By Lawrence R. Glosten and Paul Milgrom; Bid, ask and transaction prices in a specialist market Journal of Financial Economics, , vol. Dealer Markets Models. Glosten and Milgrom () sequential model. Assume a market place with a quote-driven protocol. That is, with competitive market. Glosten, L.R. and Milgrom, P.R. () Bid, Ask and Transactions Prices in a Specialist Market with Heterogeneously Informed Traders. Journal.
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Relationships, Human Behaviour and Financial Transactions. If the trading strategies are admissible, is a non-increasing function ofis a non-decreasing function ofboth value functions satisfy the conditions above, and the trading strategies are continuously differentiable on the intervalthen the trading strategies are optimal for all.
Furthermore, the aggregate level of market liquidity remains unaltered across both highly active and inactive markets, suggesting a reactive strategy by informed traders who step in to compete with market makers during high information intensity periods when their attention allocation efforts are compromised.
Notes: Glosten and Milgrom () – Research Notebook
Numerical Solution In the results below, I set and for simplicity. First, observe that since is distributed exponentially, the only relevant state variable is at time.
Journal of Financial Economics, 14, Optimal Trading Strategies I now characterize the equilibrium trading intensities of the informed traders. Let be the left limit of the price at time. Scientific Research An Academic Publisher. The algorithm updates the value function in each step by first computing how badly the no trade indifference condition in Equation 15 is violated, and then lowering the values of for near when the high type informed trader is too eager to trade and raising them when he is too apathetic about trading and vice versa for the low type trader.
Let be the closest price level to such that and let be the closest price level to such mipgrom.
At each forset and ensure that Equation 14 is satisfied. No arbitrage implies that for all with and since: Finally, I show how glosetn numerically compute comparative statics for this model.
Combining these equations leaves a formulation milgorm which contains only prices. I consider the behavior of an informed trader who trades a single risky asset with a market milgromm that is constrained by perfect competition. In order to guarantee a solution to the optimization problem posed above, I restrict the domain of potential trading strategies to those that generate finite end of game wealth.
Then, in Section I solve for the optimal trading strategy of the informed agent as a system of first order conditions and milgrm constraints. Perfect competition dictates that the market maker sets the price of the risky asset. Theoretical Economics LettersVol. It is not optimal for the informed traders to bluff. There are forces at work here.
Let and denote the bid and ask prices at time. I seed initial guesses at the values of and.
This combination 19855 conditions pins down the equilibrium. The estimation strategy uses the fixed point problem in Equation 13 to compute and given and and then separately uses the martingale condition in Equation 9 to compute the drift in the price level.
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I use the teletype style to denote the number of iterations in the optimization algorithm. I now want to derive a set of first order conditions regarding the optimal decisions of high and low type informed agents as functions of these bid and ask prices which can be used to pin down the equilibrium vector of trading intensities. At each timean equilibrium consists of a pair of bid and ask prices. This implies that informed traders may not only exploit their informational advantage against uninformed traders but they may also use it to reap a higher share of liquidity-based profits.
There is a single risky asset which pays out at a random date. Along the way, the algorithm checks that neither informed trader type has an incentive to bluff. Value function for the high red and kilgrom blue type informed trader.
Notes: Glosten and Milgrom (1985)
Bid red and ask blue prices for the risky asset. Related Party Transactions and Financial Performance: In all time periods in which the informed mipgrom does not trade, smooth pasting implies that he must be indifferent between trading and delaying an instant.
For the high type informed trader, this value includes the value change due to the price millgromthe value change due to an uninformed trader placing a buy order with probability and the value change due to an uninformed trader placing a sell order with probability.
The algorithm below computes, and. The informed trader chooses a trading strategy in order to maximize his end of game wealth at random date with discount rate.