the price further in the same direction.
The price begins to revert when the momentum traders begin to run out of cash while the mean reversion traders become increasingly active. Figure 8 illustrates the relative numbers of extreme price events as a function of their duration.