英文标题：
《A Stochastic Model of Order Book Dynamics using Bouncing Geometric
  Brownian Motions》
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作者：
Xin Liu, Qi Gong, Vidyadhar G. Kulkarni
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最新提交年份：
2016
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英文摘要：
  We consider a limit order book, where buyers and sellers register to trade a security at specific prices. The largest price buyers on the book are willing to offer is called the market bid price, and the smallest price sellers on the book are willing to accept is called the market ask price. Market ask price is always greater than market bid price, and these prices move upwards and downwards due to new arrivals, market trades, and cancellations. We model these two price processes as \"bouncing geometric Brownian motions (GBMs)\", which are defined as exponentials of two mutually reflected Brownian motions. We then modify these bouncing GBMs to construct a discrete time stochastic process of trading times and trading prices, which is parameterized by a positive parameter $\\delta$. Under this model, it is shown that the inter-trading times are inverse Gaussian distributed, and the logarithmic returns between consecutive trading times follow a normal inverse Gaussian distribution. Our main results show that the logarithmic trading price process is a renewal reward process, and under a suitable scaling, this process converges to a standard Brownian motion as $\\delta\\to 0$. We also prove that the modified ask and bid processes approach the original bouncing GBMs as $\\delta\\to0$. Finally, we derive a simple and effective prediction formula for trading prices, and illustrate the effectiveness of the prediction formula with an example using real stock price data.
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中文摘要：
我们考虑一个限价订单簿，买家和卖家登记以特定价格交易证券。书上买家愿意提供的最大价格称为市场买入价，书上卖家愿意接受的最小价格称为市场卖出价。市场要价总是大于市场买入价，这些价格会随着新来者、市场交易和取消而上下波动。我们将这两个价格过程建模为“反弹几何布朗运动（GBMs）”，它被定义为两个相互反射的布朗运动的指数。然后，我们修改这些反弹GBM，构造一个交易时间和交易价格的离散时间随机过程，该过程由正参数$\\delta$参数化。在该模型下，交易时间服从逆高斯分布，连续交易时间之间的对数收益服从正态逆高斯分布。我们的主要结果表明，对数交易价格过程是一个更新报酬过程，在适当的标度下，这个过程收敛到一个标准的布朗运动，即$\\delta\\到0$。我们还证明了修改后的询问和出价过程以$\\delta\\to0$的形式接近原始的反弹GBM。最后，我们推导了一个简单有效的交易价格预测公式，并用实际股票价格数据举例说明了该预测公式的有效性。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Trading and Market Microstructure        交易与市场微观结构
分类描述：Market microstructure, liquidity, exchange and auction design, automated trading, agent-based modeling and market-making
市场微观结构，流动性，交易和拍卖设计，自动化交易，基于代理的建模和做市
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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