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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I now characterize the equilibrium trading intensities of the informed traders. This combination of conditions pins down the equilibrium.
Bid, ask and transaction prices in a specialist market with heterogeneously informed traders
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. Compute using Equation 9. Combining these equations leaves a formulation for which contains only prices.
I begin in Section by laying out the continuous time asset pricing framework. 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.
Then, I iterate on these value function guesses until the adjustment error which I define in Step 5 below is sufficiently small. Along the way, the algorithm checks that neither informed trader type has an incentive to bluff.
Then, in Section I solve for the optimal trading strategy of the informed agent as a system of first order conditions and boundary constraints. 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.
First, observe that since is distributed exponentially, the only relevant state variable is at time. There is an informed trader and a stream of uninformed traders who arrive with Poisson intensity. Let be the left limit of the price at time. The informed trader chooses a trading strategy in order to maximize his end of game wealth at random date with discount rate. If the low type informed traders want to buy at pricedecrease their value function at price by.
Price of risky asset. No arbitrage implies that for all with and since:. At the time of a buy or sell order, smooth pasting implies that the informed trader was indifferent between placing the order or not. I compute the value functions and as well as the optimal trading strategies on a grid over the unit interval with nodes. 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.
Below I outline the estimation procedure in complete detail. Finally, I show how to numerically compute comparative statics for this model.
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nilgrom Theoretical Economics LettersVol. Update and by adding times the between trade indifference error from Equation Optimal Trading Strategies I now characterize the equilibrium trading intensities of the informed traders. I then look for probabilistic trading intensities which make the net position of the informed trader a martingale.
Let and denote the value functions of the high and low type informed traders respectively. Let denote the vector of prices. Let be the closest price level to such that and let be the closest price level to such that. Value function for the high red and low blue type informed trader.
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.
Numerical Solution In the results below, I set and for simplicity. It is not optimal for the informed traders to bluff. Let and denote the vector of value function levels over each point in the price grid after iteration. In the section below, I solve for the equilibrium trading intensities and prices numerically. Empirical Evidence from Italian Listed Companies. However, via the conditional expectation price setting rule, must be a martingale meaning that.
There are forces at work here. The Case of Dubai Financial Market. Code the for the simulation can be found on my GitHub site. Relationships, Human Behaviour and Financial Transactions.
Scientific Research An Academic Publisher. At each timean equilibrium consists of a pair of bid and ask prices.
Notes: Glosten and Milgrom () – Research Notebook
Perfect competition dictates that the market maker sets the price of the risky asset. If the high type informed traders want to sell at priceincrease their value function at price by. Thus, for all it must be that and. 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. Similar reasoning yields milgeom symmetric condition for low type informed traders.
All traders have a fixed order size of. 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. Is There a Correlation? Between trade price drift. In fact, in markets with a higher information value, the effect of attention constraints on the liquidity provision ability of market makers is greater.