Rough volatility models: small-time asymptotics and calibration

Inspired by the work of Al`os, Le ́on and Vives [ALV07] and Fukasawa [Fuk17], who showed that a volatility process driven by a fractional Brownian motion generates the power law at-the-money volatility skew observed in financial market data, Gatheral, Jaisson and Rosenbaum [GJR18a] spawned a class of models now known as rough volatility models. We study the asymptotic behaviour of such models, and investigate how convolutional neural networks can be used for their calibration. Chapter 1 serves as an introduction. We begin with implied volatility, and then intro- duce a number of model classes, starting with local volatility models and ending with rough volatility models, and discuss their associated asymptotic behaviour. We also introduce the theoretical tools used to prove the main results. In Chapter 2 we study the small-time behaviour of the rough Bergomi model, introduced by Bayer, Friz, and Gatheral [BFG16]. We prove a pathwise large deviations principle for a small-noise version of the model, and use this result to establish the small-time behaviour of the rescaled log stock price process. This, in turn, allows us to characterise the small-time implied volatility behaviour of the model. Using the same theoretical framework, we are also able to establish the small-time implied volatility behaviour of the lognormal fSABR model of Akahori, Song, and Wang [ASW17]. In Chapter 3 we present small-time implied volatility asymptotics for realised variance (RV) options for a number of (rough) stochastic volatility models via a large deviations principle. We interestingly discover that these (rough) volatility models, together with others proposed in the literature, generate linear smiles around the money. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we develop an approximation scheme for the density of the realised variance, which in turn allows the volatility swap density to be expressed in closed form. Lastly, we investigate different constructions of multi-factor models and how their construction affects the convexity of 4 the implied volatility smile. Remarkably, we identify a class of models that can generate non-linear smiles around-the-money. Additionally, we establish small-noise asymptotic behaviour of a general class of VIX options in the large strike regime. In Chapter 4, which is self-contained, we give an introduction to machine learning and neural networks. We investigate the use of convolutional neural networks to find the H ̈older exponent of simulated sample paths of the rough Bergomi model, a method which performs extremely well and is found to be robust when applied to trajectories of a fractional Brownian motion and an Ornstein-Uhlenbeck process. We then propose a novel calibration scheme for the rough Bergomi model based on our results.Open Acces

Efficient Likelihood Evaluation of State-Space Representations

We develop a numerical procedure that facilitates efficient likelihood evaluation in applications involving non-linear and non-Gaussian state-space models. The procedure employs continuous approximations of filtering densities, and delivers unconditionally optimal global approximations of targeted integrands to achieve likelihood approximation. Optimized approximations of targeted integrands are constructed via efficient importance sampling. Resulting likelihood approximations are continuous functions of model parameters, greatly enhancing parameter estimation. We illustrate our procedure in applications to dynamic stochastic general equilibrium models.Adaption, dynamic stochastic general equilibrium model, efficient importance sampling, kernel density approximation, particle filter.

High-frequency financial data modelling with hybrid marked point processes

The rise of electronic order-driven financial markets has brought a profusion of new high-frequency data to study, with an opportunity to understand the price formation mechanism at the smallest timescales. The original motivation of this thesis is to find a stochastic process that provides an accurate statistical dynamic description of this new data. A critical analysis of the literature reveals a dichotomy between two main sorts of model, Hawkes processes and continuous-time Markov chains, each having qualities that the other lacks. In particular, models of the former sort are successful at capturing excitation effects between different event types but fail to incorporate the state of the market. We resolve this dichotomy by introducing state-dependent Hawkes processes, an extension of Hawkes processes where events can now interact with an auxiliary state process. These new stochastic processes provide us with the first model that features both excitation effects and an explicit feedback loop between events and the state of the market. The application of this new model to high-quality data demonstrates that the excitation effects are indeed strongly state-dependent. State-dependent Hawkes processes come however with theoretical challenges: under which conditions do they exist, are they unique and do not explode? To answer these questions, we view state-dependent Hawkes processes as ordinary point processes of higher dimension, which we then generalise to the class of hybrid marked point processes. This class provides a framework that unifies and extends the existing high-frequency models. Since hybrid marked point processes are defined implicitly via their intensity, one can address the above questions by studying instead a Poisson-driven stochastic differential equation (SDE). We are able to solve this SDE under general assumptions that dispense with the Lipchitz condition usually required in the literature, which yields, as a corollary, the existence and uniqueness of non-explosive state-dependent Hawkes processes.Open Acces

A pure-jump market-making model for high-frequency trading

We propose a new market-making model which incorporates a number of realistic features relevant for high-frequency trading. In particular, we model the dependency structure of prices and order arrivals with novel self- and cross-exciting point processes. Furthermore, instead of assuming the bid and ask prices can be adjusted continuously by the market maker, we formulate the market maker\u27s decisions as an optimal switching problem. Moreover, the risk of overtrading has been taken into consideration by allowing each order to have different size, and the market maker can make use of market orders, which are treated as impulse control, to get rid of excessive inventory. Because of the stochastic intensities of the cross-exciting point processes, the optimality condition cannot be formulated using classical Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI), so we extend the framework of constrained forward backward stochastic differential equation (CFBSDE) to solve our optimal control problem

Modelling of and empirical studies on portfolio choice, option pricing, and credit risk

This thesis develops and applies a statistical spanning test for mean-coherent regular risk portfolios. Similarly in spirt to Huberman and Kandel (1987), this test can be implemented by means of a simple semi-parametric instrumental variable regression, where instruments have a direct link with a stochastic discount factor. Applications to different asset classes are studied. The results are compared to the conventional mean-variance approach. The second part of the thesis concerns option pricing under stochastic volatility and credit risk modelling. It is shown that modelling dynamics of the implied prices of volatility risk can improve out-of-sample option pricing performance. Finally, an equity-based structural model of credit risk with a constant elasticity of volatility assumption is discussed. This model might be particularly suitable for analysis of high yield fixed income instruments, where correlation between credit spreads and equity returns is substantial.

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Machine Learning and Portfolio Optimization: an application to Italian FTSE-MIB Stocks

A model that combines econometric ARMA model with new machine learning techniques will be developed to build an efficient portfolio, composed of Italian FTSE-MIB stocks. The goal of this portfolio is to over-perform a benchmark portfolio obtained throw traditional Markowitz optimisation.A model that combines econometric ARMA model with new machine learning techniques will be developed to build an efficient portfolio, composed of Italian FTSE-MIB stocks. The goal of this portfolio is to over-perform a benchmark portfolio obtained throw traditional Markowitz optimisation