### Schedule for the summer school (June 12-14):

 9:00-10:30 Lecture 10:30-11:00 Coffee break 11:00-12:30 Lecture 12:30-2:30 Lunch (on our own) 2:30 - 4:00 Lecture

The 3 day summer school will consist of the following three tracks:

Competitive Equilibria by Gordan Žitkovic:

We will talk about the classical general-equilibrium model, first introduced by Leon Walras in the 1874. With an emphasis on the mathematics behind it, our goal is to touch upon the developments the theory has undergone in the 20th century, due to McKenzie, Arrow, Debreu, Radner, Smale, and others. The lectures are designed to serve as a telegraphic introduction to mathematical economics for beginning graduate students in mathematics with absolutely no background in economics.
Principal-Agent by Hao Xing (Notes):
We will introduce three Principal-Agent problems in continuous time setting: Holmstrom-Milgrom (1987), Sannikov (2008), DeMarzo-Sannikov (2007). Each of them describes a situation where Agent's action is not completely observable by Principal. Taking this information asymmetry into account, Principal offers Agent a contract to maximize her expected utility. Using the theory of stochastic control, we will unify these three problems into the same framework.
Kyle's market microstructure model by Kasper Larsen (Notes):
The lectures will start with Kyle (1985) who introduces a discrete-time equilibrium model with heterogeneously informed traders. We will then discuss various extensions including continuous-time extension of Back 1992 and to models with multiple informed traders (Foster and Viswanathan 1994, 1996).
The targeted audience is 1st and 2nd year Ph.D. students with an interest in both math and financial economics. Secondly, the lectures will be interesting for researchers working in applied math (in particular, in math-finance) who seek to expand their knowledge.

### Workshop (June 15)

Chair: Scott Robertson

 9:00-9:50 D. Kramkov 10:00-10:50 M. Rindisbacher 11:00-11:50 F. Riedel 12:00-2:00 Lunch (on our own) 2:00 - 2:50 K. Back 3:00 - 3:50 D. Seppi 3:50 - 4:20 Coffee break 4:30 - 5:20 S. Viswanathan 6:00 - Dinner at the Porch (free to all participants, cash bar)

The goal of the workshop is to give the participants a taste of what cutting edge research related to equilibrium theory looks like.

#### Abstracts

Dmitri Kramkov, Carnegie Mellon University
Title: Replication under Price Impact and Martingale Representation Property (Joint work with Sergio Pulido) (Slides)
Abstract: We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand on stocks. We show that in this model every contingent claim can be replicated with an arbitrary accuracy with respect to $\mathcal{L}_\infty$-norm. The proof is based on the result of independent interest, which shows that the family of equivalent probability measures with the Martingale Representation Property is dense in $\mathcal{L}_\infty$-norm.
Marcel Rindisbacher, Boston University
Title: Information and Derivatives (Joint work with Jerome Detemple and Angie Wang)
Abstract: This paper studies a multiasset continuous time economy with heterogeneous information and a derivative market. The derivative has a general payoff written on an underlying stock paying a future dividend distributed as a weighted sum of noncentral chi-squares. The economy is populated by informed and uninformed investors as well as investors trading on noise. The noisy rational expectations equilibrium is derived in explicit form. The equilibrium stock price is positive at all times and has a stochastic volatility which is affine in the fundamentals and the endogenous information signals. The derivative, written on the underlying stock, cannot be replicated, except at rare endogenous stopping times when the market becomes incomplete. Properties of equilibrium, such as informational efficiency and its relation to dynamic completeness, volatility structure and asset holdings behavior are examined. The behavior of asset holdings in periods surrounding times of market incompleteness is studied. The model predicts an increase in trading activity, stock holdings and derivatives open interest on expiration dates.
Frank Riedel, Universität Bielefeld
Title: Knight-Walras Equilibria
Abstract: Knightian uncertainty leads naturally to nonlinear expectations. We introduce a corresponding equilibrium concept with sublinear prices and establish their existence. In general, such equilibria lead to Pareto inefficiency and coincide with Arrow–Debreu equilibria only if the values of net trades are ambiguity–free in the mean. Without aggregate uncertainty, inefficiencies arise generically. We introduce a constrained efficiency concept, uncertainty–neutral efficiency and show that Knight–Walras equilibrium allocations are efficient in this constrained sense. Arrow–Debreu equilibria turn out to be non–robust with respect to the introduction of Knightian uncertainty.
Kerry Back, Rice University
Title: Activism, Strategic Trading, and Liquidity (Slides)
Abstract: We analyze dynamic trading in an anonymous market by an activist investor who can expend costly effort to affect firm value. We obtain the equilibrium in closed form for a general activism technology, including both binary and continuous outcomes. The optimal continuous trading strategy is independent of the activism technology. Activism, prices, and liquidity are jointly determined in equilibrium. Variation in noise trading volatility can produce either positive or negative effects on both efficiency and liquidity, depending on the activism technology and model parameters, because future effort depends on the realized amount of noise trading. The `lock in' effect emphasized in previous literature (e.g., Coffee (1991), Bhide (1993) and Maug (1998)) holds only for special forms of the activism technology. Reducing the uncertainty about the activist's position improves market liquidity, but the effect on efficiency depends on the specification of the effort cost function. Variation in the activist's productivity produces a negative cross-sectional relation between efficiency and liquidity as the possibility of more activism exacerbates the risk of adverse selection.
Duane Seppi, Carnegie Mellon University
Title: Dynamic Asset Pricing with Privately Known Investor Preferences (Slides)
Abstract: We consider a dynamic market in which heterogeneous investors are symmetrically informed about the dividend process for traded assets but are privately informed about their personal current and future investment preferences. As a result, there are two types of randomness in future prices: Randomness due to future cash flow realizations and randomness in the future stochastic discount factor due to uncertainty about investors' asset demands. In our model, the amount of stochastic discount factor randomness is endogenous because investor trading decisions reveal information over time about investors' preferences and, thus, about their future asset demands. We present a rational expectations equilibrium in which we derive dynamics for stochastic discount factor uncertainty and the associate risk premia for cash flow and preference uncertainty risk.
S. Vishwanathan, Duke University
Title: Collateral and Equilibrium Insurance
Abstract: TBA