Spread Modelling Under Asymmetric Information
Abstract Bid-ask spread is a key measure of pricing efficiency in a microstructure framework. Today there is no universal model of spread formation that includes all three factors of transaction costs, inventory risk (losses in case of a changing value of a stored asset) and information asymmetry that influence the behaviour of traders and market-makers. Empirical evaluations of these three components of spread are very contradictory (Campbell et al., The econometrics of financial markets. University Press, Princeton, 1997; Easley and O’Hara, Microstructure and asset pricing. In: George MC, Milton H, Rene HS (eds) Handbook of the economics of finance. Elsevier, Amsterdam, pp 1022-1047, 2003). In our work, after the introduction of the additional uncertainty about the real asset value, we propose an algorithm of bid-ask spread formation for the market-maker, based on classical model of Glosten and Milgrom (J Financ Econ 14:71-100, 1985). Our modification allows us to the reproduce intertemporal spread dynamics under asymmetric information and limited inventory risk of a market-maker.
Keywords Asymmetric information • Bid-ask spreads • Glosten-Milgrom • Inventory risk • Market microstructure • Price formation
JEL Classification G14, D47, D82
Today there is some controversy about the asset pricing process between the microstructure approach and macro models of finance theory. In history, we can find a similar period of misunderstanding between the micro and macro economies (Ball and Romer 1990). The experience of restoring the integrity of economic theory draws attention to the efficient market hypothesis (Fama 1970). We can 
assume that the key “macro request” to microstructure analysis is a quantitative measure of information efficiency (measure of market price deviation from fair price) or mechanism designed to regulate the asset pricing process, which provides given characteristics that suits assumptions of macro models. For instance, Agarwal and Wang (2007) stated that there is an explanation of the high descriptive power paradox of the empirical three-factor Fama-French model (Fama and French 1993), which arises because transaction costs were not taken into account. In this case, the spread acts as an indirect measure of information efficiency of the asset pricing (Roll
1985) . Today there is no universal model of spread formation that includes all three factors. Generally, when authors have modelled spread formation, they took into account only one factor, as discussed above. The impact of other factors is limited. Empirical estimations of these three factors are controversial (Campbell et al. 1997; Easley and O’Hara 2003).
The mechanism of information allocation and incorporation into the market price plays a key role in bid-ask spread formation. The American scientists L. Glosten and P. Milgrom provided in 1985 the basic research in this area. The authors, hereafter referred to as GM, constructed their model to show the influence of information asymmetry on bid-ask spread. That is why they introduced strict assumptions:
• Uniqueness of informational event and common knowledge of the moment when informed traders receive information about real asset value
• Knowledge of possible real asset value (V and V, lower higher price)
• Fixed volume for one transaction
• Traders cannot refuse to perform a transaction
• Informed traders have no power over price manipulation
• The market-maker has no need to account for inventory risk
• The market-maker has zero profit and losses
• Authors exclude competition between market-makers
The key point in obtaining a complex model of bid-ask spread formation, based on the GM model, is the accounting of inventory risk. Straight incorporation of inventory risk in a bid-ask spread yields explosive growth of spread and price. In our work, we attempt to find such a relaxation of assumptions of the GM model that allows the market-maker to implement simultaneous control of inventory costs and costs from adverse selection and, at the same time, keep the key features of the GM model (i. e. the martingale property of prices and intertemporal dynamics of spread). In our study, we do not include transaction costs.
In our work, we have made following changes in assumptions of the GM model:
• We introduced uncertainty of market-maker’s expectations about real asset value (the market-maker has no knowledge about expecting higher V and lower V prices. Instead, he/she knows only the range [ VMM; VMM], where real asset value is located).
• Informed traders make errors, but they still know about the exact expected value of real asset value (V or V).
• Informed traders can refuse to perform non-profltable transactions.
• We added some statistical functions to analyze inventory risk of the market – maker and its financial result.
Other basic assumptions of the GM model remain unchanged. The proposed modification of the original assumption was influenced by studies (Das 2005; Zachariadis 2012; Gerig and Michayluk 2010). The logic of proposed changes to the GM model is as follows:
• A necessary condition of write-off of the market-maker’s inventory costs is dilution of an informed trader’s monopoly by introducing informational uncertainty for informed traders. After that, informational uncertainty for the market-maker must also be introduced.
• The correct solution for the informed trader’s informational uncertainty problem assumes introduction of the learning mechanism. However, in our study, we restricted ourselves to the introduction of a simplified version of informed traders’ information uncertainty, which involves the consideration of a certain percentage of mistakes made by informed traders.
• We accepted that the informed trader could refuse a non-profit transaction when profit from expected operation (purchase or sale) generates a loss.
• We introduced an algorithm that allows the market-maker to correct bid-ask spread by taking into account the refusals of informed traders.
Comparative analysis of the numerical example of the proposed modification shows that the speed of incorporation of information decreases, which creates opportunities for the market-maker to control some inventory risk and adverse selection risk during bid-ask spread formation.
The rest of the paper is organized as follows. In the first chapter, we provide a review of studies concerning GM model modifications and distinguishing features of our extension. In the second chapter, we describe and analyze two stages of GM model modifications: market-maker’s uncertainty about real asset value and errors made by informed traders. In the third chapter, we conduct a comparative analysis of results of modelling the basic GM model and the modified GM model. The findings are attached.
Our violation of modification logic is connected with conservation of research chronology, when we first searched for a solution for the market-maker strategy, as the most complicated stage of GM model modification.