Post/Doctoral Seminar in Mathematical Finance

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Spring Semester 2021

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Speaker

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9 March 2021
15:15-16:15

Dr. Philippe Casgrain
ETH Zurich, Switzerland

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Optimizing Optimizers: Regret-optimal gradient descent algorithms
Speaker, Affiliation	Dr. Philippe Casgrain, ETH Zurich, Switzerland
Date, Time	9 March 2021, 15:15-16:15
Location	Zoom
Abstract	The need for fast and robust optimization algorithms are of critical importance in all areas of machine learning. This paper treats the task of designing optimization algorithms as an optimal control problem. Using regret as a metric for an algorithm's performance, we study the existence, uniqueness and consistency of regret-optimal algorithms. By providing first-order optimality conditions for the control problem, we show that regret-optimal algorithms must satisfy a specific structure in their dynamics which we show is equivalent to performing dual-preconditioned gradient descent on the value function generated by its regret. Using these optimal dynamics, we provide bounds on their rates of convergence to solutions of convex optimization problems. Though closed-form optimal dynamics cannot be obtained in general, we present fast numerical methods for approximating them, generating optimization algorithms which directly optimize their long-term regret. Lastly, these are benchmarked against commonly used optimization algorithms to demonstrate their effectiveness. Authors: Philippe Casgrain, Anastasis Kratsios

Optimizing Optimizers: Regret-optimal gradient descent algorithmsread_more

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23 March 2021
15:15-16:15

David Pires Tavares Martins
ETH Zurich, Switzerland

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Mean-variance hedging: theory, techniques and application to the rough Heston model
Speaker, Affiliation	David Pires Tavares Martins, ETH Zurich, Switzerland
Date, Time	23 March 2021, 15:15-16:15
Location	Zoom
Abstract	In recent years, a great volume of mathematical finance research has been devoted to asset price models more complex than the Black-Scholes and Bachelier. Many of those models, such as stochastic volatility, imply the incompleteness of financial markets, which creates challenges for the pricing and hedging of non-liquidly traded derivatives. Mean-variance hedging has emerged as a powerful tool to address this, replacing a perfect hedge with the optimal approximate hedge (in a quadratic sense). Besides its intuitiveness, mean-variance hedging offers other advantages, such as analytical tractability in a number of models, incorporating information from historical probability measures, and reasonable pricing of illiquid derivatives on which buyers and sellers can agree. The study of mean-variance hedging in depth has made use of a wide array of techniques from stochastic calculus and beyond. We present some of the theory and structures behind our current understanding of the problem, emphasising the techniques used for computing solutions in practice. In particular, we will consider their application to the rough Heston model. Rough volatility models have become quite popular recently as they capture both the historic and the implied volatility remarkably well. Because the volatility process is neither a Markov process nor a semimartingale in these models, they are mathematically more difficult to handle. We will show how to overcome this in the rough Heston model, where the affine structure still allows an effective application of mean-variance hedging. Joint work with Christoph Czichowsky (London School of Economics).

Mean-variance hedging: theory, techniques and application to the rough Heston modelread_more

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29 March 2021
14:00-15:00

Dr. Anastasis Kratsios
ETH Zurich, Switzerland

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Universal Probability Measure-Valued Deep Neural Networks
Speaker, Affiliation	Dr. Anastasis Kratsios, ETH Zurich, Switzerland
Date, Time	29 March 2021, 14:00-15:00
Location	Zoom
Abstract	We introduce deep neural architecture types with inputs from a separable and locally-compact metric space Xand outputs in the Wasserstein-1 space over a separable metric space Y. We establish the density of our architecture type in C(X,P1(Y)), quantitatively. NB that our results are new even in the case where Xand Y are Euclidean, in which case, we find that many commonly used types such as MDNs and MGANs are universal special cases of our model type. We show that our models approximate functions in C(X,P1(Y)) by implementing ε-metric projections in the Wasserstein-metric onto the hull of certain finite families of measures therein. If the target function can be represented as a mixture of finitely many functions, each taking values in a finite-dimensional topological submanifold of the Wasserstein space, we find that the approximating networks can be assumed to have bounded width. As applications of our results, we address the following problems. We show that, under mild conditions, our architecture can approximate any regular conditional distribution of an X-valued random element X depending on a Y-valued random element Y with arbitrarily high probability. Consequentially, we show that once our approximation of this regular conditional distribution is learned, any conditional expectation of the form 𝔼[f(X, Y)\|Y=y] for Carathéodory f with uniformly-Lipschitz first component and a uniformly-bounded second component, is approximable by standard Monte-Carlo sampling against the learned measure. We illustrate our theory in the context of stochastic processes.

Universal Probability Measure-Valued Deep Neural Networksread_more

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30 March 2021
14:00-15:00

Dr. Jorge Yslas

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Inhomogeneous phase-type distributions: Fitting and applications to survival analysis
Speaker, Affiliation	Dr. Jorge Yslas,
Date, Time	30 March 2021, 14:00-15:00
Location	Zoom
Abstract	A phase-type (PH) distribution is the distribution of the time until absorption of an otherwise transient time-homogeneous pure-jump Markov process. Moreover, the class of PH distributions is known to be dense on the distributions on the positive half-line. When it comes to applications, classical PH models might not be suitable to fit data where heavy tails are present, as PH tails are always light. In order to tackle such limitation, the class of inhomogeneous phase-type (IPH) distributions was recently introduced in Albrecher and Bladt [Journal of Applied Probability, 56(4):1044-1064, 2019] as a dense extension of classical PH distributions, which leads to more parsimonious models in the presence of heavy tails. In this talk, we propose a fitting procedure for the IPH class to given data. We furthermore consider an analogous extension of Kulkarni’s multivariate phase-type class to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. Finally, we propose new regression models in survival analysis based on homogeneous and inhomogeneous phase-type distributions and provide fitting procedures.

Inhomogeneous phase-type distributions: Fitting and applications to survival analysisread_more

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6 April 2021
15:00-16:00

Dr. Hampus Engsner

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Least Squares Monte Carlo applied to Dynamic Monetary Utility Functions
Speaker, Affiliation	Dr. Hampus Engsner,
Date, Time	6 April 2021, 15:00-16:00
Location	Zoom
Abstract	In this talk, we explore ways of numerically computing recursive dynamic monetary risk measures and utility functions. Computationally, this problem suffers from the curse of dimensionality and nested simulations are unfeasible if there are more than two time steps. The approach considered in this paper is to use a Least Squares Monte Carlo (LSM) algorithm to tackle this problem, a method which has been primarily considered for valuing American derivatives, or more general stopping time problems, as these also give rise to backward recursions with corresponding challenges in terms of numerical computation. We give some overarching consistency results for the LSM algorithm in a general setting as well as explore numerically its performance for a recursive Cost-of-Capital valuation, a special case of a dynamic monetary utility function.

Least Squares Monte Carlo applied to Dynamic Monetary Utility Functionsread_more

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4 May 2021
15:15-16:15

Florian Rossmannek
ETH Zurich, Switzerland

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	Approximation Capacities of ReLU Neural Networks
Speaker, Affiliation	Florian Rossmannek, ETH Zurich, Switzerland
Date, Time	4 May 2021, 15:15-16:15
Location	Zoom
Abstract	It is often said that neural networks can break the curse of dimensionality. Such a general statement can be misleading. Thus, we review known approximation results, in which neural networks actually suffer from the curse, and discuss an Ansatz for breaking the curse in a different setting. Our approach is based on mimicking the structure of a neural network, but without using the same activation function for each neuron. Instead, the activation function is replaced by a collection of "basis" functions that are easy to approximate. Then, one can construct a neural network approximation of the mimicking network. Along the way, we develop an improved method for realizing these constructions.

Approximation Capacities of ReLU Neural Networksread_more

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25 May 2021
15:15-16:15

Despoina Makariou
LSE

Event Details

Post/Doctoral Seminar in Mathematical Finance

Title	A random forest based approach for predicting spreads in the primary catastrophe bond market
Speaker, Affiliation	Despoina Makariou, LSE
Date, Time	25 May 2021, 15:15-16:15
Location	Zoom
Abstract	We introduce a random forest approach to enable spreads' prediction in the primary catastrophe bond market. In a purely predictive framework, we assess the importance of catastrophe spread predictors using permutation and minimal depth methods. The whole population of non-life catastrophe bonds issued from December 2009 to May 2018 is used. We find that random forest has at least as good prediction performance as our benchmark-linear regression in the temporal context, and better prediction performance in the non-temporal one. Random forest also performs better than the benchmark when multiple predictors are excluded in accordance with the importance rankings or at random, which indicates that random forest extracts information from existing predictors more effectively and captures interactions better without the need to specify them. The results of random forest, in terms of prediction accuracy and the minimal depth importance are stable. There is only a small divergence between the drivers of catastrophe bond spread in the predictive versus explanatory framework. We believe that the usage of random forest can speed up investment decisions in the catastrophe bond industry both for would-be issuers and investors. Despoina Makariou (D.Makariou@lse.ac.uk), Pauline Barrieu (P.M.Barrieu@lse.ac.uk), and Yining Chen (Y.Chen101@lse.ac.uk) Department of Statistics, London School of Economics and Political Science

A random forest based approach for predicting spreads in the primary catastrophe bond marketread_more

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