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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Combining these equations leaves a formulation for which contains milgeom prices. Given thatwe can interpret as the probability of the event at time given the information set. In all time periods in which the informed trader does not trade, smooth pasting implies that he must be indifferent between trading and delaying an instant.
Related Party Transactions and Financial Performance: Optimal Trading Strategies I now characterize the equilibrium trading intensities of the informed traders.
All traders have a fixed order size of. The informed trader chooses a trading strategy in order to maximize his end of game wealth at random date with discount rate. The equilibrium trading intensities can be derived from these values analytically.
In the section below, I solve for the equilibrium trading intensities and prices numerically. Let and denote the value functions of the high and low type informed traders respectively. I compute the value functions and as well as the optimal trading strategies on a grid over the unit interval with nodes.
Let denote the vector of prices. Journal of Financial Economics, 14, 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. The algorithm below computes, and. I then look for probabilistic trading intensities which make the net position of the informed trader a martingale.
First, observe that since is distributed exponentially, the only relevant state variable is at time. Finally, I show how to numerically compute comparative statics for this model. 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.
Notes: Glosten and Milgrom (1985)
Below I outline the estimation procedure in complete detail. This combination of conditions pins down the equilibrium. Relationships, Human Behaviour and Financial Transactions. Then, in Section I solve for the optimal trading strategy of the informed agent as a system of first order conditions and boundary constraints.
Notes: Glosten and Milgrom () – Research Notebook
Compute using Equation 9. Numerical Solution In the results below, I set and for simplicity.
There is a single risky asset which pays out at a random date. Value function for the high red and low blue type informed trader.
Along the way, the algorithm checks that neither informed trader type has an incentive to bluff. 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.
Substituting in the formulas for and from above yields an expression for the price change that is purely in terms of the trading intensities and the price.
Price of risky asset. I use the teletype style to denote the number of iterations in the optimization algorithm. For the high type informed trader, this value includes the value change due to the price driftthe value change due to an uninformed trader placing gosten buy order with probability and the value change due to an uninformed trader placing a sell order with probability.
Theoretical Economics LettersVol. 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 glostfnthen miogrom trading strategies are optimal for all. Thus, in the equations below, I drop the time dependence wherever it causes no confusion.
It is not optimal for the informed traders to gloshen. At each timean equilibrium consists of a pair of bid and ask prices. 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.
So, for example, denotes the trading intensity at some time in the buy direction of an informed trader who knows that the value of the asset is. Perfect competition glksten that the market maker sets the price of the risky asset. At each forset and ensure that Equation 14 is satisfied.
Bid, ask and transaction prices in a specialist market with heterogeneously informed traders
Is There a Correlation? Bid red and ask blue prices for the risky asset. Let and denote the bid and ask prices at time. Application to Pricing Using Bid-Ask. If the high type informed traders want to sell at priceincrease their value function at price by.
At the time of a buy or sell order, smooth pasting implies that the informed trader was indifferent between placing the order or not. There are forces at work here.