Abstracts

Short Course Abstracts

Umut Cetin:  Asymmetric Information Models

One of the main goals of the Market Microstructure Theory is to understand the temporary and permanent impacts of trade on the asset price and how the price-setting rules evolve in time. While the temporary impacts arise as a result of inventory considerations of the market makers or dealers, the permanent impacts are due the asymmetric information in the market. 

The lectures will introduce the Kyle model as a canonical set up for markets with asymmetric information, where the orders are batched, and study its several extensions. This will require some understanding of linear and nonlinear filtering theory, enlargement of filtrations, and conditioning of Markov processes. If time permits, the connection with the sequential order model of Glosten and Milgrom will also be discussed.

Frank Riedel:  Knightian Uncertainty

Knightian Uncertainty has emerged as a major research topic in recent years.  Frank Knight's pioneering dissertation on "Risk, Uncertainty, and Profit"  distinguishes risk—a situation that allows for an objective probabilistic description—from uncertainty—a situation that cannot be modeled by a single probability distribution. By now, it is widely acknowledged that such Knightian uncertainty is crucial in many fields, including financial markets, climate economics, and pandemics.

The lectures introduce the main decision-theoretic models that have been developed. Applications to finance (risk measures and management, absence of arbitrage, robust portfolio choices) and economics (game theory, mechanism design, climate change) are discussed in detail.

Kim Weston:  Tackling Radner Equilibrium with BSDEs

Radner equilibria are equilibria where the economic agents can trade in a financial market.  The agents seek to maximize their expected utility from consumption, subject to market clearing constraints.  In complete financial market settings, Radner equilibria implement Arrow-Debreu equilibria and can be solved using a representative agent approach.  This approach reduces the question of equilibrium existence to a finite-dimensional optimization problem.  In incomplete financial market settings, there is no such known simplification.  One successful tool for studying the existence of Radner equilibria in incomplete market settings is to characterize equilibria in terms of solutions to (systems of) backward stochastic differential equations (BSDEs).

In these lectures, we will derive systems of BSDEs that characterize Radner equilibria.  We will use Mathematica and show how Mathematica can be useful in efficiently deriving systems.  Will discuss the structure of the BSDE systems and how their structure relates to the current literature of existence results for BSDE systems.

Jianfeng Zhang:  Mean-Field Games

Mean field games, and the related mean field control problems, characterize the asymptotic behavior of certain large symmetric systems with weak interaction. One main feature is the involvement of the law of the representative's state process and hence the theory builds on the analysis on the Wasserstein space of probability measures. 

In these introductory lectures we will start from the standard stochastic control problems and finite player games, and discuss heuristically how the $N$-player problems lead to the mean field problems. We then introduce briefly the stochastic analysis on the Wasserstein space. The main theory will consist of three parts: (i) HJB equation for mean field control problems; (ii) Master equation for mean field games, when the mean field equilibrium is unique; and (iii) The convergence from the $N$-player problem to the mean field problem, with or without the uniqueness.

Invited Talk Abstracts

Peter Bank:  t.b.d.

Umut Cetin:  t.b.d.

Ibrahim Ekren:  t.b.d.

Sergey Nadtochiy:  Cascade equation for Stefan problem as a mean field game

The solutions to Stefan problem with Gibbs-Thomson law (i.e., with surface tension effect) are well known to exhibit singularities which, in particular, lead to jumps of the associated free boundary along the time variable. The correct times, directions and sizes of such jumps are only well understood under the assumption of radial symmetry, under which the free boundary is a sphere with varying radius. The characterization of such jumps in a general multidimensional setting has remained an open question until recently. In our ongoing work with M. Shkolnikov and Y. Guo, we have derived a separate (hyperbolic) partial differential equation — referred to as the cascade equation — whose solutions describe the jumps of the solutions to the Stefan problem without any symmetry assumptions. It turns out that a solution of the cascade equation corresponds to a maximal element of the set of all equilibria in a family of (first-order local) mean field games. In this talk, I will present and justify the cascade equation, will show its connection to the mean field games, and will prove the existence of a solution to the cascade equation. If time permits, I will also show how these results can be used to construct a solution to the Stefan problem itself.

Frank Riedel:  Insuring Model Uncertainty

Model uncertainty, also known as Knightian uncertainty, has become a major research topic in recent years. On the decision-theoretic side, various approaches show how one can successfully capture model uncertainty with the help of mathematical models. The lecture reviews recent  model of preferences under Knightian uncertainty. These approaches are closely related to attempts to quantify risk in finance. A particular focus will be on the so-called smooth model, an ambiguity-averse version of a second-order Bayesian Ansatz,  that goes back to Klibanoff, Marinacci, and Mukerji (Econometrica 2005). We will  study its axiomatic foundations and discuss the relationship of this approach with statistics, in particular the issue of identification of models (Denti, Pomatto, Econometrica 2022). Moreover, we show how the smooth model is related to variational and coherent risk measures.

We then investigate consequences of model uncertainty for the insurance market. We study the case where the typical consumer in the economy is ambiguity-averse with smooth ambiguity preferences  and the set of priors   is point identified, i.e., the true law  can be recovered ex post empirically from observed events. The   identifiability of models allows to write insurance contracts on models, with    important economic consequences. We are able to construct a representative agent who, in general, has also smooth ambiguity preferences, yet with a model-dependent ambiguity attitude. We illustrate our results in the classic Wilson framework of risk sharing where the representative agent has model-independent ambiguity attitude and insurance against ambiguity can be explicitly computed.

Jianfeng Zhang:  Set Values for Nonzero Sum Games

Nonzero sum games typically have multiple Nash equilibria, and more importantly, different equilibria may induce different values. In this talk we propose to study the set of values over all possible equilibria, called the set value of games. This set value is by nature unique, and it shares many nice properties of the value function for a standard control problem. In particular, it satisfies the dynamic programming principle and the regularity/stability. We shall also discuss how to characterize the dynamic set value function of games through set valued Hamiltonians and set valued PDEs. The talk is based on a series of works, joint with Feinstein, Iseri, Qiao, and Rudloff.

Contributed Talk Abstracts

Maxim Bichuch:  Hedging the Divergence Loss of the Constant Product Market Maker in Decentralized Finance

Automated Market Makers (AMMs) are a decentralized approach for creating financial markets by allowing investors to invest in liquidity pools of assets against which traders can transact. Liquidity providers are compensated for making the market with fees on transactions. The collected fees, along with the final value of the pooled portfolio, act as a derivative of the underlying assets with price given by the price of the initially pooled assets. Following this notion, we study the hedging portfolio for the divergence loss, also called the impermanent loss, of the liquidity providers.

Nikolaos Constantinou:  Equilibria in incomplete markets – an FBSDE approach

Starting with a complete-market specification, we study equilibrium asset pricing over infinite time horizon in an incomplete market, where the incompleteness stems from an extra source of randomness for the dividend stream. We consider two heterogeneous agents with either CARA or CRRA preferences. In both cases, the equilibrium condition leads to a system of strongly coupled Forward-Backward Stochastic Differential Equations (FBSDEs). This talk is based on joint work in progress with Martin Herdegen.

Justin Gwee:  Equilibrium Asset Pricing with Proportional Transaction Costs in a Stochastic Factor Model

We consider an economy with two agents, each of whom re- ceive a random endowment flow. This cumulative flow is mod- elled as a stochastic integral of a deterministic function of the economy’s state, and the economy’s state is modelled by means of a general Itoˆ diffusion. Each of the two agents have mean- variance preferences with different risk-aversion coefficients. The two agents can also trade a risky asset to hedge against the fluc- tuations of their endowment streams. We determine the agents’ optimal (Radner) equilibrium trading strategies in the presence of proportional transaction costs. In particular, we derive a new free boundary problem that provides the solution to the agents’ optimal equilibrium problem. Furthermore, we derive the ex- plicit solution to this free boundary problem when the problem data is such that the frictionless optimiser is a strictly increasing or a strictly increasing and then strictly decreasing function of the economy’s state. Finally, we derive small transaction cost asymptotics.

Eduardo Ferioli-Gomes:  Directional High Frequency Trading in the Kyle-Back Model

In traditional Kyle-Back models, the only source of information comes from the insider’s signal. We consider a more realistic version of the Kyle-Back model with a private and a public signal. The insider observes both signals. The private signal, that is only directly observed by the insider, may be static, when the insider knows the value of the asset in advance, or dynamic, when it converges to the true value of the asset at the end of the trading period. The market maker receives a dynamic signal that also converges to the true value of the asset at the end of the trading period.

In the dynamic case, we prove that the insider’s valuation of the asset is given by a linear combination of both the public and private signals and it is a martingale the insider’s filtration.

Furthermore, we show that the price - which is the market maker’s valuation of the asset - is also given by a linear combination of the public signal and the weighted demand. In addition, it is proven that it converges to the true price of the asset as it is expected in the traditional theory.

An interesting fact that is observed is that it is possible to see an increase in the volatility of the price in the end of the trading period when trading becomes aggressive due to the convergence of both signals to the true price of the asset.

Melih Iseri:  Set Valued Hamilton-Jacobi-Bellman Equations

Building upon the dynamic programming principle for set valued functions arising from many applications, in this paper we propose a new notion of set valued PDEs. The key component in the theory is a set valued It\^{o} formula, characterizing the flows on the surface of the dynamic  sets. In the contexts of multivariate control problems, we establish the wellposedness of the set valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, a moving scalarization for certain time inconsistent problems is constructed by using the classical solution of the set valued HJB equation.

Bin Zou:  Strategic underreporting and optimal deductible insurance

This paper proposes a theoretical insurance model to explain well-documented loss underreporting and to study how strategic underreporting affects insurance demand. We consider a utility-maximizing insured who purchases a deductible insurance contract and follows a barrier strategy to decide whether she should report a loss. The insurer adopts a bonus-malus system with two rate classes, and the insured will move to or stay in the more expensive class if she reports a loss. First, we fix the insurance contract (deductibles) and obtain the equilibrium reporting strategy in semi-closed form. A key result is that the equilibrium barriers in both rate classes are strictly greater than the corresponding deductibles, provided that the insured economically prefers the less expensive rate class, thereby offering a theoretical explanation to underreporting. Second, we study an optimal deductible insurance problem in which the insured strategically underreports losses to maximize her utility. We find that the equilibrium deductibles are strictly positive, suggesting that full insurance, often assumed in related literature, is not optimal. Moreover, in equilibrium, the insured underreports a positive amount of her loss. Finally, we examine how underreporting affects the insurer’s expected profit.