Past Mathematical and Computational Finance Internal Seminar

10 June 2021
16:00
FELIX PRENZEL
Abstract

 

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.

  • Mathematical and Computational Finance Internal Seminar
3 June 2021
16:00
Abstract


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.

 

  • Mathematical and Computational Finance Internal Seminar
3 June 2021
16:00
Abstract


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.

 

  • Mathematical and Computational Finance Internal Seminar
27 May 2021
16:00
JUSTIN SIRIGNANO
Abstract


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.


 

  • Mathematical and Computational Finance Internal Seminar
20 May 2021
16:00
PHILIIPPE CASGRAIN
Abstract

 

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.

 

 

 

 

  • Mathematical and Computational Finance Internal Seminar
13 May 2021
16:00
GIACOMO CALZOLARI
Abstract

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

  • Mathematical and Computational Finance Internal Seminar
6 May 2021
16:00
Alain Rossier
Abstract

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.
 

  • Mathematical and Computational Finance Internal Seminar
29 April 2021
16:00
Abstract

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)
 

  • Mathematical and Computational Finance Internal Seminar
11 March 2021
16:00
Abstract

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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  • Mathematical and Computational Finance Internal Seminar
4 March 2021
16:00
HUINING YANG
Abstract

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.

  • Mathematical and Computational Finance Internal Seminar

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