Modelling stock prices has been a research topic for many decades and it is still an open question. Different approaches have been used in the literature, the majority of which can be classified within the so-called econometric framework and sometimes also referred to as the macro-to-micro approach. Another strand of literature relies on the modelling of directly observable quantities, the so-called micro-to-macro approach. Based on this second line of research, we propose a new multivariate stochastic process to model simultaneously price returns, trading volumes and the time interval between changes in trades, price and volume. The proposed model is based on a generalization of semi-Markov chain models and copulas and is motivated by empirical evidence that the three mentioned variables are correlated and long-range autocorrelated. Utilizing Monte Carlo simulations, we compared our model with real data from the Italian stock market and show that it can reproduce many empirical pieces of evidence. The proposed model can be used in the field of portfolio optimization, development of risk measure and volatility forecasting.

中文翻译：

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退货、数量和等待时间的微观到宏观方法

几十年来，股票价格建模一直是一个研究课题，但仍然是一个悬而未决的问题。文献中使用了不同的方法，其中大部分可以归类在所谓的计量经济学框架内，有时也称为宏观到微观的方法。另一类文献依赖于直接可观察量的建模，即所谓的微观到宏观方法。基于第二条研究，我们提出了一种新的多元随机过程，以同时对价格回报、交易量以及交易、价格和交易量变化之间的时间间隔进行建模。所提出的模型基于半马尔可夫链模型和 copula 的泛化，并受到三个提到的变量相关和长程自相关的经验证据的启发。利用蒙特卡罗模拟，我们将我们的模型与意大利股票市场的真实数据进行了比较，并表明它可以重现许多经验证据。所提出的模型可用于投资组合优化、风险度量的开发和波动率预测领域。