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Ergodic transition in a simple model of the continuous double auction.

Radivojević T, Anselmi J, Scalas E - PLoS ONE (2014)

Bottom Line: We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues.In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns.On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.

View Article: PubMed Central - PubMed

Affiliation: BCAM - Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain.

ABSTRACT
We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.

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Time series of prices and log-returns.Time series of prices and log-returns in a system of  prices, length of interval for placing orders , with initial price  and number of simulated events . (a) ergodic case (); (b) non-ergodic case for  () and (c) non-ergodic case for  ().
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pone-0088095-g002: Time series of prices and log-returns.Time series of prices and log-returns in a system of prices, length of interval for placing orders , with initial price and number of simulated events . (a) ergodic case (); (b) non-ergodic case for () and (c) non-ergodic case for ().

Mentions: For our further analysis, we shall focus on the case of a symmetric auction, assuming and and we shall consider the ratio as the basic order parameter of the model. In fact, there is no reason for a random auction to be unbalanced towards selling or buying. As discussed above, if , we are in the ergodic regime, whereas for , we are in a regime where the orders accumulate and , for . The two regimes give rise to two radically different behaviours for the tick-by-tick log-returns (1). This is qualitatively shown in Fig. 2, where we report the behavior of prices and log-returns in a Monte Carlo simulation for (Fig. 2 (a)), (Fig. 2 (b)) and for (Fig. 2 (c)). One can see by eye that, in the ergodic regime, high and low log-returns are clustered, whereas, in the non-ergodic one, such a volatility clustering does not occur. Fig. 2 (a) clarifies the origin of clustering. When the price is lower, log-returns are higher and the price process has the persistence behavior typical of random walks which immediately leads to clusters of low and high volatility as the price slowly moves up and down, respectively. The comparison between Fig. 2 (b) and Fig. 2 (c) shows that there are two sub-regimes in the non-ergodic case. If , even if and diverge, the limit orders belonging to the best bid and the best ask can be removed by market orders and prices can fluctuate among a set, whereas if , then after a transient, the number of limit orders belonging to the best bid and the best ask diverges and prices can only fluctuate between two values. In this condition, the price process becomes a random telegraph process. This behavior is justified by the fact that the process of the number of orders at the best bid price (respectively, best ask) can be coupled with the state of an queue with arrival rate () and service rate (); this is so because limit orders, upon arrival, distribute uniformly over the best prices. If , then the number of orders at the best bid price converges to infinity, as , meaning that all trades will occur at the price where the bids accumulate. If , then the queue of the best bids eventually empties with probability one, meaning that there is a positive probability that the trading price changes. However, if we are in the region , then we know that , as . This means that the queue of the bids at some price will eventually never empty, which means that trades at lower prices will never occur. By symmetry, the same argument holds for asks.


Ergodic transition in a simple model of the continuous double auction.

Radivojević T, Anselmi J, Scalas E - PLoS ONE (2014)

Time series of prices and log-returns.Time series of prices and log-returns in a system of  prices, length of interval for placing orders , with initial price  and number of simulated events . (a) ergodic case (); (b) non-ergodic case for  () and (c) non-ergodic case for  ().
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC3928121&req=5

pone-0088095-g002: Time series of prices and log-returns.Time series of prices and log-returns in a system of prices, length of interval for placing orders , with initial price and number of simulated events . (a) ergodic case (); (b) non-ergodic case for () and (c) non-ergodic case for ().
Mentions: For our further analysis, we shall focus on the case of a symmetric auction, assuming and and we shall consider the ratio as the basic order parameter of the model. In fact, there is no reason for a random auction to be unbalanced towards selling or buying. As discussed above, if , we are in the ergodic regime, whereas for , we are in a regime where the orders accumulate and , for . The two regimes give rise to two radically different behaviours for the tick-by-tick log-returns (1). This is qualitatively shown in Fig. 2, where we report the behavior of prices and log-returns in a Monte Carlo simulation for (Fig. 2 (a)), (Fig. 2 (b)) and for (Fig. 2 (c)). One can see by eye that, in the ergodic regime, high and low log-returns are clustered, whereas, in the non-ergodic one, such a volatility clustering does not occur. Fig. 2 (a) clarifies the origin of clustering. When the price is lower, log-returns are higher and the price process has the persistence behavior typical of random walks which immediately leads to clusters of low and high volatility as the price slowly moves up and down, respectively. The comparison between Fig. 2 (b) and Fig. 2 (c) shows that there are two sub-regimes in the non-ergodic case. If , even if and diverge, the limit orders belonging to the best bid and the best ask can be removed by market orders and prices can fluctuate among a set, whereas if , then after a transient, the number of limit orders belonging to the best bid and the best ask diverges and prices can only fluctuate between two values. In this condition, the price process becomes a random telegraph process. This behavior is justified by the fact that the process of the number of orders at the best bid price (respectively, best ask) can be coupled with the state of an queue with arrival rate () and service rate (); this is so because limit orders, upon arrival, distribute uniformly over the best prices. If , then the number of orders at the best bid price converges to infinity, as , meaning that all trades will occur at the price where the bids accumulate. If , then the queue of the best bids eventually empties with probability one, meaning that there is a positive probability that the trading price changes. However, if we are in the region , then we know that , as . This means that the queue of the bids at some price will eventually never empty, which means that trades at lower prices will never occur. By symmetry, the same argument holds for asks.

Bottom Line: We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues.In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns.On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.

View Article: PubMed Central - PubMed

Affiliation: BCAM - Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain.

ABSTRACT
We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.

Show MeSH
Related in: MedlinePlus