Kerry Back

Imperfect Competition among Informed Traders

We analyze competition among informed traders in the continuous-time Kyle(1985) model, as Foster and Viswanathan (1996) do in discrete time. We explicitly describe the unique linear equilibrium when signals are imperfectly correlated and confirm the conjecture of Holden and Subrahmanyam (1992) that there is no linear equilibrium when signals are perfectly correlated. One result is that at some date, and at all dates thereafter, the market would have been more informationally efficient had there been a monopolist informed trader instead of competing traders. The relatively large amount of private information remaining near the end of trading causes the market to approach complete illiquidity. Copyright The American Finance Association 2000.

Asset pricing for general processes

Abstract This paper presents some asset pricing results for the general case in which asset prices can jump. Asset gains (price plus cumulative dividends) processes are assumed to be special semimartingales. This is the broadest class of processes for which local risk premia exist. The risk premium can be separated into a premium for the continuous part of the return and a premium for the jump part. The jump premium is non-zero only for jumps which occur simultaneously with jumps in the state price density process. The CCAPM holds if consumption rates vary continuously with time, in which case all jump risks are unpriced. More generally, the CCAPM always characterizes the continuous part of the security return.

Auctions of divisible goods with endogenous supply

Abstract Uniform-price auctions are studied in which the seller may cancel part of the supply after observing the bids. This feature eliminates many of the ‘collusive seeming’ equilibria of the auction. In equilibrium the seller always sells the full quantity.

On the fundamental theorem of asset pricing with an infinite state space

An example is given of a securities market in which there is no arbitrage and a risk-neutral agent has an optimal demand subject to a minimum wealth constraint, yet there is no risk-neutral probability measure and no state price density. Also, there is no linear pricing rule on Lp for any p < ∞. This failure of the ‘Fundamental Theorem of Asset Pricing’ is due to a lack of countable additivity of the pricing operator in the market. Some sufficient conditions are also given for the existence of a risk-neutral probability measure and state price density for pricing L∞ claims.

The Shadow Price of Information in Continuous Time Decision Problems

We formulate a continuous time stochastic control problem and establish the existence of the shadow price of information. This shadow price is the Lagrange multiplier for the constraint that the control be adapted or predictable; it is a stochastic process of integrable variation, and, in one formulation, it is a martingale. The results are applied to problems of security investment, selling an asset, and economic growth. In the last application, it is shown that the existence of the shadow price of information implies the validity of the stochastic maximum principle

Concepts of similarity for utility functions

Abstract A generalization of the compact-open topology is defined for a space of utility functions with different choice sets. The space is a complete separable metric space. A continuous representation theorem of Levin (1983) gives a homeomorphism between this space and the space of preference relations, topologized by closed convergence. A map into the space is measurable with regard to the Borel algebra iff the choice set correspondence is measurable and the utility function is measurable on its graph. A coarser topology (that of subgraph convergence) is also studied. This topology is coarser than the compact-open topology when the utility functions are defined on the same choice set. However the demand correspondence is still upper-hemicontinuous. A homeomorphism is given between this space and the space of preference relations, with the latter given a certain topology coarser than closed convergence.

Structure of consumption sets and existence of equilibria in infinite-dimensional spaces

The price supportability of Pareto optima and existence of competitive equilibrium are established when consumption sets as well as preferences are uniformly proper consumption sets can be different from the non-negative cone. A sufficient condition for uniform properness of a consumption set is that it be the upper-contour set of a well behaved function. Bewley’s (1972) theorem on the existence of equilibrium prices in L’ is also extended by allowing for somewhat more genera1 consumption sets. A class of examples is given to demonstrate the need for some restrictions on the structure of consumption sets.

Strategic Liquidity Provision in Limit Order Markets

We characterize and prove the existence of Nash equilibrium in a limit order market with a finite number of risk-neutral liquidity providers. We show that if there is sufficient adverse selection, then pointwise optimization (maximizing in p for each q) in a certain nonlinear pricing game produces a Nash equilibrium in the limit order market. The need for a sufficient degree of adverse selection does not vanish as the number of liquidity providers increases. Our formulation of the nonlinear pricing game encompasses various specifications of informed and liquidity trading, including the case in which nature chooses whether the market-order trader is informed or a liquidity trader. We solve for an equilibrium analytically in various examples and also present examples in which the first-order condition for pointwise optimization does not define an equilibrium, because the amount of adverse selection is insufficient.

Incomplete and Asymmetric Information in Asset Pricing Theory

These notes could equally well be entitled “Applications of Filtering in Financial Theory.” They constitute a selective survey of incomplete and asymmetric information models. The study of asymmetric information, which emphasizes differences in information, means that we will be concerned with equilibrium theory and how the less informed agents learn in equilibrium from the more informed agents. The study of incomplete information is also most interesting in the context of economic equilibrium.

The Informational Role of Stock and Bond Volume

In a Kyle (1985) model, the sign of the correlation between a firms debt and equity returns is the same as the sign of the cross-market Kyles lambda. The sign is positive (negative) if private information concerns the mean (risk) of the firms assets. We show empirically that information conveyed by order flows is primarily about asset means. The cross-market lambdas are quite large; consequently, the portions of bond and stock returns explained by order flows are highly correlated, even though the order flows themselves are virtually uncorrelated.