Simulation Results

Our numerical simulation includes the GM model, first-stage modification (the market-maker’s uncertainty about real asset value) and second-stage modification (simultaneous uncertainty of the market-maker and the informed trader about real asset value).

We made only one trial simulation of model modification to study changes in inventory risk, price, spread and the market-maker’s financial result.[8] This simulation is only the first attempt and we recognize the need for further simulations to test our results.

We performed the simulation according to the following conditions: V = 10$ (low value), V = 20$ (high value), 1 = 0.5 (starting possibility of V = V), д = 0.2 (share of informed traders), T = 1,000 (time periods). At the beginning of the trading period, informed traders receive a signal that real asset value will be V = V = 20$. _

For the first-stage modification, we added conditions that VMM = 13$, VMM = 22$ (the range where the market-maker assumes the location of real asset value). Under these conditions at the beginning of trading, the informed traders receive a signal that real asset value will be V = V = 20$ and the market-maker learns the real asset value by taking into account the traders’ actions. For the second-stage modification, we introduce an additional parameter: the probability of informed traders’ error, жj = 0.85.

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Fig. 3 Price change for the first stage (Price modi) and the second stage of (Price mod2) modification

We repeated our tests 500 times. After that, we took the averaged value for each observed variable.

Price change graphs for the first – and second-stage modifications are shown in Fig. 3.

Simulation results prove that the market-maker manages to determine real asset value in spite of the absence of knowledge about higher and lower prices (V or V). Thus, we manage to maintain the key properties of the GM model (incorporation of informed trader information in market price) even when the market-maker has no information about possible real asset value.

One can note that the quantity of trades needed for market price to become equal to real asset price significantly increases from the first stage to the second stage of modification. This result is quite predictable, because informed traders start to make errors and trade in the opposite direction from real asset value. Thus, the share of buys increases more slowly and the market-maker gradually changes bid and ask quotes.

Graphs of bid-ask spread change for the GM model for the first – and second – stage modification are shown in Fig. 4.

With an increase in number of transactions and refusals, the spread in our modifications tends toward the lower. Consequently, knowledge of possible real asset value (V or V) is not a necessary assumption for the market-maker to find real asset value during the trading process. The spread decreases more slowly in the first-stage modification than in the GM model. This can be explained by increased uncertainty for the market-maker about real asset value. The spread in the second – stage modification decreases even more slowly, because actions of informed traders bring less information through errors.

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image130t (nominal time intervals)

Fig. 5 Market-maker’s inventory change for GM model (Inventory GM) with the first-stage (Inventory modi) and the second-stage (Inventory mod2) modification

Graphs of the market-maker’s inventory change for the GM model, for the first – and the second-stage modifications, are shown in Fig. 5:

Due to the decrease of price learning speed, inventory accumulation becomes smoother and the total volume significantly reduces in both the first and the second stages of modification in contrast to the GM model. It should be noted that we have no inventory control function, as did Das (2005). However, we reserve the right

to introduce an additional inventory control function and to include the remaining inventory costs into the spread.

Graphs of a market-maker’s financial result for the GM model, and for the first and the second stages of modification, are shown in Fig. 6:

Market-maker’s financial result for the first stage significantly increases in comparison with the GM model. The rate of profit growth is proportional to spread decrease. We can conclude that our algorithm of bid-ask spread correction is very attractive to the market-maker, because he/she earns profits. On the other hand, this result shows that our algorithm is non-optimal, because the market-maker must earn zero profit and has no losses. Due to the increased period of spread decrease in the second-stage modification, the market-maker earns even bigger profits than in the first-stage modification.

It should be noted that a positive financial result is not the purpose of our work. Without additional study, we cannot affirm that this financial result is the outcome of relaxation assumptions of the GM model. In any event, GM’s proposition about the introduction of competition between market-makers should deprive them from profits. Even in the absence of competition between market-makers, conclusions Gerig and Michayluk (2010) show that some aspects of the competition effect are created by HFT, which gradually drive out market-makers. Thus, we returned to our starting point. However, we made a spiral motion instead of a circular motion, because we achieved significant approximation of reality of the GM model.

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Fig. 6 Financial result for GM model (Fin. Res. GM), with the first (Fin. Res. modi) and the second (Fin Res. mod2) stages of modification

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Conclusion

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 (1985). Our modification allows us to reproduce the intertemporal spread dynamics under the asymmetric information and limited inventory risk of a market-maker.

Our modification of the GM model is divided in two steps. During the first step of modification we introduced uncertainty of market-maker’s expectations about real asset value. The market-maker knows only the range, where real asset value is located and learns it only by actions of traders. The second step introduces information uncertainty of informed traders. We modelled this process by introduction of a certain percentage of errors of informed traders during transactions.

Results, obtained after the modification of the GM model, allow us to make a step closer in building complex model of bid-ask spread formation. This model will include all three factors of spread formation: transaction costs, inventory risk and information asymmetry.

Our study can be extended in the following areas:

• Development of a model with the introduction of uncertainty about frequency and arrival time of new information for traders and the market – maker

• Development of the learning algorithm of informed traders

• Introduction of HFT in the trading process

• Incorporation of transaction costs

 

A.1 Appendix

A.1.1 Calculation of Probabilities, Which Are Included in Bid and Ask Quotes

 

According to Bayes’ rule:

 

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P

 

P

 

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P [Vmm]

 

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P

 

P

 

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P BuyJ

 

Full probabilities of making buy and sell can be described by formulas:

P [Buy] = AL^S C 0.5 (1 — д) l

C (1 — AH) (1 — BH) д (1 — l) C 0.5 (1 — д)(1 — l)

P [Sell] = (1 — BL) (1 — AL) дl C 0.5(1 — д)l C ВНд (1 — l) C 0.5 (1 — д) (1 — l)

Integrating full probabilities into formulas, obtained according to Bayes’ rule, yields final formulas for calculating probabilities, which are included in bid and ask quotes. For example, the probability that real asset value asset is equal to lower limit of range V_mmI VMM] can be calculated through this formula:

P vmm Buy

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l (ALn C 0.5 (1 — д))

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A.1.2 Event Tree for the First-Stage GM Model Modification

A.1.3 Event Tree for the Second-Stage GM Model Modification

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Acknowledgments The author would like to thank Igor Zakharov for his helpful comments, suggestions and support throughout the work on this paper.

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