Schedule

We consider a market with a certain number of agents and goods. Each agent has a position with respect to each good—either to receive it or to give it away. The objective is to construct an exchange plan that maximize the total value of the goods exchanged. In its discrete form, this problem can be formulated as a linear programming problem. However, solving it directly requires substantial computational resources. Therefore, we propose addressing a regularized version of the problem using an iterative algorithm.

The talk will focus on the structure of the solution to the regularized problem, how to obtain it, and why it converges to the solution of the original formulation.

Recently, reinforcement learning has gained remarkable attention in the finance community. Deep Hedging expressed the problem of derivative hedging and pricing in the RL language, which paved the way for employing a wide range of risk-averse RL algorithms for this problem. In the seminar, we will implement the Deep Hedging algorithm from scratch and test it on numerous problems. I will assume that you are familiar with the following topics (if not, read the recommended literature):

- Basics of deep learning: you know how to write a basic train loop [PyTorch tutorial].

- Basics of RL: Bellman equations, value function [Sutton & Barto (eng) chapters 1, 3, 4 OR Ivanov (rus) chapters 1, 3.1-3.3].

- Expected utility theory: utility functions, exponential utility [wiki1, wiki2].

- Basics of finance: discrete-time model of stock dynamics, European options [Follmer & Schied chapter 1].

We investigate the weak limit of the hyper-rough square-root process as the Hurst index H goes to -1/2. This limit corresponds to the fractional kernel t^{H - 1 / 2} losing integrability. We establish the joint convergence of the couple (X, M), where X is the hyper-rough process and M the associated martingale, to a fully correlated Inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod J1 topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod M1 topology for X and in the non-Skorokhod S topology for M. Additionally, we show that an Inverse Gaussian approximation leads to a new numerical scheme for simulating Volterra square-root processes, which performs better as H decreases to -1/2.

The seminar will cover the construction of a local volatility model that precisely matches given marginal distributions while avoiding computationally expensive numerical integration schemes.

The seminar is based on a paper by Antoine Conze and Pierre Henry-Labordere.

Motivated by optimal execution with stochastic signals, concave market impact and almost sure constraints in financial markets, we formulate and solve an optimal trading problem with a general nonlinear propagator model under linear functional inequality constraints. In the general concave transient impact case, the first-order condition reduces to the resolution of a nonlinear stochastic Fredholm equation whose source-term is an effective signal process dependent on the Lagrange multipliers capturing the corresponding constraints. In the particular case of a linear transient impact, such Fredholm equation can be semi-explicitly solved in terms of the Lagrange multipliers and their conditional expectations. Leveraging both stochastic Karush-Kuhn-Tucker optimality conditions and such semi-explicit solution in the linear case, we present novel numerical schemes to build sample paths of the optimal trading strategy facing almost sure constraints or concave transient market impact. We illustrate our findings on various applications including: (i) an optimal execution problem with an exponential or a power law decaying transient impact, with either a `no-shorting' constraint in the presence of a `sell' signal, a `no-buying' constraint in the presence of a `buy' signal or a stochastic `stop-trading' constraint whenever the exogenous price drops below a specified reference level; (ii) a trader facing concave transient market impact for various types of kernel decays, including the power-law decay.

 In the Black-Scholes model with small transaction costs, Leland proposed an asymptotic super-hedging strategy for the call option as the number of revision dates tends to +∞.  The idea is to use a delta-hedging strategy with an adjusted volatility to compensate for transaction costs.

A natural question is how to solve the same problem in a more general setting where the number of discrete dates is fixed, the proportional transaction cost coefficient does not tend to zero, and no martingale measure is assumed. We propose a new, easily computable method to address this problem.

We will understand what the multi-armed bandit problem is, what methods exist for solving it, and why it falls under reinforcement learning (RL). We will discuss the exploration-exploitation dilemma and consider the portfolio selection problem as a clear example of its application. And, of course, we will answer the question of why the multi-armed bandit has so many arms.

In this seminar, we’ll develop intuition for everything that resembles an integral of the logarithm of a density function—from KL divergence to Girsanov’s theorem. No tedious proofs—just the essence, mixed with crude references and dumb jokes.

We will dive into the fascinating world of optimal stopping and free-boundary problems. We  start with a classical problem on Brownian motion hitting a boundary. Then, we explore Markov processes and their generators. This helps to formulate a free-boundary problem solving the optimal stopping problem. We also touch on how these ideas come into play when pricing American options.

This talk will focus on the concept of the renormalization group. We’ll start by trying to understand what it is using simple models, then analyze a specific example from machine learning. After that, I’ll explain how this concept, in its less trivial form, connects to optimal transport theory. Finally, we’ll discuss how a potentially optimal inversion of this semigroup could be useful for financial theory.

We will revisit the fundamentals of optimal transport. We’ll examine the optimal transport problem in the formulations of Monge, Kantorovich, and the weak formulation. Then, we’ll move on to the dual form and explore its connection to the Schrödinger bridge. After that, I’ll discuss various problems where optimal transport arises, including generative modeling, robust finance, and the calibration of stochastic volatility models to market data.