Abstracts

Guillermo Alonso Alvarez: Optimal Contract Design via Relaxation: Application to the Problem of Brokerage Fees

I will present a new approach to show the existence of optimal contracts based on the relaxation of the agent's optimal control problem. Introducing the notion of "relaxed" controls of the agent we prove the existence of optimal contracts in models where the state is given by a diffusion process with linearly controlled drift. Under concavity assumptions we show that the "relaxed" optimal contracts solve their associated strong optimal contract problem. The main advantage of our model is that it allows us to (i) write an optimal contract as a limit of epsilon-optimal contracts, ii) show the existence of optimal contracts for non-standard Principal-Agent problems (state constraints or difference in flirtations between Principal and Agent). These advantages make this approach well suited for the problem of optimal brokerage fees, in which a client of a broker has access to a larger filtration (representing the client's trading signal). I will show how the latter problem can be solved using the relaxed control approach. This is a joint work with Sergey Nadtochiy.

Ofelia Bonesini:  A McKean-Vlasov game of commodity production, consumption and trading

We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on λ units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. 

We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean-Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity λ such that the players agree on the price. 

Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price. This is joint work with René Aïd, Giorgia Callegaro, and Luciano Campi.

Alexandra Chronopoulou: Discrete-time Approximation of Rough Volatility Models  

“Rough” volatility models (RVM) have been introduced to describe the anti-persistent behavior of the volatility of financial assets. These are models in which the stock follows a geometric Brownian motion, with volatility described by a fractional Ornstein-Uhlenbeck process with Hurst parameter less than 1/2. In the first part of this talk, we will introduce a new framework for the estimation of the volatility process of an asset using low frequency daily option trading entries. We will apply this method to S&P 500 data and obtain estimates of the Hurst parameter that motivate the need for RVM. In the second part of the talk, we will establish the weak convergence of a novel Donsker-type scheme for RVM, which leads naturally to a Binomial tree for option pricing.

Albina Danilova: Order routing and market quality: Who benefits from internalization? 

The practice of retail internalization has been a controversial topic since the late 1990s. The crux of this debate is whether this practice benefits, via the price improvement relative to exchange, or disadvantages, via the reduced liquidity on exchange, retail traders.

To answer this question we set two models of market design that differ in their mode of liquidity provision: in the model capturing retail order internalization the liquidity is provided by market makers (representing wholesalers) competing for the retail order flow in a Bertrand fashion, whereas in the model characterizing the open exchange the price-taking competitive agents act as liquidity providers.  We discover that, when liquidity providers in both market designs are risk averse, routing of the marketable orders to the wholesalers is preferred by all retail traders: informed, uninformed and noisy.  

In addition to addressing optimal order routing problem, we identify a  universal parameter that allows comparison of market liquidity, profit and value of information across different markets and demonstrate that  the risk aversion of liquidity providers fundamentally changes market outcomes. In particular, we observe  mean reverting inventories, price reversal, and lower market depth as the result of retail investor (informed or not) absorbing large shocks in their inventory to compensate for the unwillingness of liquidity providers to bear risk.

Gökçe Dayanikli: A Single-level Machine Learning Approach to Solve Stackelberg Mean Field Game Problems  

In this talk, we discuss a single-level numerical approach that uses machine learning techniques to solve bi-level Stackelberg mean field game problems between a principal and a mean field of agents. In Stackelberg mean field game, the mean field of agents play a non-cooperative game and choose their controls to optimize their individual objectives by interacting with the principal and other agents in the society through the population distribution. The principal can influence the resulting mean field game Nash equilibrium through incentives to optimize its own objective. This creates a bi-level problem where at the lower level, the equilibrium of the population given the policies of the principal is found and at the upper level, policy optimization of the principal is executed given the reactions of agents. We rewrite this bi-level problem between the principal and mean field of agents as a single-level problem to implement an efficient numerical approach to solve it.  We discuss the convergence of the solution of the single-level problem to the original problem. Later, we extend our numerical approach to a more generalized problem setting and discuss the experiment results on different financial applications such as regulating systemic risk and finding an optimal contract for a large number of agents.

(This is a joint work with Mathieu Lauriere.)

Filippo De Angelis:  Multilevel Function Approximation

We consider function approximations for which the synthetic training set is generated by means of expensive numerical methods and is, thus, the dominant part of the computational cost. We show that multilevel ideas can reduce the computational cost by generating most samples with low accuracy at a corresponding low cost, with relatively few high-accuracy samples at a high cost.

As an application of the multilevel approach, we consider learning the function that maps the parameters of the model and of the financial product to the price of the financial product. In the simple case of one-layer neural networks and second-order accurate finite difference methods, the computational cost to achieve accuracy O(ε) is reduced from O(ε−4−dX/2) to O(ε−4), where dX is the dimension of the underlying pricing PDE. The analysis is supported by numerical results showing significant computational savings.

This is joint work with Prof. Mike Giles and Prof. Christoph Reisinger.

Xin Guo: Mathematics of transfer learning and transfer risk:  from medical to financial data analysis 

Transfer learning is an emerging and popular paradigm for utilizing existing knowledge from  previous learning tasks to improve the performance of new ones. In this talk, we will first present transfer learning in the early diagnosis of eye diseases: diabetic retinopathy and retinopathy of prematurity.  We will discuss how this empirical  study leads to the mathematical analysis of the feasibility and transferability  issues in transfer learning. We show how a mathematical framework for the general procedure of transfer learning helps establish  the feasibility of transfer learning as well as  the analysis of the associated transfer risk, with applications to financial time series data.

Anran Hu: MF-OMO: An Optimization Formulation of Mean-Field Games 

The literature on theory and computation of mean-field games (MFGs) has grown exponentially recently, but current approaches are limited to contractive or monotone settings, or with an a priori assumption of the uniqueness of the Nash equilibrium (NE). In this talk, we present MF-OMO (Mean-Field Occupation Measure Optimization), a mathematical framework that analyzes MFGs without these restrictions. MF-OMO reformulates the problem of finding NE solutions in MFGs as a single optimization problem. This formulation thus allows for directly utilizing various optimization tools, algorithms and solvers to find NE solutions of MFGs in practice. We also provide convergence guarantees for finding (multiple) NE solutions using popular algorithms like projected gradient descent. For MFGs with linear rewards and mean-field independent dynamics, solving MF-OMO can be reduced to solving a finite number of linear programs, hence solved in finite time.

Emma Hubert: Continuous-time incentives in hierarchies 

In this talk, we will study a model of continuous–time optimal contracting in a hierarchy, which generalizes the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs. We will also discuss some possible extensions of this model, in particular when the outcome processes can be impacted by negative random jumps, representing accidents, and the workers can control their intensity. 

Joint work in progress with Sarah Bensalem and Nicolás Hernández-Santibáñez.

Laura Körber: Optimal execution and speculation with trade signals 

We propose a price impact model where changes in prices are purely driven by the market order flow. The stochastic price impact of market orders and the arrival rates of limit and market orders are functions of the market liquidity process which reflects the balance of liquidity providing and liquidity taking orders. Limit and market orders mutually excite each other so that liquidity is mean reverting. We use the theory of Meyer-$\sigma$-fields to introduce a short-term signal process from which a trader learns about imminent changes in order flow. In this setting, we examine an optimal execution problem and derive the Hamilton--Jacobi--Bellman (HJB) equation for the value function. The HJB equation is solved numerically and we illustrate how the trader uses the signal to enhance the performance of execution problems and to execute speculative strategies.

This talk is based on joint work with Peter Bank and Álvaro Cartea.

Heeyoung Kwon: Trading Constraints in continuous-time Kyle models 

In a continuous-time Kyle setting, we prove global existence of an equilibrium when the insider faces a terminal trading constraint. We establish the explicit equilibrium by deriving an autonomous system of first-order nonlinear ordinary differential equations (ODEs). Moreover, we obtain results associated with empirical findings, such as autocorrelated aggregate holdings, decreasing price impacts, and U-shaped trading patterns.

Peiyao Lai: The Convergent Rate of The Equilibrium Measure for The LQG Mean Field Game with A Common Noise 

The convergence rate of equilibrium measures of N-player Games with Brownian common noise to its asymptotic Mean Field Game system is known as O(N^{−1/9}) with respect to 1-Wasserstein distance, obtained by the monograph [Cardaliaguet, Delarue, Lasry, Lions (2019)]. In this work, we study the convergence rate of the N-player LQG game with a Markov chain common noise towards its asymptotic Mean Field Game. The approach relies on an explicit coupling of the optimal trajectory of the N-player game driven by N dimensional Brownian motion and Mean Field Game counterpart driven by one-dimensional Brownian motion. As a result, the convergence rate is O(N^{−1/2}) with respect to 2-Wasserstein distance. 

Adriana Ocejo: The effect of fees on optimal allocation and utility of payoff with financial guarantees 

We propose a novel fee structure of variable annuities with a guaranteed minimum maturity benefit. The fee structure adjusts to the investment mix maximizing policyholder’s utility while keeping the contract fairly priced. This yields a constrained non-concave utility maximization problem. We assume that the policyholder is risk seeking towards perceived losses and risk-averse towards perceived gains, relative to a reference level. We solve the associated constrained stochastic control problem using a martingale approach and analyze the impact of the fee structure on the optimal investment strategies and payoff. Numerical results show that it is possible to find an optimal portfolio for a wide range of fees, while keeping the contract fairly priced.

Aldaïr Petronilia:  Construction of Solutions to Models of Systemic Risk with Endogenous Contagion through Smoothed Approximations

We propose a new method to construct solutions to McKean–Vlasov equations which originate from the mean-field limit of a particle system with positive feedback. Interacting diffusions on the positive half-line with an absorbing boundary can be used as a simplified model for contagion in large financial networks or portfolios with defaultable entities. Each particle has a drift, an independent Brownian motion and a contagion term. The contagion is an impulse felt by all entities in the system when an entity defaults, that is the entity takes a non-positive value. Due to the instantaneous nature of the contagion, cascades of defaults can occur. Hence, the mean-field limit lives in the space of cadlag functions. Current methods to construct solutions to the limiting equation are through the mean-field limit of interacting particle systems or a fixed-point approach.

In our work, we look at smoothed approximations to the limiting equation where the contagion is mollified with a kernel. Due to the kernel smoothening the feedback, these approximations do not jump. By employing a sequence of kernels which converge towards the Dirac delta, we can construct solutions to the limiting McKean–Vlasov equation with more general drift and diffusion coefficients than those that have been established in the literature. Lastly, provided with suitable regularity on the contagion, we have obtained a rate of convergence prior to the first jump time of the smoothed equation to the limiting equation.

Co-authors: Prof. Ben Hambly, Prof. Christoph Reisinger, Dr Andreas Sojmark, Dr Stefan Rigger

Alejandra Quintos:  Stopping Times Occurring Simultaneously

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In this talk, we present two ways in which one can create a family of stopping times that are not necessarily conditionally independent, allowing for a positive probability for them to be equal. The first way uses a modified Cox construction and the second one uses phase-type distributions. We also present a series of results exploring the special properties of this construction and we provide an application of our model to Credit Risk.

Renyuan Xu: Reversible and Irreversible Decisions under Costly Information Acquisition 

Many real-world analytics problems involve two significant challenges: estimation and optimization. Due to the typically complex nature of each challenge, the standard paradigm is estimate-then-optimize. By and large, machine learning or human learning tools are intended to minimize estimation error and do not account for how the estimations will be used in the downstream optimization problem (such as decision-making problems). In contrast, there is a line of literature in economics focusing on exploring the optimal way to acquire information and learn dynamically to facilitate decision-making. However, most of the decision-making problems considered in this line of work are static (i.e., one-shot) problems which over-simplify the structures of many real-world problems that require dynamic or sequential decisions. 

As a preliminary attempt to introduce more complex downstream decision-making problems after learning and to investigate how downstream tasks affect the learning behavior, we consider a simple example where a decision maker (DM) chooses between two products, an established product A with known return and a newly introduced product B with an unknown return. The DM will make an initial choice between A and B after learning about product B for some time. Importantly, our framework allows the DM to switch to Product A later on at a cost if Product B is the initial choice. We establish the general theory and investigate the analytical structure of the problem through the lens of the Hamilton—Jacobi—Bellman equation and viscosity solutions. We then discuss how model parameters and the opportunity to reverse affect the learning behavior of the DM.

This is based on joint work with Thaleia Zariphopoulou and Luhao Zhang from UT Austin.

Thaleia Zariphopoulou: Mean-field games with unbounded controls and general payoffs 

I will present a mean-field game arising in optimal portfolio choice under relative performance concerns. Both the utilities and the competition functions are general, extending all existing works with homothetic utilities and linear competition modeling. The main contribution is the solution of a mean -field game with unbounded controls that appear in both the drift and the volatility. Explicit solutions are given for the value of the game as well as for the optimal state and control processes (joint work with P. Souganidis).