Poisson Process

The Hidden Constant of Market Rhythms: How $1-1/e$ Defines Scaling in Intrinsic Time ArXiv ID: 2511.14408 “View on arXiv” Authors: Thomas Houweling Abstract Directional-change Intrinsic Time analysis has long revealed scaling laws in market microstructure, but the origin of their stability remains elusive. This article presents evidence that Intrinsic Time can be modeled as a memoryless exponential hazard process. Empirically, the proportion of directional changes to total events stabilizes near $1 - 1/e = 0.632$, matching the probability that a Poisson process completes one mean interval. This constant provides a natural heuristic to identify scaling regimes across thresholds and supports an interpretation of market activity as a renewal process in intrinsic time. ...

Simulating and analyzing a sparse order book: an application to intraday electricity markets ArXiv ID: 2410.06839 “View on arXiv” Authors: Unknown Abstract This paper presents a novel model for simulating and analyzing sparse limit order books (LOBs), with a specific application to the European intraday electricity market. In illiquid markets, characterized by significant gaps between order levels due to sparse trading volumes, traditional LOB models often fall short. Our approach utilizes an inhomogeneous Poisson process to accurately capture the sporadic nature of order arrivals and cancellations on both the bid and ask sides of the book. By applying this model to the intraday electricity market, we gain insights into the unique microstructural behaviors and challenges of this dynamic trading environment. The results offer valuable implications for market participants, enhancing their understanding of LOB dynamics in illiquid markets. This work contributes to the broader field of market microstructure by providing a robust framework adaptable to various illiquid market settings beyond electricity trading. ...

Modelling financial volume curves with hierarchical Poisson processes ArXiv ID: 2406.19402 “View on arXiv” Authors: Unknown Abstract Modeling the trading volume curves of financial instruments throughout the day is of key interest in financial trading applications. Predictions of these so-called volume profiles guide trade execution strategies, for example, a common strategy is to trade a desired quantity across many orders in line with the expected volume curve throughout the day so as not to impact the price of the instrument. The volume curves (for each day) are naturally grouped by stock and can be further gathered into higher-level groupings, such as by industry. In order to model such admixtures of volume curves, we introduce a hierarchical Poisson process model for the intensity functions of admixtures of inhomogenous Poisson processes, which represent the trading times of the stock throughout the day. The model is based on the hierarchical Dirichlet process, and an efficient Markov Chain Monte Carlo (MCMC) algorithm is derived following the slice sampling framework for Bayesian nonparametric mixture models. We demonstrate the method on datasets of different stocks from the Trade and Quote repository maintained by Wharton Research Data Services, including the most liquid stock on the NASDAQ stock exchange, Apple, demonstrating the scalability of the approach. ...