Digital Techniques Share Price Modeling based on a Time-varying Walrasian Equilibrium under Exchange Processes in the Financial Market

This paper aims to develop an analytical theory of share pricing in a financial market environment. The proposed approach corresponds to an actual auction mechanism in which an electronic stock exchange terminal processes real-time data. The theoretical framework is based on the microeconomic model of an individual investor’s net demand. Equity resources and resources of “free” capital (exchanged for shares) owned by traders and the investors’ perception of the structure of the target portfolio are considered the initial variables of the model. The model differs from the classical theory of asset pricing in its notion of fundamental variables. The relations derived for the aggregated net demand in the stock market describe share pricing as a market exchange that results in the Walrasian equilibrium approximation. The authors offered an appropriate econometric technique for estimating the parameters of instantaneous aggregated net demand. The developed approach was tested using the Walrasian equilibrium concept, which demonstrated that the modeled share price corresponded with the observed share price for the Russian financial market. As an example, the authors presented the results of the investment strategy based on the developed approach for the Russian financial market during the 2008–2009 crash. The authors based their approach on identifying the expectations of the stock market participants through an analysis of high-frequency trading platform information. The microeconomic model describes the motives of traders for placing limit orders in the stock market and associates the price and volume of a particular limit order within the parameters of the capital capacity of the net demand of the trader. The application of the algorithm allows for the monitoring of the financial market situation and reveals the market expectations of traders based on the analysis of information transmitted by an order book of a trading platform.

Capital asset pricing; Stock exchange; Stock prices; Trading algorithm; Walrasian equilibrium in the Financial Market

    The need to use digital methods and tools to assess various economic phenomena associated with economic risk and uncertainty has been repeatedly emphasized in several modern papers (Lyukevich et al., 2020; Polyanin et al., 2020; Berawi, 2021). Platforms for online trading have made it possible to monitor the environment of an order-driven market and analyze it in real time. Additionally, they have opened access to valuable information on intra-market processes, which is much richer than commonly used data on prices and trading volumes. The use of this information permits one to track the harbingers and the emergence of new market price trends. In this context, the most important and interesting information, as we see it, pertains to the flows of buy and sell limit orders. Theoretical models, methods, and some results of processing this information can be found in many contributions investigating the market microstructure (O'Hara, 2015; Easley et al., 2016).

There are well-established classical models of capital asset pricing, typically categorized into utility-based and arbitrary-based models (Sharpe, 1964). At the same time, these models used by academics and practitioners have been subject to serious criticism for at least two reasons. First, many authors have noted that the basic assumptions of classical models are not realistic (Dionne and Li, 2011; Zhang and Dong, 2015). Most issues arise from the "self-contradictoriness" of classical models that consider market asset pricing as a solution to a problem of portfolio optimization (in other words—a problem of intertemporal choice) by an "aggregated" investor that is thought of as an assembly of all security holders. Second, classical models do not adequately explain systematic biases, such as stock return excess volatility. Further studies of these effects have led to the appearance of new areas of research in financial economics: behavioral finance (Shiller, 2014), forecasting of stock market volatility (Engle, 2001; Lansing and LeRoy, 2014), and studies of information asymmetry and market manipulation (Aggarwal and Wu, 2006; Zhang and Dong, 2015). Moreover, global financial crisis transmission effects add to the complexity of capital asset pricing models (Mendoza and Quadrini, 2010; Reinhart and Rogoff, 2014). Cochrane (2011) and suggest that refocusing the analysis on prices and long-run payoff streams rather than one-period returns may mitigate the difficulties associated with the classical theory.

Microeconomic models of equilibrium prices for interacting markets are the subject of general equilibrium theory. The Walrasian equilibrium concept (Walras, 1874) has been used in several papers to solve applied asset pricing problems for different markets (Huberman and Stanzl, 2005; Hammond, 2017; Beloso and Garcia, 2020; Paes and Wong, 2020; Ruiter, 2020). Nevertheless, not enough attention has been paid to the possibility of using this concept for predicting prices in the stock market and its practical implementations in the form of investment strategies in the conditions of modern online trading. This is further complicated by the financial market’s frequent price changes. In addition, economists have repeatedly faced the problem of designing mechanisms or processes to ensure that Walrasian equilibrium in the market system can actually be achieved (Hammond, 2017). In particular, Huberman and Stanzl (2005) studied the effect of market liquidity on asset pricing. These authors, however, adhered to classical frameworks in which market share prices are established due to shareholders' intertemporal choice. Microeconomic and phenomenological models have explained the structure of the bid-ask spread (Huang and Stoll, 2001) and the influence of investor heterogeneity on market demand and supply fluctuations. They also examined the relationship between the flows of limit orders, bid–ask spread, market depth, and price volatility (Chordia et al., 2000), etc.

Petrov et al. (2013) suggested an alternative version of the Walrasian approach, which focused on share pricing phenomena under exchange among trading participants. The framework does not turn to the principle of intertemporal choice; thus, the condition ofl utility maximization by individuals becomes irrelevant. On the contrary, classical asset pricing theory (Cochrane, 2011) uses other original variables: expected future payoff on shares, statistical characteristics of their random returns, allocation of risky and risk-free assets among shareholders, and the degree of their risk aversion. It must also be noted that, in accordance with the classical point of view, share prices are settled when all investors aspire to optimize their portfolios. In addition, the classical approach posits a time period for investments.

This paper proposes an analytic description of the time-varying Walrasian equilibrium in the stock exchange by constructing a momentary aggregate net demand function. We perform such an experiment in two different ways. First, a direct comparison of momentary calculated share price (estimated per the analytical model) with the observed market price is possible. Second, we explain the observational dependence between stock market demand and supply features and the direction of the ensuing share price trend, which Petrov et al. (2017) previously detected. The method may be considered valid over the long term if active investment strategies based on regularities that the theory provides demonstrate promising results. The authors’ approach involves the development of a theory and a practical technique of asset pricing in the financial market, which corresponds to the real mechanisms of an auction in the stock exchange’s electronic terminal. This allows for the real-time processing of stock data and allows the prediction of prices and construction of investment strategies in comparison with other approaches based on the concept of Walrasian equilibrium. 

We propose a model that describes the share pricing phenomena under stock exchange trading using the Walrasian concept of momentary market equilibrium. The model is substantially different from the classical theory of asset pricing in its notion of the fundamental variables responsible for share evaluation. In the classical model, every individual solves the problem of intertemporal choice; in contrast, our model posits an equality of simultaneous opposite flows of orders to buy and sell. It is not an optimization problem per se; there is no objective function for the problem of exchange (Neumann and Morgenstern, 2007). Finally, in describing the market exchange phenomena, the suggested model operates directly with share prices; in contrast, the classical approach typically evaluates the expected return on an asset.

We have developed a technique for verifying the model; its test results on highly liquid Russian shares demonstrates that share price modeling per the Walrasian equilibrium concept is in good agreement with the observed share prices in the stock exchange. However, the closeness of agreement of the suggested model decreases slightly if we consider less liquid shares. In a practical sense, the model opens the way for online diagnostics of the behavior of investors who trade stocks based on information transmitted by trading terminals. Our experimental procedure permits the analysis of the dynamics of the trading behavior of either "free capital holders" or shareholders on the demand and supply sides. The dynamic behavior that we uncover may reflect insiders’ activity in the stock exchange. Active portfolio strategies monitoring the activity of large investors using the model may yield significant returns.

    The study was carried out within the framework of the basic part of the state assignment of the Ministry of Education and Science of the Russian Federation, project N?. 0729-2020-0056 “Modern methods and models for diagnosing, monitoring, preventing and overcoming crisis phenomena in the economy in the context of digitalization as a way to ensure the economic security of the Russian Federation”.

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