When buying or selling a security, investors typically are interested in the following three questions: How likely am I to transact? What price am I likely to receive? How much does this transaction price vary? We therefore estimate the following three observables in the model both with and without HFT: (1) the probability that a submitted order will result in a transaction, (2) the average transaction price of filled orders, and (3) the volatility of the transaction price of filled orders (the standard deviation of the transaction price). We run the simulation 100 times for 10000 time steps both with and without HFT, and we record average values of the relevant observables. The results are shown in Figs. 3 and 4 and in Table 1, which we discuss in more detail below.

The model reproduces several empirical findings that have otherwise been diffi­cult to explain (Hendershott et al. 2011; Hasbrouck and Saar 2013; Brogaard et al. 2013):

1. Transaction prices are more accurate when HFT is present, i. e., they are closer to the equilibrium value.

2. Volatility is reduced when HFT is present.

3. Liquidity is increased when HFT is present.

The equilibrium price, defined as the intersection of the expected aggregate supply and demand curve in the simulation, is just the mean of the uniform distribution of prices, i. e., 100.5. As seen in Table 1, the average transaction price both with and without HFT converges to the equilibrium value within the 2 standard error range that defines a 95 % confidence interval. However, the variance around the equilibrium value is reduced when HFT is added. The reduction in variance is

Подпись: a x 104 AI1 Transaction Prices - Base Case Price Fig. 3 Histogram of transaction prices in the simulation. (a) shows the market without HFT; (b) shows the market with HFT, including transactions over both exchanges. Note that without HFT the average transaction price is not observed as often as prices at the extremes

shown in Table 1 and can be seen in the comparison of the histogram of transaction prices in Fig. 3a,b. As seen in the figure, HFT causes more transactions to occur near the equilibrium price. This result matches previous empirical studies that have shown algorithmic trading in general and HFT specifically increases the accuracy of prices in markets (Brogaard et al. 2013; Hendershott et al. 2011).

Empirical studies have also found that HFT reduces intraday volatility (Has – brouck and Saar 2013). Our simulation reproduces this result as well (see Fig. 4b). Because the equilibrium price is constant in the model, any variance in transaction price can be interpreted as excess volatility. Because HFT reduces the variance of execution prices, it also reduces the volatility of the market.

The final metric we consider is liquidity. An asset is liquid if “it is more certainly realizable at short notice without loss” (Keynes 1930). Liquidity can be defined quantitatively in a number of ways. However, our model accounts for the requirement of short notice, as orders are canceled if they do not result in a transaction, and when they do transact, the price must satisfy the reservation price initially generated. As a result, our measure of liquidity is the number of transactions that take place per simulation, or the probability that an order transacts. As shown

Fig. 4 Comparison plots of (a) average transaction price, (b) volatility, (c) transaction probability, and (d) volume (number of trades) in the simulation both with and without HFT. Note that the introduction of HFT has no discernible effect on price, but statistically significant reduction of volatility, along with an increase in the number of trades and the likelihood of a given order being filled. Error bars denote 95 % confidence intervals

Подпись: Parameter Base case HFT Price 100.5 100.5 (1.38) (0.71) Volatility 60.1 47.4 (0.6) (0.4) Volume 2255 3580 (33) (6) Probability ofTransaction 0.226 0.358


Table 1 Average of parameters over 100 runs of 10000 iterations of the simulation. Standard deviations are shown in parentheses below the average. For the HFT case, the average is taken over both markets

Подпись: Conclusions In this chapter, we analyzed the effects of high-frequency trading in a simu-lated environment. With the premise that HFT activity connects orders across markets, we found that prices are closer to their equilibrium value, volatility is reduced, and liquidity is increased when HFT is present. These results suggest that connecting order flow across similar securities is important for investor welfare, and to the extent that HFT performs this function, it serves an important purpose in modern financial markets.

in Table 1 and Fig. 4c,d, orders are more likely to be filled when HFT activity is present in our model. Again, this result matches empirical findings (Hasbrouck and Saar 2013).

Acknowledgements This chapter is a modified version of Benjamin Myers MPhys thesis originally entitled “Agent Based Simulations of High-Frequency Trading in Financial Markets.” This work was supported by the European Commission FP7 FET-Open Project FOC-II (no. 255987).

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