Distinguishing Features of Glosten and Milgrom Model Modification
There are numerous studies devoted to analysis of changing or relaxing assumptions of the GM model. Back and Baruch (2004) investigate relations between two major models of market mircostructure: the GM model and the Kyle model (Kyle 1985). The authors show that, under certain conditions, the equilibrium of the GM model converges into one of the equilibrium states of the Kyle model. Thus, an opportunity arises to introduce the concepts of a strategic informed trader, i. e. volume and classical characteristics of market microstructure (tightness, depth and resiliency) in bid-ask spread models. However, the authors emphasize that they managed to construct equilibrium in the GM model only for a special case and in numerical form. It should be noted that most GM and Kyle models studies use the same limitations. Takayama (2013) confirms this tendency in his 2013 paper: research of microstructure price dynamics and information disclosure is not complete, because there is no closed-form solution for equilibrium in a GM model; moreover, it is not yet known if the equilibrium is unique in the Kyle model analytical solution.
Questions about a market-maker’s existence and regulation, and necessity and conditions of competition, are closely related with the issue of High Frequency Trading (HFT). HFT replaces classic market intermediaries by providing liquidity to markets. Today, when there is no valid constraint on HFT activity, they have a number of advantages over classic market-makers: information processing and decision-making speed, instant arbitrage on many financial markets and the exclusive right to stop trading at any moment, because they do not have any commitment to maintain liquidity or pricing stability. Gerig and Michayluk (2010) tried to update the GM model by adding multiple assets and introducing HFT into the list of market participants. The authors conclude that the bid-ask spread on a market with a large share of uninformed traders is lower than in the classic model. The opposite situation is observed in a market with a high share of informed traders: bid-ask spread is higher than in the classic model. Thus, HFT activity increases informed traders’ transaction costs. After adding elasticity of liquidity traders’ demand, Gerig and Michayluk concluded that HFT helps to increase trading volume and generally decreases the transaction costs of other market participants.
Zachariadis (2012) studied the issues of information allocation in time and between market participants. In the GM model, information is distributed evenly and simultaneously between informed traders. Zachariadis (2012) modified the GM model by reducing the difference between informed and uninformed traders. This renders the GM model more realistic, because in reality every market participant has information about the real value of an asset, which changes over time. Thus, information efficiency of asset pricing is not constant over time and does not depend on the ratio of noise and informed traders. The author suggested giving every market participant the ability to learn new information about the real value of an asset from price, spread and volume dynamics, and showed that in spite of eliminating pure noise trading, the main conclusions of the GM model are still correct.
Das (2005) studied the GM model modification when the market-maker has no information about possible real asset value (V or V). At the same time, the market – maker knows the exact time when information that could change the asset price comes to market. Informed traders are still the same, as in the standard GM model. The market-maker is forced to learn real asset value from the actions of market participants (buying, selling). To do that, the author suggests a numerical algorithm for explicitly computing approximate solutions to the expected-value equations for setting prices in an extension of the GM model. Moreover, Das (2005) trained the market-maker in inventory control. Empirical analysis of the artificial time series, obtained during the author’s modification of the GM model and real market data, sufficiently differ from one another.
The main and distinguishing features of the GM model modifications discussed above and our work are described in Table 1.
When GM (1985) discussed assumptions about the market-maker’s zero profit, they accepted that a specialist accumulates inventory risk. The long-term asset market is rising; thus, according to GM (1985), the market-maker will accumulate shorts on the rising market and will not be able to sell them without losses. The authors argue that the market-maker may remain at break even in the case of competition between market-makers. In GM’s opinion, the addition of competition between market-makers must yield the Nash equilibrium (Nash 1951) and so market-makers will remain at break-even. However, competition between market – makers is not always possible. For instance, NYSE assigns only one market-maker for each asset and GM’s assumption about market-maker’s break-even cannot be satisfied.
Gerig and Michayluk (2010) created grounds for the possible relaxation of GM’s assumption about market-makers’ competition by taking into account HFT influence. Competition between market-makers, needed to create the Nash equilibrium, to some extent can be replaced with competition between the market-maker and HFT, or only between HFT. This hypothesis needs additional verification and goes beyond the scope of our work.
In contrast to straight incorporation of inventory costs into spread, as Das (2005) did, in our work the introduction of information uncertainty for market participants about real asset value allows us to change dynamic characteristics of the market – maker’s inventory costs.