In this paper, we develop a Markovian model that deals with the volume offered at the best quote of an electronic order book. The volume of the first limit is a stochastic process whose paths are periodically interrupted and reset to a new value, eit
her by a new limit order submitted inside the spread or by a market order that removes the first limit. Using applied probability results on killing and resurrecting Markov processes, we derive the stationary distribution of the volume offered at the best quote. All proposed models are empirically fitted and compared, stressing the importance of the proposed mechanisms.
The call auction is a widely used trading mechanism, especially during the opening and closing periods of financial markets. In this paper, we study a standard call auction problem where orders are submitted according to Poisson processes, with rando
m prices distributed according to a general distribution, and may be cancelled at any time. We compute the analytical expressions of the distributions of the traded volume, of the lower and upper bounds of the clearing prices, and of the price range of these possible clearing prices of the call auction. Using results from the theory of order statistics and a theorem on the limit of sequences of random variables with independent random indices, we derive the weak limits of all these distributions. In this setting, traded volume and bounds of the clearing prices are found to be asymptotically normal, while the clearing price range is asymptotically exponential. All the parameters of these distributions are explicitly derived as functions of the parameters of the incoming orders flows.
We study the analytical properties of a one-side order book model in which the flows of limit and market orders are Poisson processes and the distribution of lifetimes of cancelled orders is exponential. Although simplistic, the model provides an ana
lytical tractability that should not be overlooked. Using basic results for birth-and-death processes, we build an analytical formula for the shape (depth) of a continuous order book model which is both founded by market mechanisms and very close to empirically tested formulas. We relate this shape to the probability of execution of a limit order, highlighting a law of conservation of the flows of orders in an order book. We then extend our model by allowing random sizes of limit orders, hereby allowing to study the relationship between the size of the incoming limit orders and the shape of the order book. Our theoretical model shows that, for a given total volume of incoming limit orders, the less limit orders are submitted (i.e. the larger the average size of these limit orders), the deeper is the order book around the spread. This theoretical relationship is finally empirically tested on several stocks traded on the Paris stock exchange.
It has been suggested that marked point processes might be good candidates for the modelling of financial high-frequency data. A special class of point processes, Hawkes processes, has been the subject of various investigations in the financial commu
nity. In this paper, we propose to enhance a basic zero-intelligence order book simulator with arrival times of limit and market orders following mutually (asymmetrically) exciting Hawkes processes. Modelling is based on empirical observations on time intervals between orders that we verify on several markets (equity, bond futures, index futures). We show that this simple feature enables a much more realistic treatment of the bid-ask spread of the simulated order book.
We highlight a very simple statistical tool for the analysis of financial bubbles, which has already been studied in [1]. We provide extensive empirical tests of this statistical tool and investigate analytically its link with stocks correlation structure.
