We revisit Kyle’s (Econometrica 53:1315–1335, 1985) model of price formation in the presence of p... more We revisit Kyle’s (Econometrica 53:1315–1335, 1985) model of price formation in the presence of private information. We begin by using Back’s (Rev Financ Stud 5(3):387–409, 1992) approach, demonstrating that if standard assumptions are imposed, the model has a unique equilibrium solution and that the insider’s trading strategy has a martingale property. That in turn implies that the insider’s strategies are linear in total order flow. We also show that for arbitrary prior distributions, the insider’s trading strategy is uniquely determined by a Doob $$h$$-transform that expresses the insider’s informational advantage. This allows us to reformulate the model so that Kyle’s liquidity parameter $$\lambda $$ is characterized by a Lagrange multiplier that is the marginal value or shadow price of information. Based on these findings, we can then interpret liquidity as the marginal value of information.
We analyze a one-period Kyle model (Kyle (1985)) where the risk-neutral informed trader can use a... more We analyze a one-period Kyle model (Kyle (1985)) where the risk-neutral informed trader can use arbitrary (linear or non-linear) deterministic strategies, and the market maker can use arbitrary pricing rules. We show that the standard linear insider’s strategy, and correspondingly, the linear pricing rule, lead to the unique equilibrium in the model, even if the possible strategies are extended to arbitrary nonlinear piece-wise continuously di¤erentiable functions of the fundamental. This means that there is a unique equilibrium in Kyle (1985), achieved on the standard linear insider’s strategy, and the linear pricing rule.
Capital Markets: Market Microstructure eJournal, 2007
Price quotes are a valuable commodity by themselves. This is a conundrum in the standard asset pr... more Price quotes are a valuable commodity by themselves. This is a conundrum in the standard asset pricing framework. We study the value of access to accurate and timely prices in a market economy explicitly taking into account that in the U.S., exchanges have property rights in the price quotes they generate. The fact that typically large institutions and sophisticated individuals obtain real time price quotes motivates us to propose a simple model based on complementarity of private information on the fundamentals and information on price. We find that granting the public access to real time pricing data has benefits such as stimulating the role of stock market monitoring. Since the effect on liquidity can be negative, exchanges need to be able to charge a fee for this service. In other situations, the exchange can also benefit from free public disclosure of price quotes. We explicitly derive an equilibrium for differentially informed traders and a profit maximizing exchange. We confi...
ABSTRACT We study the policy implications that arise from introducing a disaster relief fund to a... more ABSTRACT We study the policy implications that arise from introducing a disaster relief fund to an insurance monopoly. Such a form of government intervention can promote higher demand and lower prices in the market for catastrophe insurance. Our model predicts buyers to increase their demand by 15% and the seller to lower prices by 30%. The optimal design of the fund contains two components. On average, we find that 20% of the fund serves as complementary insurance while 80% effectively serves as catastrophe reinsurance. The model also predicts a positive correlation between the size of the disaster relief fund and the fraction that serves as complementary insurance. The downside of the design is that the seller may be overhedged, and it becomes more costly for a potential new firm to enter the market.
ABSTRACT This paper analyzes the equilibrium trading strategies of informed traders in the presen... more ABSTRACT This paper analyzes the equilibrium trading strategies of informed traders in the presence of market closures defined as periodic predictable stops of trading. We construct a dynamic auction model based on rational strategic behavior with asymmetric information across the agents. Empirical evidence indicates that market closures have important impact on the information structure of financial markets, in particular the private information flow. In our model, the insiders repeatedly increase their informational advantage over other agents by receiving private signals about fundamentals when the market is closed. In a continuous-time setting, we solve a dynamic programming problem and derive closed-form solutions for optimal intertemporal strategies of both insiders and the market maker. The key feature of insiders' optimal strategy is that they act strategically by anticipating future market closures. Because of this, even though market closures are periodic, the intertemporal pattern of optimal trading strategies is \textit{not} periodic. This aperiodicity of trading is quite important since while it is a definitive feature of the data, it has been missing from the existing theoretical literature on market closures. In agreement with broad empirical evidence, we obtain a U-shaped pattern of trading volume during the periods when the market is open, superimposed on a U-shaped pattern during the lifetime of the economy, before all information about the asset is revealed
This paper analyzes the equilibrium trading strategies of large heterogeneously informed traders ... more This paper analyzes the equilibrium trading strategies of large heterogeneously informed traders who repeatedly increase their informational advantage over other agents by receiving private signals about fundamentals. In our model, which is based on rational strategic behavior, the exact timing of private information acquisition is public knowledge and all informed traders receive their signals simultaneously. In a continuous-time setting, we derive closed-form solution for the equilibrium intertemporal trading strategies of informed traders. While our solution is quite general, it can be naturally linked with a literature on market closures by assuming that new private information arrives at the beginning of each trading day, just before the stock market reopens. The key feature of the solution is that informed traders strategically anticipate future arrivals of new information. Because of this, the intertemporal pattern of optimal trading strategies is not periodic even when the t...
We analyze the Dennert (1993) model and allow for arbitrary non- linear trading strategies of the... more We analyze the Dennert (1993) model and allow for arbitrary non- linear trading strategies of the insider and correspondingly nonlinear pricing rules set by market makers. We claim that the symmetric Nash equilibrium in this game is unique and is still given by the one presented in Dennert (1993). We also analyze the Stackelberg setting when the market makers play the roles of the leaders, and show that the standard linear equilibrium is unique if and only if a certain non- linear ODE has no bifurcation points. Our numerical analysis shows that this is indeed the case, and therefore there is a uniqueness also in the Stackelberg setting. Importantly, the static Kyle (1985) model can be viewed as a limiting case of the Dennert (1993) model in the limit when the number of competing dealers J innitely increases, J ! 1. In this limit, the dealers' rents go to zero and they set an informationally-ecient (regret-free) price. Using this and the above results on the Dennert (1993) model,...
We revisit Kyle’s (Econometrica 53:1315–1335, 1985) model of price formation in the presence of p... more We revisit Kyle’s (Econometrica 53:1315–1335, 1985) model of price formation in the presence of private information. We begin by using Back’s (Rev Financ Stud 5(3):387–409, 1992) approach, demonstrating that if standard assumptions are imposed, the model has a unique equilibrium solution and that the insider’s trading strategy has a martingale property. That in turn implies that the insider’s strategies are linear in total order flow. We also show that for arbitrary prior distributions, the insider’s trading strategy is uniquely determined by a Doob $$h$$-transform that expresses the insider’s informational advantage. This allows us to reformulate the model so that Kyle’s liquidity parameter $$\lambda $$ is characterized by a Lagrange multiplier that is the marginal value or shadow price of information. Based on these findings, we can then interpret liquidity as the marginal value of information.
We analyze a one-period Kyle model (Kyle (1985)) where the risk-neutral informed trader can use a... more We analyze a one-period Kyle model (Kyle (1985)) where the risk-neutral informed trader can use arbitrary (linear or non-linear) deterministic strategies, and the market maker can use arbitrary pricing rules. We show that the standard linear insider’s strategy, and correspondingly, the linear pricing rule, lead to the unique equilibrium in the model, even if the possible strategies are extended to arbitrary nonlinear piece-wise continuously di¤erentiable functions of the fundamental. This means that there is a unique equilibrium in Kyle (1985), achieved on the standard linear insider’s strategy, and the linear pricing rule.
Capital Markets: Market Microstructure eJournal, 2007
Price quotes are a valuable commodity by themselves. This is a conundrum in the standard asset pr... more Price quotes are a valuable commodity by themselves. This is a conundrum in the standard asset pricing framework. We study the value of access to accurate and timely prices in a market economy explicitly taking into account that in the U.S., exchanges have property rights in the price quotes they generate. The fact that typically large institutions and sophisticated individuals obtain real time price quotes motivates us to propose a simple model based on complementarity of private information on the fundamentals and information on price. We find that granting the public access to real time pricing data has benefits such as stimulating the role of stock market monitoring. Since the effect on liquidity can be negative, exchanges need to be able to charge a fee for this service. In other situations, the exchange can also benefit from free public disclosure of price quotes. We explicitly derive an equilibrium for differentially informed traders and a profit maximizing exchange. We confi...
ABSTRACT We study the policy implications that arise from introducing a disaster relief fund to a... more ABSTRACT We study the policy implications that arise from introducing a disaster relief fund to an insurance monopoly. Such a form of government intervention can promote higher demand and lower prices in the market for catastrophe insurance. Our model predicts buyers to increase their demand by 15% and the seller to lower prices by 30%. The optimal design of the fund contains two components. On average, we find that 20% of the fund serves as complementary insurance while 80% effectively serves as catastrophe reinsurance. The model also predicts a positive correlation between the size of the disaster relief fund and the fraction that serves as complementary insurance. The downside of the design is that the seller may be overhedged, and it becomes more costly for a potential new firm to enter the market.
ABSTRACT This paper analyzes the equilibrium trading strategies of informed traders in the presen... more ABSTRACT This paper analyzes the equilibrium trading strategies of informed traders in the presence of market closures defined as periodic predictable stops of trading. We construct a dynamic auction model based on rational strategic behavior with asymmetric information across the agents. Empirical evidence indicates that market closures have important impact on the information structure of financial markets, in particular the private information flow. In our model, the insiders repeatedly increase their informational advantage over other agents by receiving private signals about fundamentals when the market is closed. In a continuous-time setting, we solve a dynamic programming problem and derive closed-form solutions for optimal intertemporal strategies of both insiders and the market maker. The key feature of insiders' optimal strategy is that they act strategically by anticipating future market closures. Because of this, even though market closures are periodic, the intertemporal pattern of optimal trading strategies is \textit{not} periodic. This aperiodicity of trading is quite important since while it is a definitive feature of the data, it has been missing from the existing theoretical literature on market closures. In agreement with broad empirical evidence, we obtain a U-shaped pattern of trading volume during the periods when the market is open, superimposed on a U-shaped pattern during the lifetime of the economy, before all information about the asset is revealed
This paper analyzes the equilibrium trading strategies of large heterogeneously informed traders ... more This paper analyzes the equilibrium trading strategies of large heterogeneously informed traders who repeatedly increase their informational advantage over other agents by receiving private signals about fundamentals. In our model, which is based on rational strategic behavior, the exact timing of private information acquisition is public knowledge and all informed traders receive their signals simultaneously. In a continuous-time setting, we derive closed-form solution for the equilibrium intertemporal trading strategies of informed traders. While our solution is quite general, it can be naturally linked with a literature on market closures by assuming that new private information arrives at the beginning of each trading day, just before the stock market reopens. The key feature of the solution is that informed traders strategically anticipate future arrivals of new information. Because of this, the intertemporal pattern of optimal trading strategies is not periodic even when the t...
We analyze the Dennert (1993) model and allow for arbitrary non- linear trading strategies of the... more We analyze the Dennert (1993) model and allow for arbitrary non- linear trading strategies of the insider and correspondingly nonlinear pricing rules set by market makers. We claim that the symmetric Nash equilibrium in this game is unique and is still given by the one presented in Dennert (1993). We also analyze the Stackelberg setting when the market makers play the roles of the leaders, and show that the standard linear equilibrium is unique if and only if a certain non- linear ODE has no bifurcation points. Our numerical analysis shows that this is indeed the case, and therefore there is a uniqueness also in the Stackelberg setting. Importantly, the static Kyle (1985) model can be viewed as a limiting case of the Dennert (1993) model in the limit when the number of competing dealers J innitely increases, J ! 1. In this limit, the dealers' rents go to zero and they set an informationally-ecient (regret-free) price. Using this and the above results on the Dennert (1993) model,...
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