Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard "quadratic" backward stochastic differential equation. The main result of the paper is the existence of a Markovian solution to this forward-backward system on an arbitrary time interval, which is obtained via a fixed-point argument on the space of absolutely continuous distribution functions. Moreover, the equilibrium obtained in this paper is able to explain several stylized facts which are not captured by the current asymmetric information models.Comment: Published at http://dx.doi.org/10.1214/15-AAP1138 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Markov bridges: SDE representation

Let $X$ be a Markov process taking values in $\mathbf{E}$ with continuous paths and transition function $(P_{s,t})$. Given a measure $\mu$ on $(\mathbf{E}, \mathscr{E})$, a Markov bridge starting at $(s,\varepsilon_x)$ and ending at $(T^*,\mu)$ for $T^* <\infty$ has the law of the original process starting at $x$ at time $s$ and conditioned to have law $\mu$ at time $T^*$. We will consider two types of conditioning: a) {\em weak conditioning} when $\mu$ is absolutely continuous with respect to $P_{s,t}(x,\cdot)$ and b) {\em strong conditioning} when $\mu=\varepsilon_z$ for some $z \in \mathbf{E}$. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.Comment: A missing reference is adde

Optimal investment with inside information and parameter uncertainty

An optimal investment problem is solved for an insider who has access to noisy information related to a future stock price, but who does not know the stock price drift. The drift is filtered from a combination of price observations and the privileged information, fusing a partial information scenario with enlargement of filtration techniques. We apply a variant of the Kalman-Bucy filter to infer a signal, given a combination of an observation process and some additional information. This converts the combined partial and inside information model to a full information model, and the associated investment problem for HARA utility is explicitly solved via duality methods. We consider the cases in which the agent has information on the terminal value of the Brownian motion driving the stock, and on the terminal stock price itself. Comparisons are drawn with the classical partial information case without insider knowledge. The parameter uncertainty results in stock price inside information being more valuable than Brownian information, and perfect knowledge of the future stock price leads to infinite additional utility. This is in contrast to the conventional case in which the stock drift is assumed known, in which perfect information of any kind leads to unbounded additional utility, since stock price information is then indistinguishable from Brownian information

Information asymmetries, volatility, liquidity, and the Tobin Tax

We develop a tractable model in which trade is generated by asymmetry in agents' information sets. We show that, even if news are not generated by a stochastic volatility process, in the presence of information treatment and/or order processing costs, the (unique) equilibrium price process is characterised by stochastic volatility. The intuition behind this result is simple. In the presence of trading costs and dynamic information, agents strategically choose their trading times. Since new (constant volatility) information is released to the market at trading times, the price process sampled at trading times is not characterised by stochastic volatility. But since trading and calendar times differ, the price process at calendar times is the time change of the price process at trading times – i.e. price movements on the calendar time scale are characterised by stochastic volatility. Our closed form solutions show that: i) volatility is autocorrelated and is a non-linear function of both number and volume of trades; ii) the relative informativeness of numbers and volume of trades depends on the sampling frequency of the data; iii) volatility, the limit order book, and liquidity, in terms of tightness, depth, and resilience, are jointly determined by information asymmetries and trading costs. The model is able to rationalise a large set of empirical evidence about stock market volatility, liquidity, limit order books, and market frictions, and provides a natural laboratory for analysing the equilibrium effects of a financial transaction tax

On pricing rules and optimal strategies in general Kyle-Back models

The folk result in Kyle-Back models states that the value function of the insider remains unchanged when her admissible strategies are restricted to absolutely continuous ones. In this paper we show that, for a large class of pricing rules used in current literature, the value function of the insider can be finite when her strategies are restricted to be absolutely continuous and infinite when this restriction is not imposed. This implies that the folk result doesn’t hold for those pricing rules and that they are not consistent with equilibrium. We derive the necessary conditions for a pricing rule to be consistent with equilibrium and prove that, when a pricing rule satisfies these necessary conditions, the insider’s optimal strategy is absolutely continuous, thus obtaining the classical result in a more general setting. This, furthermore, allows us to justify the standard assumption of absolute continuity of insider’s strategies since one can construct a pricing rule satisfying the necessary conditions derived in the paper that yield the same price process as the pricing rules employed in the modern literature when insider’s strategies are absolutely continuous

Explicit construction of a dynamic Bessel bridge of dimension 3

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V(t) satisfies V(t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V(τ), where τ:= inf {t > 0: Zt = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V(τ). We call this a dynamic Bessel bridge since V(τ) is not known in advance. Our study is motivated by insider trading models with default risk, where the insider observes the firm's value continuously on time. The financial application, which uses results proved in the present paper, has been developed in the companion paper [6]

Stock market insider trading in continuous time with imperfect dynamic information

In this paper, I study the equilibrium pricing of asset shares in the presence of dynamic private information. The market consists of a risk-neutral informed agent who observes the firm value, noise traders and competitive market makers who set share prices using the total order flow as a noisy signal of the insider's information. I provide a characterization of all optimal strategies and prove the existence of both Markovian and non-Markovian equilibria by deriving closed-form solutions for the optimal order process of the informed trader and the optimal pricing rule of the market maker. The consideration of non-Markovian equilibrium is relevant since the market maker might decide to re-weight past information after receiving a new signal. Also, I show that (1) there is a unique Markovian equilibrium price process that allows the insider to trade undetected and (2) the presence of an insider increases the market's informational efficiency, in particular, for times close to dividend payment

Dynamic Markov bridges and market microstructure: theory and applications

This book undertakes a detailed construction of Dynamic Markov Bridges using a combination of theory and real-world applications to drive home important concepts and methodologies. In Part I, theory is developed using tools from stochastic filtering, partial differential equations, Markov processes, and their interplay. Part II is devoted to the applications of the theory developed in Part I to asymmetric information models among financial agents, which include a strategic risk-neutral insider who possesses a private signal concerning the future value of the traded asset, non-strategic noise traders, and competitive risk-neutral market makers. A thorough analysis of optimality conditions for risk-neutral insiders is provided and the implications on equilibrium of non-Gaussian extensions are discussed. A Markov bridge, first considered by Paul Lévy in the context of Brownian motion, is a mathematical system that undergoes changes in value from one state to another when the initial and final states are fixed. Markov bridges have many applications as stochastic models of real-world processes, especially within the areas of Economics and Finance. The construction of a Dynamic Markov Bridge, a useful extension of Markov bridge theory, addresses several important questions concerning how financial markets function, among them: how the presence of an insider trader impacts market efficiency; how insider trading on financial markets can be detected; how information assimilates in market prices; and the optimal pricing policy of a particular market maker. Principles in this book will appeal to probabilists, statisticians, economists, researchers, and graduate students interested in Markov bridges and market microstructure theory