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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Numerical Solution In the results below, I set and for simplicity.
At each forset and ensure that Equation 14 is satisfied. The model end date is distributed exponentially with intensity. Asset Pricing Framework There is a single risky asset which pays out at a random date. Related Party Transactions and Financial Gllsten I use the teletype style to denote the number of iterations in the optimization algorithm. If the low type informed traders want to buy at price mlgrom, decrease their value function at price by.
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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.
I consider the behavior of an informed trader who trades a single risky asset with a market maker that is constrained by perfect competition. Let and denote the value functions of the high and low type informed traders respectively. This combination of conditions pins down the equilibrium. At each timean equilibrium consists of a pair of bid and ask prices.
There are forces at work here. There is an informed trader and a stream of uninformed traders who arrive with Poisson intensity. The equilibrium trading intensities can be derived from these values analytically. This cost has to be offset by the value delaying. 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.
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. Then, I iterate on these value function guesses until the adjustment error which I define in Step 5 below is sufficiently small.
Finally, I show how to numerically compute comparative statics for this model. Relationships, Human Behaviour and Financial Transactions. I compute the value functions and as well as glostem optimal trading strategies on a grid over the unit interval with nodes. The algorithm below computes, and. Thus, for all it must be that and.
Along the way, the algorithm checks that neither informed trader type has an g,osten to bluff. This effect is only significant in less active markets.
Notes: Glosten and Milgrom () – Research Notebook
At the time of a buy or sell order, smooth pasting implies that the informed trader was indifferent between placing the order or not. Scientific Research An Academic Publisher. Theoretical Economics LettersVol. Let and denote the vector of value function levels over each point in the price grid after iteration.
Optimal Trading Strategies I now characterize the equilibrium trading intensities of the informed traders. First, observe that since is distributed exponentially, the only relevant state variable is at time. Journal of Financial Economics, 14, I interpolate the value function levels at and linearly.
I then look gllsten probabilistic trading intensities which make the net position glostdn the informed trader a martingale. I seed initial guesses at the values of and.
Price of risky asset. Given thatwe can interpret as the milfrom of the event at time given the information set. Thus, in the equations below, I drop the time dependence wherever it causes no confusion. In the definition above, the and subscripts denote the realized value and trade directions for the informed traders. Let be the closest price level to such that and let be the closest price level to such that.
In fact, in markets with a higher information value, the effect of attention constraints on the liquidity provision ability of market makers is greater. Between trade price drift. Bid red and ask blue prices for the risky asset. No arbitrage implies that for all with and since:. I now characterize the equilibrium trading intensities of the informed traders.
Combining these equations leaves a formulation for which contains only prices. Compute using Equation 9. The informed trader chooses a trading strategy in order to maximize his end of game wealth at random date with milhrom rate. Let be the left limit of the price at time. Perfect competition dictates that the market maker sets the price of the risky asset. Application to Pricing Using Bid-Ask. It is not optimal for the informed traders 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 glostej and raising them when he is too apathetic about trading and vice versa for the low type trader.
In the section below, I solve for the equilibrium trading intensities and prices numerically.