Quantizing arbitrage | Acciaio Beatrice (ETH Zurich)
Abstract : In this talk I will present a way to quantize arbitrage, that allows to deal with model uncertainty without imposing the no-arbitrage condition. In markets that admit "small arbitrage", we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close to being martingales, and the hedging strategies will need to cover some additional cost. We show a quantitative version of the Fundamental Theorem of Asset Pricing and of the Super-replication theorem. We study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability with respect to a strong adapted Wasserstein distance. Based on joint work with J. Backhoff and G. Pammer.

Boundary-free free boundary problems and particle systems with selection | Atar Rami (Technion, Haifa)
Abstract: Hydrodynamic limits of particle systems with selection are closely related to control problems that involve the heat (or more general parabolic) equation(s), which can also be posed as free boundary problems (FBP). A rigorous relation to classical FBP solutions has been established for some selection models while it remains open for others, where the difficulty lies in questions of regularity of the free boundary and of the solution near it. We propose a weak formulation of a class of FBP that does not involve a free boundary at all, but is based instead on the heat equation with measure-valued right-hand side. This allows us to avoid regularity questions and prove hydrodynamic limits far beyond cases where classical solutions are known to exist. Unlike in all earlier treatment of particle systems with selection, the PDE serves as a tool for establishing limits.

Prediction problems and second order equations | Bayraktar Erhan (University of Michigan)

Abstract: We study the long-time regime of the prediction problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to second order parabolic partial differential equations in the Euclidian space. We exhibit solvable cases for this equation. In the adversarial bandit feedback setting, we show that the problem leads to a second order equation in the Wasserstein space. Based on joint works with Ibrahim Ekren and Xin Zhang.

The space of stochastic processes in continuous time | Beiglbock Mathias (University of Vienna)

Abstract: Researchers from different areas have independently defined extensions of the usual weak topology between laws of stochastic processes. This includes Aldous' extended weak convergence, Hellwig's information topology and convergence in adapted distribution in the sense of Hoover-Keisler. We show that on the set of continuous processes with canonical filtration these topologies coincide and are metrized by a suitable adapted Wasserstein distance AW. Moreover we show that the resulting topology is the weakest topology that guarantees continuity of optimal stopping. While the set of processes with natural filtration is not complete, we establish that its completion consists precisely in the space processes with filtration FP. We also observe that (FP, \AW) exhibits several desirable properties. Specifically, (FP, AW) is Polish, Martingales form a closed subset and approximation results like Donsker's theorem extend to \AW.

Markovian Equilibria In Ergodic Many-Player Games and Mean-Field Games | Cohen Asaf (University of Michigan)

Abstract: We consider a symmetric stochastic game with weak interactions between many players. Time is continuous, the number of states is finite, and costs are ergodic. We prove the existence of a unique Nash equilibrium in the game and show that its limiting behavior (as the number of players goes to infinity) is governed by the unique mean-field equilibrium of the corresponding mean-field game. This is joint work with Ethan Zell.

Utility Maximisation with Model-Independent Trading Constraints | Cox Alexander (University of Bath)

Abstract: In this work we consider the classical utility maximization problem for a trader who is constrained by a model-independent portfolio constraint. Specifically, the trader aims to maximize her utility subject to the constraint that her portfolio value is bounded below when any derivative contracts are valued at their intrinsic value. Here, the intrinsic value is taken to be the model-independent super/sub-hedging price of the derivative.

Using ideas from El Karoui and Meziou (2006), we are able to find explicit strategies for the trader in examples where the trader takes positions in call options and also for positions in certain path dependent options.

(Joint work with Daniel Hernandez-Hernandez, CIMAT).

Signature methods in stochastic portfolio theory | Cuchiero Christa (University of Vienna)
Abstract: In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider (random) signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous (possibly path-dependent) portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the log-optimal portfolio in several classes of non-Markovian models arbitrarily well and illustrate numerically that the trained signature portfolios are remarkably close to the theoretical log-optimal portfolios. This applicability to non-Markovian markets makes these portfolios much more general than classical functionally generated portfolios usually considered in stochastic portfolio theory. Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing expected logarithmic utility or mean-variance optimization within the class of linear path-functional portfolios reduces to a convex quadratic optimization problem, thus making it computationally highly tractable. We apply our method to real market data and show generic out-performance on out-of-sample data even under transaction costs. The talk is based on joint work with Janka Möller.

Reinforcement Learning Algorithm for Mixed Mean Field Control Games | Fouque Jean-Pierre (University of California Santa Barbara)
Abstract: We present a new combined Mean Field Control Game (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between the groups. Within each group the players coordinate their strategies. An example of such a situation is a modification of the classical trader's problem. Groups of traders maximize their wealth. They are faced with transaction cost for their own trades and a cost for their own terminal position. In addition they face a cost for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a reinforcement learning algorithm to approximate the solution of such mixed Mean Field Control Game problems. We test the algorithm on benchmark linear-quadratic specifications for which we have analytic solutions.

Joint work with A. Angiuli, N. Detering, Mathieu Laurière, and J. Lin

Limit distributions for the discretization error of stochastic Volterra equations with a fractional kernel | Fukasawa Masaaki (Osaka University)

Abstract: Our study aims to specify the asymptotic error distribution in the discretization of a stochastic Volterra equation with a fractional kernel. It is well-known that for a standard stochastic differential equation, the discretization error, normalized with its rate of convergence $1/\sqrt{n}$, converges in law to the solution of a certain linear equation. Similarly to this, we show that a suitably normalized discretization error of the Volterra equation converges in law to the solution of a certain linear Volterra equation with the same fractional kernel.

A General Theory of Option Pricing | Gershon David (Hebrew University)
Abstract: We present a generic formalism for option pricing which does not require specifying the stochastic process of the underlying asset price, undergoing a Markovian stochastic behavior. We first derive a consistency condition that the risk neutral density function to maturity must satisfy in order to guarantee no arbitrage. As an example, we show that when the underlying asset price undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black-Scholes-Merton pricing formula. We provide data from the market to prove that the price of European options is independent of the term structure of the volatility prior to maturity. Based on this observation we offer a method to calculate the density function to maturity that satisfies the consistency condition we derived. In the general case the price of European options depends only on the moments of the price return of the underlying asset. When the underlying asset undergoes a continuous time process then only moments up to second order contribute to the European option price. In this case any set of option prices on three strikes with the same maturity contains the information to determine the whole volatility smile for this maturity. Using a great amount of data from the option markets we show that our formalism generates European option prices that match the markets prices very accurately in all asset classes: currencies, equities, interest rates and commodities. Finally, using bootstrapping method with market data of the whole term structure we determine the probability transfer density function from inception to maturity, thus allowing the calculation of path dependent options. Comparison of the results of the model to the market shows a very high level of accuracy.

Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents | Guasoni Paolo (Dublin City University)

Abstract: We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.

Higher rank signatures and their applications to optimal stopping problems in finance | Horvath Blanka (Oxford)
Abstract: Bayer et al (2021) propose a new signature-based nonparametric method for solving optimal stopping problems (such as American option pricing) in finance, which applies beyond the Markovian or semimartingale setting. There, the method, based on an estimation of the expected signature of the payoff for some truncation level, combined with a numerical approximation of the supremum, is found to be challenging due to difficulties connected to the truncation level of the signature.

Recently, a kernel-based learning procedure has been introduced in Lemercier et al (2020), which is based on the notion of path-signature to approximate weakly continuous functionals, such as the pricing functionals of path–dependent options’ payoffs. The use of kernel methods allow us to bypass the question of truncation altogether, however the procedure in its original form fails for Optimal Stopping problems, such as the pricing of American options, because the corresponding value functions are in general discontinuous with respect to the weak topology. In this talk we present a rigorous mathematical framework to resolve this issue by recasting an Optimal Stopping problem as a higher order kernel mean embedding regression based on the notion of higher rank signatures of measure–valued paths. We also present other applications of this procedure which rely on the calculation of conditional distributions.

Quantum algorithms in Finance | Jacquier Antoine (Imperial College London)

Abstract: We introduce several algorithms from Quantum Computing technologies aimed at providing speedup compared to their classical counterparts. We shall highlight in particular how quantum entanglement provides potential expressive explanatory power for neural networks. Time permitting, we will showcase further applications to linear systems and PDEs and to optimization problems.

ERROR ESTIMATES FOR DISCRETE APPROXIMATIONS OF GAME OPTIONS WITH MULTIVARIATE GEOMETRIC DIFFUSION ASSET PRICES | Kifer Yuri (Hebrew University)

Abstract: We obtain error estimates for strong moment approximations of a diffusion with a diffusion matrix $\sigma$ and a drift $b$ by the discrete time process defined recursively $$X_N ((n+1)/N) = X_N (n/N)+N^{-1/2}\sigma(X_N (n/N))\xi(n+1)+N^{-1} b(X_N (n/N))$$ where $\xi(n), n ≥ 1$ are i.i.d. random vectors, and apply this in order to approximate the fair price of game options with assets prices evolving as exponents of diffusions by values of Dynkin’s games with payoffs based on the above discrete time processes. This provides a more effective tool for computations of fair prices of game options with path dependent payoffs in multi asset diffusion markets than the standard Euler-Maruyama method because the dynamic programming algorithm involved there requires computation of conditional expectations with respect to relatively small $σ$-algebras.

TBA | Lehalle Charles-Albert (Abu Dhabi Investment Authority)

An efficient method for pricing barrier options on assets with stochastic volatility | Lipton Alexander (Abu Dhabi Investment Authority)

Abstract: By combining a one-dimensional Monte Carlo simulation and a semi-analytical one-dimensional heat potential method, we design an efficient technique for pricing barrier options on assets with stochastic volatility. Our approach utilizes two loops: first, we run the outer loop by generating volatility paths via the Monte Carlo method; second, we condition the price dynamics on a given volatility path and apply the method of heat potentials to solve the conditional problem in closed form in the inner loop. We compute state probabilities (Green’s function), survival probabilities, and values of call options with barriers. Our approach provides better accuracy and is an order of magnitude faster than existing methods. As a by- product of our analysis, we generalize well-known Willard’s conditioning formula for valuation of path-independent options to path-dependent options. Besides, we derive a novel expression for the joint probability density for the value of drifted

Brownian motion and its running extrema. Potentially, the method can be extended to cover assets with rough volatility. Joint work with Artur Sepp.

OPTIMAL TRANSPORT FOR MODEL CALIBRATION | Loeper Gregoire (Monash University)

Abstract: We provide a survey of recent results on model calibration by Optimal Transport. We present the general framework and then discuss the

calibration of local, and local-stochastic, volatility models to European options, the joint VIX/SPX calibration problem as well as calibration to some pathdependent options. We explain the numerical algorithms and present examples both on synthetic and market data.

Consistency of MLE for partially observed diffusions, with application in market microstructure modeling | Nadtochiy Sergey (Illinois Institute of Technology)

Abstract : In this talk, I will present a tractable sufficient condition for the consistency of maximum likelihood estimators (MLEs) in partially observed

diffusion models, stated explicitly via the stationary distribution of a fully observed system. This result is then applied to a model of market microstructure with latent (unobserved) price process, for which the estimation is performed using real market data for liquid NASDAQ stocks. In particular, we obtain an estimate of the price impact coefficient, as well as the micro-level volatility and drift of the latent price process (this drift is responsible for the concavity of expected price impact of a large meta-order). Joint work with Y. Yin.

Optimality in General Propagator Models with Alpha Signals | Neuman Eyal (Imperial College London)

Abstract: We consider an optimal liquidation problem with a linear transient price impact, induced by a Volterra-type kernel, while also taking into account a general price predicting signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. By identifying the problem as an infinite dimensional stochastic control problem, we characterize the value function in terms of a solution to an $L^2$-valued backward stochastic differential equations and an to an operator-valued Riccati equation, and derive an explicit optimal trading strategy. Our results also includes the case of a singular power-law price impact kernel. This is a joint work with Eduardo Abi-Jaber.

Wasserstein distributionally robust optimization with ML applications | Obloj Jan (Oxford)

Abstract: We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present several applications and extensions of the above results in decision theory and mathematical finance. We then focus on applications in machine learning. Specifically, we propose a measure of robustness of a trained NN to adversarial data and benchmark our metric against the standard methods. We include over 60 trained models from the Robust Bench. We then consider the issue of robust training of DNNs.

Based on joint works with Daniel Bartl, Samuel Drapeau and Johannes Wiesel, and Xingjian Bai, Guangyi He and Yifan Jiang.

Moral hazard for time-inconsistent agents and BSVIEs | Possamai Dylan (ETH Zurich)

Abstract: We address the problem of Moral Hazard in continuous time between a Principal and an Agent that has time-inconsistent preferences. Building upon previous results on non-Markovian time-inconsistent control for sophisticated agents, we are able to reduce the problem of the principal to a novel class of control problems, whose structure is intimately linked to the representation of the problem of the Agent via a so-called extended Backward Stochastic Volterra Integral equation. We will present some results on the characterization of the solution to problem for different specifications of preferences for both the Principal and the Agent.

Singular Exotic Perturbation | Reghai Adil (Natixis)

Abstract: The PnL explanation exercise is at the heart of model performance assessment in Financial Markets. It is the interplay of information from the Position, the Market and the Model. It gives the strategy to calibrate a model and opens up new formulae and new techniques. In this presentation, we introduce a new methodology that combines Singular Perturbation analysis and Exotic greek computation. We show how to manage advanced hedging costs (Stochastic Interest Rates, Correlation skew, Dynamic of the smile etc) relying on inferior models and their exotic greeks. We detail this approach for the Local Stochastic Volatilty model which measures the smile dynamic cost for path dependent options. Tests are performed on the mostly traded Autocalls in the equity derivatives business.

THE STRUCTURE OF MARTINGALE BENAMOU–BRENIER IN $R^d$ | Schachermayer Walter (University of Vienna)

Abstract: In classical optimal transport, the contributions of Benamou–Brenier and Mc-

Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. Stretched Brownian motion provides an analogue for the martingale version of this problem. In this talk we provide a characterization in terms of gradients of convex functions, related to the characterization of optimizers in the classic transport problem for quadratic distance cost. Based on joint work with J. Backhoff, M. Beiglböck and B. Tschiderer.

Stopping Games and Random Stopping Times | Solan Eilon (Tel-Aviv University)

Abstract: I will discuss different notions of random stopping times, both in discrete and continuous time, and then talk on the value of two-player zero-sum stopping games with random stopping times.

Optimal Stopping in High-dimensions | Soner Mete (Princeton University)

Abstract: A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several examples related to financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

How to learn constraint dynamics: an example from volatility surface modeling | Teichmann Josef (ETH Zurich)

Abstract: It is a well-known, challenging problem to write consistent dynamic models for implied volatility surfaces. We present a geometric viewpoint of the problem going back to, e.g., Tomas Bjoerk, which, however, still suffers from challenging drift constraints. One way to resolve these constraints is to learn appropriate drifts by machine learning technology. (joint work with Matteo Gambara).