Past Mathematical and Computational Finance Internal Seminar

This talk presents a new framework for Merton’s optimal investment problem which uses the theory of Meyer $\sigma$-fields to allow for signals that possibly warn the investor about impending jumps. With strategies no longer predictable, some care has to be taken to properly define wealth dynamics through stochastic integration. By means of dynamic programming, we solve the problem explicitly for power utilities. In a case study with Gaussian jumps, we find, for instance, that an investor may prefer to disinvest even after a mildly positive signal. Our setting also allows us to investigate whether, given the chance, it is better to improve signal quality or quantity and how much extra value can be generated from either choice.
This talk is based on joint work with Peter Bank.

Theory of mean-field games (MFGs) has recently experienced an exponential growth. Existing analytical approaches to find Nash equilibrium (NE) solutions for MFGs are, however, by and large restricted to contractive or monotone settings, or rely on the uniqueness of the NE. We propose a new mathematical paradigm to analyze discrete-time MFGs without any of these restrictions. The key idea is to reformulate the problem of finding NE solutions in MFGs as solving an equivalent optimization problem, called MF-OMO (Mean-Field Occupation Measure Optimization), with bounded variables and trivial convex constraints. It is built on the classical work of reformulating a Markov decision process as a linear program, and by adding the consistency constraint for MFGs in terms of occupation measures, and by exploiting the complementarity structure of the linear program. This equivalence framework enables finding multiple (and possibly all) NE solutions of MFGs by standard algorithms such as projected gradient descent, and with convergence guarantees under appropriate conditions. In particular, analyzing MFGs with linear rewards and with mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This optimization reformulation of MFGs can be extended to variants of MFGs such as personalized MFGs.

The possibility of `tacit collusion', in which interactions across market-making algorithms lead to an outcome similar to collusion among market makers, has increasingly received regulatory scrutiny. 
    We model the interaction of market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, which we characterise in terms of a system of coupled Hamilton-Jacobi-Bellman (HJB) equations, while Pareto optima correspond to collusion. 
    Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers learn to adjust their quotes, we show how the interaction of market-making algorithms may lead to tacit collusion with spread levels strictly above the competitive equilibrium level, without any explicit sharing of information.
 

TBC

We develop a new online algorithm for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm optimizes over the parameters in the multi-dimensional SDE model in order to minimize the distance between the model's stationary distribution and the target statistics. We rigorously prove convergence for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the unbiased descent direction under the stationary distribution. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds on a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm. This presentation is based upon research with Ziheng Wang.
 

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.

We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and it satisfies  a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the losses, the agent optimises a terminal loss function of the asset values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. Joint work with Christa Cuchiero and Stefan Rigger.

We introduce a method for estimating the roughness of a function based on a discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We discuss the consistency of this estimator in a pathwise setting under high-frequency asymptotics. We investigate its finite sample performance for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of Fractional Brownian motion and other fractional processes.
We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations.
Through a detailed numerical experiment based on a stochastic volatility model, we show that even when instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rougher behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Similar behaviour is observed in financial data, which suggests that the origin of the roughness observed in realized volatility time-series lies in the `microstructure noise' rather than the volatility process itself.

Abstract: In this work, we analyze a class of stochastic inverse optimal control problems with entropy regularization. We first characterize the set of solutions for the inverse control problem. This solution set exemplifies the issue of degeneracy in generic inverse control problems that there exist multiple reward or cost functions that can explain the displayed optimal behavior. Then we establish one resolution for the degeneracy issue by providing one additional optimal policy under a different discount factor. This resolution does not depend on any prior knowledge of the solution set. Through a simple numerical experiment with deterministic transition kernel, we demonstrate the ability of accurately extracting the cost function through our proposed resolution.

Joint work with Sam Cohen (Oxford) and Lukasz Szpruch (Edinburgh).

Orders in major electronic stock markets are executed through centralised limit order books (LOBs). Large amounts of historical data have led to extensive research modeling LOBs, for the purpose of better understanding their dynamics and building simulators as a framework for controlled experiments, when testing trading algorithms or execution strategies.Most work in the literature models the aggregate view of the limit order book, which focuses on the volume of orders at a given price level, using a point process. In addition to this information, brokers and exchanges also have information on the identity of the agents submitting the order. This leads to a more granular view of limit order book dynamics, which we attempt to model using a heterogeneous model of order flow.

We present a granular representation of the limit order book that allows to account for the origins of different orders. Using client order flow from a major broker, we analyze the properties of variables in this representation. The heterogeneity of order flow is modeled by segmenting clients into different clusters, for which we identify representative prototypes. This segmentation appears to be stable both over time as well as over different stocks. Our findings can be leveraged to build more realistic order flow models that account for the diversity of the market participants.

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

Model-free machine learning is a tabula rasa method, estimating parametric functions purely from the data. In contrast, model-driven machine learning augments mathematical models with machine learning. For example, unknown terms in SDEs and PDEs can be represented by neural networks. We compare these two approaches, discuss their mathematical theory, and present several examples. In model-free machine learning, we use reinforcement learning to train order-execution models on limit order book data. Event-by-event simulation, based on the historical order book dataset, is used to train and evaluate limit order strategies. In model-driven machine learning, we develop SDEs and PDEs with neural network terms for options pricing as well as, in an application outside of finance, predictive modeling in physics. We are able to prove global convergence of the optimization algorithm for a class of linear elliptic PDEs with neural network terms.


 

Abstract: We consider the problem of testing statistical hypotheses and building confidence sequences for elicitable and identifiable functionals, a broad class of statistics which are of particular interest in the field of quantitative risk management. Assuming a sequential testing framework in which data is collected in sequence, where a user may choose to accept or reject a hypothesis at any point in time, we provide powerful distribution-free and anytime-valid testing methods which rely on controlled test supermartingales. Leveraging tools from online convex optimization, we show that tests can be optimized to improve their statistical power, with asymptotic guarantees for rejecting false hypotheses. By "inverting the test", these methods are extended to the task of confidence sequence building. Lastly, we implement these techniques on a range of simple examples to demonstrate their effectiveness.

I will discuss the following papers in my talk:
(1) Protecting consumers from collusive prices due to AI, 2020 with E. Calvano, V. Denicolò, J. Harrington, S.  Pastorello.  Nov 27, 2020, SCIENCE, cover featured article.
(2) Artificial intelligence, algorithmic pricing and collusion, 2020 with E. Calvano, V. Denicolò, S. Pastorello. AMERICAN ECONOMIC REVIEW,  Oct. 2020.
(3) Algorithmic Collusion with Imperfect Monitoring, 2021, with E. Calvano, V. Denicolò, S.  Pastorello

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
 

Abstract: We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact on the risky asset's execution price along with taking into account a common general price predicting signal. In contrast to an interaction of the agents through purely permanent price impact as it is typically considered in the literature on multi-player price impact games, accrued transient price impact does not persist but decays over time. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal for the accumulated transient price distortion induced by all other agents' price impact; and thus unfolds a direct and natural link in equilibrium between the trading signal and the agents' trading activity. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents and show how the latter provides an approximate Nash-equilibrium for the finite-player game. Specifically we prove the convergence of the N-players game optimal strategy to the optimal strategy of the mean field game.     (Joint work with Moritz Voss)
 

It is well known that expected signatures can be used as the “moments” of the law of stochastic processes. Inspired by this fact, we introduced higher rank expected signatures to capture the essences of the weak topologies of adapted processes, and characterize the information evolution pattern associated with stochastic processes. This approach provides an alternative perspective on a recent important work by Backhoff–Veraguas, Bartl, Beiglbock and Eder regarding adapted topologies and causal Wasserstein metrics.

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We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The first example is the optimal liquidation of a holding in an asset. We show results for the case where we assume a model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework and that it is more robust with respect to model mis-specification when compared to a model-based approach. The second example is an LQR system in a higher-dimensional setting with synthetic data.

In this talk, we will introduce two problems of contract theory, in continuous–time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one–period framework of Sung (2015). The hierarchy is modeled 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. Both agents and the manager each independently control a stochastic process representing their outcome. We will see through a simple example that even if the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Even this first simple 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, one extension consists in the elaboration of more general contracts, indexing the compensation of one worker on the result of the others. This increase in the complexity of contracts is beneficial for the principal, and constitutes a first approach to even more complex contracts, in the case, for example, of a continuum of workers with mean–field interactions. This will lead us to introduce the second problem, namely optimal contracting for demand–response management, which consists in extending the model by Aïd, Possamaï, and Touzi (2019) to a mean–field of consumers. Finally, we will conclude by mentioning that this principal-agent approach with a multitude of agents can be used to address many situations, for example to model incentives for
lockdown in the current epidemic context.
 

TBC
 

Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics. While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions. While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed. For this reason, we introduce a widely-applicable Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.
 

Abstract: Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e. removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimise prices' changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.
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Abstract: We reconsider the idea of trend-based predictability using methods that flexibly learn price patterns that are most predictive of future returns, rather than testing hypothesized or pre-specified patterns (e.g., momentum and reversal). Our raw predictor data are images—stock-level price charts—from which we elicit the price patterns that best predict returns using machine learning image analysis methods. The predictive patterns we identify are largely distinct from trend signals commonly analyzed in the literature, give more accurate return predictions, translate into more profitable investment strategies, and are robust to a battery of specification variations. They also appear context-independent: Predictive patterns estimated at short time scales (e.g., daily data) give similarly strong predictions when applied at longer time scales (e.g., monthly), and patterns learned from US stocks predict equally well in international markets.

This is based on joint work with Jingwen Jiang and Bryan T. Kelly.