Unknown Real Asset Value and Informed Traders’ Errors During the Trading Process
Our GM model extension is divided into two stages. During the first stage, the marketmaker loses its knowledge about possible real asset value, while conditions for informed and uninformed traders remain the same. During the second stage, in addition to uncertainty for the marketmaker about possible real asset value, we introduce errors of informed participants.
The disruption of the modification sequence is motivated by conservation of research chronology, when in the beginning our task was to search for a market maker’s strategy, since it was the hardest stage of GM model modification.
The first stage of modification is a relaxation of the assumption about the market maker’s knowledge of possible real asset value (V and V). This is the same as in the work of Das (2005). In our study, informed traders can refuse unprofitable transactions. Thus, the marketmaker takes into account these actions and, with some additional algorithms, reduces the range in which the real asset value is located.
Indeed, original assumptions of GM about the marketmaker’s knowledge of possible real asset value moves the GM model away from reality. In the real asset market, especially shortterm and mediumterm markets, marketmakers do not have time to calculate possible real asset value in the event of the arrival of new information. Therefore, the assumption that the marketmaker knows only the range in which the real asset value is located seems more realistic.
In contrast to GM’s mediumterm market, our modification characterizes bidask spread formation on the shortterm market, where information arrives frequently and at different times, (Gerig and Michayluk 2010). Market is still pure dealership, so all orders are market orders. There are informed traders, uninformed traders and only one marketmaker. The trading process is separated into T periods. In every period there is only one deal.
At the beginning of trade, informed traders receive real asset value V, which will be publically known at the moment of time T. The marketmaker knows that real asset value is located in the range from V_MM until VMM, i. e. V є V_MMI Vмм]. Limits of this range are used by the marketmaker to calculate bid and ask price instead of values V and V of the standard GM model. The marketmaker sets the bid and asks price, using knowledge of VMM, VMM, estimated share of informed traders д and direction of price movement 1. In the standard GM model, the 1 parameter determines the probability of the real asset value equalling the higher or lower price. In our modification the 1 parameter determines the probability that real asset value will be above or below midrange VMM; VMM].
Subsequently, based on the actions of market participants (buy, sell, refusal of a transaction) the marketmaker corrects the higher or lower limit of the range X_mm I VMM] so that one of these limits becomes equal to the real asset value.
After the marketmaker has established bid and ask, a random trader observes quotes and makes a decision: to buy or sell the asset or refuse the transaction. We assume that only informed traders can refuse a transaction and they can make a refusal only from nonprofitable transactions. There is no timing refusal in our model. Uninformed traders must make a deal at any price.
As in the GM model, we assume that in every moment of time, there is only one transaction and trading volume is limited to one block of assets, e. g. one share.
Informed traders are prohibited from performing manipulative strategies, due to random selection of traders. They cannot evaluate how many times they will participate in trading. Thus, we exclude volume and price manipulation from our modification.
Informed participants buy if real asset value is higher than the ask price of the marketmaker. If real asset value is lower than bid price, informed traders sell the asset. Finally, if real asset value is located between bid and ask, informed traders refuse the transaction. In official terms, informed traders’ actions can be described by the formula (Eq. (1)).
Sell, if V < Bid
Informed! Refuse, if Bid < V < Ask, (1)
і Buy, if V > Ask
where V—real asset value, Bid, Ask—marketmaker’s quotes to sell and buy, Sell, Buy, Refuse—informed traders’ actions.
After each transaction or refusal of a transaction, the marketmaker reviews the bid and ask prices. A specialist knows that a refusal can only be made by an informed trader when real asset value is between bid and ask quotes. Therefore, the marketmaker takes a refusal as a signal to correct the bid and ask quotes in a special way.
After initial designations, we have constructed an event tree for the first stage modification. For the full event tree, please see the appendix. The part of the event tree that corresponds with the interaction of the marketmaker and informed trader, when real asset value is located below midrange _VMM; VMM], is shown on Fig. 1.
Ask < V_MM is the event when the ask is lower than the low limit of the range, where, according to the marketmaker’s suggestions, the real asset value is located. AL is the possibility of event Ask < VMM. Bid < V_MM is the event when the bid is lower than the low limit of the range, where, according to the marketmaker’s suggestions, real asset value is located. BL—is the possibility of event Bid < VMM. Events Bid > VMM, Ask > VMM and their possibilities can be described in the same manner.
Fig. 1 Part of the event tree of the first stage modification 
The difference between event trees of the standard GM model and our modification of the first stage consists of the informed trader’s behaviour. The informed trader, as before, uses knowledge of real asset value to maximize his/her profit from every transaction. Consequently, he/she will sell if real asset value is lower than the bid. This happens when Bid > VMM, or Bid > V_MM and Ask > V_MM take place at one time. Moreover, the informed trader will buy if the ask is lower than real asset value. This happens when Ask < VMM or Bid < VMM and Ask < VMM take place at one time. In the classic GM model, V = V real asset value is above or equal to ask during the entire trading process. In our modification, bidask spread can be above or below real asset value. Thus, informed traders, by making buys and sells, give signals to the marketmaker about spread location relative to real asset value.
Moreover, in our first stage modification, it is possible that real asset value is located within bid and ask. This happens when two conditions take place at one time: Bid < VMM and Ask > VMM or Ask > V_MM and Bid < V_MM. In this case, the informed trader has no incentives to trade, because he/she will incur losses, which he/she cannot accept due to his/her utility function. Therefore, the informed trader refuses the transaction. Consequently, refusal from trade is a signal to the market – maker that real asset value lies between bid and ask.
We calculated the probability of traders’ buys and sells depending upon possible events and placed them in Table 2.
As in the GM model, in our first stage modification, to calculate ask and bid, we use formulas (Eqs. (2) and (3)).
Ask = 
E 
V 
Buy 
= VMMP 
VMM 
Buy 
+ V mm P 
V mm 
Buy 
(2) 
Bid = 
E 
V 
Sell 
= V. mm P 
^MM 
Sell 
+ V mm P 
V mm 
Sell 
(3) 
Table 2 Probability of traders’ buys and sells

Using Bayes’ rule and probability from Table 2, we derive final formulas for bid and ask calculation. For final formulas of probability from Eqs. (2) and (3), please see the appendix.
After each transaction, the marketmaker should review his/her suggestions about 1 and the probability of real asset value falling above or below midrange [VMM; V mm] . Through 1t we denote the probability of real asset value being located relative to midrange [VMM; VMM] after transaction (buy or sell) in step t. For example, if during t period the trader bought an asset from the marketmaker, then 1t will be calculated through the following formula (Eq. (4)).
A similar expression can be written for 1t(Sellt).
We did not attempt to construct an analytical form of equilibrium, because during our review of literature we generally found that for GM model modifications, the solution can be written only in numerical form. To simulate reduction of bidask spread after the trader’s refusal of a transaction, we have chosen a simple and intuitive bisection method. After refusal, the marketmaker determines the direction of the last transaction and corrects the corresponding limit of the range [ VMM; VMM] with half the size of the spread. For instance, if the last transaction was closer to the lower limit of range [ VMM; VMM], then correction can be calculated by the formula (Eq. (5)).
vmm(t) = vmm (t – 1) C (A^ – Bidt) /2, (5)
where VMM(t) is the new value of the lower limit of the range, VMM (t – 1) is previous value of lower limit of range.
For a higher limit of range correction will be (Eq. (6)).
Vmm(t) = Vmm (t – 1) – (Ask, – Bid,) /2 (6)
In addition to the formulas of bidask spread, we have calculated the inventory accumulation of the marketmaker and his/her financial result to study inventory risk. Inventory accumulation of the marketmaker is calculated through formula (Eq. (7)).
t
rsNTt =^2,tradek, t 2 [1, T], (7)
k=1
where rsNTt is total specialist inventory after t transactions, T is the whole number of operations that the marketmaker made before public disclosure of real asset
value; tradek is the quantitative result of the transaction that reflects changes in the specialist’s cash. The quantitative result of the transaction is calculated through formula (Eq. (8)).
1, if Sell
where Sell, Buy are the marketmaker’s sells or buys of an asset.
The financial result of the marketmaker is the difference between committed transactions and inventory liquidation at current price (Eq. (9)).
rsPTt = Pricek * tradek — rsNTt * Pricet, t 2 [1, T], (9)
k = 1
where rsPTt is the financial result of the specialist after committing t transactions; Pricet is price of t transaction.
Thus, our first stage of modification can be presented as follows:
• Unit of the marketmaker’s valuations and decisions (we use Eqs. (2)–(6)).
• Unit of traders’ valuations and decisions (we use probabilities at the end of nodes of the event tree).
• Unit of statistical computations (we use Eqs. (7)–(9)).
During the second stage of modification, to simulate information uncertainty of informed traders, we introduce a determined share of informed traders’ errors. Moreover, we assume that additional information uncertainty of informed traders will help the marketmaker to reduce inventory during the trading process.
Before the beginning of trade, the informed trader assumes that real asset value will be equal to V or V. At the beginning of trading, the informed trader receives a signal that future real asset value will be V with the probability nj. Consequently, the probability that real asset value will be V is equal to (1 — nj). The informed trader takes into account this signal during transactions and trades in the direction of V with the probability nj and in the direction of V with the probability of (1 — nj).
Part of the tree that corresponds to the event when real asset value V = V is located above midrange [V_MM; VMM] is shown on Fig. 2. For the full event tree, see the appendix.
Designations on Fig. 2 are equal to Fig. 1.
Probability of buys and sells of the secondstage modification can be calculated similarly to the probabilities in Table 2.
Formulas of bid and ask calculation can be derived using the same algorithm as in the firststage modification, taking into account the probability of informed traders’ errors.
The marketmaker corrects limits of range [VMM; VMM] using the same algorithm as in the firststage modification (Eqs. (5) and (6)). After correction of the limit, the marketmaker reviews bid and ask.
Fig. 2 Part of the event tree of the secondstage modification 
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