Statistical analysis of financial returns for a multiagent order book model of asset trading

We recently introduced a realistic order book model [T. Preis et al., Europhys. Lett. 75, 510 (2006)] which is able to generate the stylized facts of financial markets. We analyze this model in detail, explain the consequences of the use of different groups of traders, and focus on the foundation of a nontrivial Hurst exponent based on the introduction of a market trend. Our order book model supports the theoretical argument that a nontrivial Hurst exponent implies not necessarily long-term correlations. A coupling of the order placement depth to the market trend can produce fat tails, which can be described by a truncated Lévy distribution.

Figure 1

(Color online) Structure of the order book: the limit buy ($p⩽p_0$) and limit sell orders ($p>p_0$) are added chronologically to the appropriate discrete price level $p$. Here a matching algorithm with price time priority is implemented. In this example, a new sell order is placed at price $p_0$, at which there is already a buy order stored in the order book. The two orders at price $p_0$, which are on the demand and offer site of the order book, are executed against each other. Thus, a trade is performed at price $p_0$, which is then called the last traded price. The spread $s$, which is the difference between the best ask and best bid prices, was two ticks before the arrival of the executable sell order at price level $p_0$ and increases to three ticks after this trade.

Figure 2

(Color online) Liquidity providers vs liquidity takers: Averaged wealth shift of the liquidity providers and the liquidity takers, respectively, as a function of $t$ for $\alpha =0.15$, $\mu =0.1$, $\delta =0.025$, and $N_A=250$. These dependences can be fitted with the functions $f_{\mathrm{T}}\left(t\right)=a_{\mathrm{T}}t$ and $f_{\mathrm{P}}\left(t\right)=a_{\mathrm{P}}t$ with the parameters $a_{\mathrm{T}}\approx -29.45$and $a_{\mathrm{P}}\approx 29.75$. The difference between the absolute values of these two gradients is due to statistical inaccuracies occurring during the determination of the averages of the distributions.

Figure 3

(Color online) Exponential order placement depth: with the parameter selection $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, ${\lambda }_0=100$, $N_A=250$, and $q_{\text{provider}}=q_{\text{taker}}=0.5$, one gets the order book depth shown here. The results were averaged over ${10}^4$ MCS. A fit to the log-normal distribution function ${\varphi }_{\mathrm{LN}}(p-p_m)=a∕x\exp \{-{[\ln (p-p_m)-b]}^2∕c\}$ leads to the parameters $a=527\pm \left(1\right)$, $b=4.483\pm (4\times {10}^{-3})$, and $c=1.79\pm \left({10}^{-2}\right)$.

Figure 4

(Color online) Exponential order placement depth: for $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, ${\lambda }_0=100$, $N_A=250$, and $q_{\text{provider}}=q_{\text{taker}}=0.5$, an exemplary price sequence of ${10}^6$ MCS is shown.

Figure 5

(Color online) Exponential order placement depth: for the same parameters as in Fig. 4, the Hurst exponent is shown here for different numbers of agents $N_A$, averaged over 50 simulation runs each. The arrow points in the direction of larger values of the parameter $N_A$.

Figure 6

(Color online) Exponential order placement depth: distributions of price changes for different $\Delta \tau$ for the same parameter values as in Fig. 4. Again the results were averaged over 50 simulation runs each. For $\Delta \tau =200$, the fit parameters for the Gaussian distribution function $\varphi \left(\Delta p\right)=a\exp (-b\Delta p^2)$ are given by $a=2.122\times {10}^{-2}\pm (4\times {10}^{-5})$ and $b=1.388\times {10}^{-3}\pm \left({10}^{-6}\right)$. The arrow points in the direction of larger values of the parameter $\Delta \tau$.

Figure 7

(Color online) Parameter space: Fit parameter $A$ as a function of the market order rate $\mu$ for different values of the order placement depth ${\lambda }_0$. The arrow points in the direction of larger values of the parameter ${\lambda }_0$.

Figure 8

(Color online) Parameter space: fit parameter $M$ as a function of the market order rate $\mu$ for different values of the order placement depth ${\lambda }_0$. The arrow points in the direction of larger values of the parameter ${\lambda }_0$.

Figure 9

(Color online) Parameter space: fit parameter $S^2$ as a function of the market order rate $\mu$ for different values of the order placement depth ${\lambda }_0$. The arrow points in the direction of larger values of the parameter ${\lambda }_0$.

Figure 10

(Color online) Deterministic perturbation realized by a sawtooth modulation of $q_{\text{taker}}$: Hurst exponent $H\left(\Delta \tau \right)$ for different agent numbers $N_A$ as a function of $\Delta \tau$. The parameter values used are $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, and ${\lambda }_0=100$. The arrow points in the direction of larger values of the parameter $N_A$.

Figure 11

(Color online) Deterministic perturbation realized by a sawtooth modulation of $q_{\text{taker}}$: Return distributions for different values of $\Delta \tau$ and for $N_A=500$. For $\Delta \tau =200$, the function $\varphi \left(\Delta p\right)=a\exp (-b\Delta p^2)$ can be numerically fitted with $a=2.251\times {10}^{-2}\pm (3\times {10}^{-5})$ and $b=1.573\times {10}^{-3}\pm (5\times {10}^{-6})$. The arrow points in the direction of larger values of the parameter $\Delta \tau$.

Figure 12

(Color online) Stochastic perturbation realized by the feedback random walk modulation of $q_{\text{taker}}$: Hurst exponent $H\left(\Delta \tau \right)$ for different agent numbers $N_A$ as a function of $\Delta \tau$. The parameters used are $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, and ${\lambda }_0=100$. The arrow points in the direction of larger values of the parameter $N_A$.

Figure 13

(Color online) Stochastic perturbation realized by the feedback random walk modulation of $q_{\text{taker}}$: price change distributions for different values of $\Delta \tau$ and for $N_A=500$. The other parameters are the same as in Fig. 12. Fitting the function $\varphi \left(\Delta p\right)=a\exp (-b\Delta p^2)$ to the data for $\Delta \tau =200$, one gets $a=1.79\times {10}^{-2}\pm \left({10}^{-4}\right)$ and $b=9.9\times {10}^{-4}\pm \left({10}^{-5}\right)$. The arrow points in the direction of larger values of the parameter $\Delta \tau$.

Figure 14

(Color online) Autocorrelation $\varrho (\delta p\left(t\right),\Delta \tau )$ of price changes, autocorrelation $\varrho (\mid \delta p\left(t\right)\mid ,\Delta \tau )$ of absolute price changes, and autocorrelation $\varrho (\delta p{\left(t\right)}^2,\Delta \tau )$ of squared price changes with $\delta p\left(t\right)=p(t+1)-p\left(t\right)$. The parameters are the same as for the results in Fig. 12 with $N_A=125$.

Figure 15

(Color online) Dynamic order placement depth: Hurst exponent $H\left(\Delta \tau \right)$ for different values of $N_A$ with ${\lambda }_0=100$, $C_{\lambda }=10$, $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, and $\Delta S=0.001$. The arrow points in the direction of larger values of the parameter $N_A$.

Figure 16

(Color online) Dynamic order placement depth: distributions of price changes for the same parameter values as in Fig. 15. One clearly finds fat tails. A Gaussian approximation $\varphi \left(\Delta p\right)=a\exp (-b\Delta p^2)$ for $\Delta \tau =200$ with the parameters $a=2.54\times {10}^{-3}\pm \left({10}^{-5}\right)$ and $b=2.8\times {10}^{-5}\pm (2\times {10}^{-7})$ strongly underestimates the probability for large price changes. The arrow points in the direction of larger values of the parameter $\Delta \tau$.

Figure 17

(Color online) Dynamic order placement depth: Hurst exponent $H(\Delta \tau ,q)$ for different values of $q$ with ${\lambda }_0=100$, $C_{\lambda }=10$, $\alpha =0.15$, $\mu =0.025$, $\delta =0.025$, $\Delta S=0.001$, and $N=125$ for a single simulation over $5\times {10}^7$ MCS.

Figure 18

(Color online) Dynamic order placement depth: Distributions of shuffled price changes for the same parameter values as in Fig. 16. After shuffling the returns on the time horizon of one MCS the longer term aggregate returns appear Gaussian. For $\Delta \tau =200$, the function $\varphi \left(\Delta p\right)=a\exp (-b\Delta p^2)$ can be numerically fitted with $a=2.9505\times {10}^{-4}\pm (4\times {10}^{-8})$ and $b=2.7419\times {10}^{-7}\pm (9\times {10}^{-11})$. The arrow points in the direction of larger values of the parameter $\Delta \tau$.

Figure 19

(Color online) Dynamic order placement depth $\lambda$: Lévy exponent ${\alpha }_L$ as a function of $\Delta \tau$ for different $C_{\lambda }$.