Elective modules

Topics in Derivatives Pricing: MATH70121  (Dr V. Piterbarg)

Option markets are extremely diverse, spanning several different asset classes and many pricing and hedging strategies. The goal of this module is to complement the other option-flavoured modules, focusing on the specificities of Foreign Exchange and Fixed Income markets. For each of these markets, the module will study their specific characteristics and evolutions, develop the technical tools needed to understand the pricing of derivatives, and explain how to set up trading and hedging strategies therein. A strong emphasis will be given on the actual implementation of the models and their calibration to real data.

Selected Topics in Quantitative Finance: MATH70128 (Prof. J. Jacquier and Dr V. Lucic)

Derivatives pricing is the core area quantitative finance which is relevant to various roles in the industry such as quant, trader, structurer and risk manager. The goal of this module is to introduce the required theoretical tools to understand the pricing and hedging of different financial derivatives.

While the exposition of the topics will be done in a theoretical manner, the module will also emphasise on the practical aspects of derivatives trading (e.g. pricing of structured products traded in real life, backtesting of hedging strategies via numerical studies, etc).

Numerical Methods: MATH70119 (Dr P. Jettkant)

The goal of this module is to complement the Core module on Simulation Methods to investigate other techniques that are widely spread among the financial industry. We shall investigate two popular techniques, namely PDE methods and Fourier methods.

For each approach, we will start with a theoretical framework, explaining how an option pricing problem can be turned into a dynamic programming problem, a PDE or a Fourier integration. We shall then focus on the numerical methods to solve these problems. Practical implementations on real models/data will be emphasised.

Advances in Machine Learning: MATH70120 (Dr Y. Zhang)

The module introduces the latest advances in machine learning. We start with reinforcement learning and demonstrate how it can be combined with neural networks in deep reinforcement learning, which has achieved spectacular results in recent years, such as outplaying the human champion at Go. We also demonstrate how advanced neural networks and tree-based methods, such as decision trees and random forests, can be used for forecasting financial time series and generating alpha. We explain how these advances are related to Bayesian methods, such as particle filtering and Markov chain Monte Carlo. We apply these methods to set up a profitable algorithmic trading venture in cryptocurrencies using Python and kdb+/q (a top technology for electronic trading) along the way.

Rough Paths and Signatures in Machine Learning: MATH70124 (Dr C. Salvi)

Building AI systems capable of extracting information from complex streams of data and
reliably perform inference is an important challenge in many areas of data science. For example summarising patients’ health records to evaluate the efficacy of a treatment, or extracting information from the stock market to design successful trading strategies.

Dealing with streamed data involves numerous challenges including: irregular sampling and missing data; multimodality (i.e. data from different sources); multiple time-scales; randomness and noise; high dimensionality/number of channels.

Classical time series analysis techniques, such as Fourier or Wavelets methods, might provide efficient bottom-up summaries of univariate time series, but because they treat each channel independently, they are not well designed to capture non-linear interactions between different channels of a multivariate stream.

This is where rough path theory, a modern mathematical framework describing complex evolving systems, is incredibly useful. The signature, a centrepiece of the theory, provides a top down description of a stream capturing crucial information over an interval of time, such as the order of events happening across different channels, and filtering out superfluous information. Thanks to its numerous analytic and algebraic properties, the signature is an ideal “feature map” for streamed data that can be used to enhance traditional machine learning models for time series prediction, classification and generation.

The theoretical footprint of rough path theory has been substantial in the study of random phenomena, notably through its presence in Martin Hairer’s Fields medal-winning work on regularity structures, which develops a rigorous framework to solve certain ill-posed stochastic PDEs.

Over the last decade, rough path theory has seen a rapidly increasing interest from the
data science and machine learning communities due to its potency and relevance to de-
scribe/predict/summarise/generate complex streams of data. This course will provide an in-depth survey of the field.

Deep Learning: MATH70116 (Dr C. Salvi)

Deep learning is subfield of Machine Learning that applies deep neural nets to represent and predict complex data. It has recently revolutionised several areas such as image recognition and artificial intelligence and it is currently gaining traction also in the financial industry. The module will first introduce the multi-layer neural nets and explain their universal approximation property. Subsequently, the module proceeds to the training of neural nets, starting from the derivation of the gradient of a neural net and its evaluation through backpropagation, culminating in the stochastic gradient descent and related modern optimisation methods. Techniques to avoid overfitting in training are also elucidated. The remainder of the module focuses on the practical implementation and training of deep neural nets using Keras and TensorFlow, with examples in computational and statistical finance. Time permitting, elements of recurrent neural nets are also sketched.

Quantum Computing (Prof. J. Jacquier and Dr A. Kondratyev)

Quantitative Finance is a rapidly changing environment, and the financial industry is always on the lookout for new techniques and new technologies able to harness the rise of big data and the availability of computing power. Quantum computing, though not a recent field, has gained huge popularity in the past few years with the development of small-scale quantum computers and quantum annealers. These have in turn pushed for new algorithms, hybrid between classical and quantum, and tailored for such computers. The financial industry is now looking at such developments and there is a common agreement that this will be one of the leading advances in the coming decade.

 The goal of this new Elective (so far not given in any similar MSc programmes around the world) is to introduce students to this new technology and these new algorithms and show them how they can be used to solve financial problems, in particular

• For portfolio optimisation,
• For data generation,
• For Machine learning and neural network.

The module will strike a fair balance between theoretical concepts of Quantum Computing, their implementation (in Python using IBM’s Qiskit framework) and their application to real financial problems. 

Fintech Regtech Suptech: MATH70117 (Dr J. Cambe)

Data Science for Fintech Regtech and Suptech: Methodological Foundations and Key Applications

Advances and innovations in computational technology have allowed data scientists to explore and understand increasingly complex financial problems. However, emerging opportunities require financial professionals to update their analytical skills and embrace new technologies, methods, and data sources. The goal of this module is to provide students with an interest in quantitative finance an overview of the evolution of data science in the context of Fintech, RegTech and Suptech, as well as to equip them with the skills to apply new analytical techniques to real world challenges. The emphasis will be on practical applications; and to this end, the module will be led by industry experts and include regular hands-on exercises involving the use of advanced data analytics

Convex Optimisation: MATH70122 (Dr A. Coache)

The module covers both the theoretical underpinnings of convex optimisation and its applications to important problems in mathematical finance. A brief outline of the course reads as follows:

• Fundamental properties of convex sets and convex functions
• The basics of convex optimisation with special emphasis on duality theory
• Markowitz portfolio theory and the CAPM model
• Expected utility maximisation and no arbitrage
• Convexity in continuous time hedging

Stochastic Control in Finance: MATH70126 (Prof H. Zheng)

Many problems in mathematical finance (and in other areas) are essentially optimisation problems subject to random perturbations, where some controls play the role of a performance criterion. The goal of this module is to bring the main concepts and techniques from dynamic stochastic optimisation and stochastic control theory to the realm of quantitative finance. It will therefore naturally start with a theoretical part focusing on required elements of stochastic analysis, and with a motivation through several examples of control problems in Finance. We will then turn to the classical PDE approach of dynamic programming, including controlled diffusion processes, dynamic programming principle, the Hamilton-Jacobi-Bellman equation and its verification theorem. We will finally see how to derive an solve dynamic programming equations for various financial problems such as the Merton portfolio problem, pricing under transaction costs, super-replication with portfolio constraints, and target reachability problems.

Quatitative Trading and Price Impact: MATH70127 (Prof J. Muhle-Karbe)

The increase in computer power over the last decades has given rise to prices being quoted and stocks being traded at an ever-increasing pace. Since humans are not able to place orders at this speed, algorithms have replaced classical traders to optimise portfolios and investments. In this module, we will study specificities of this market, and in particular, we shall develop the mathematical tools required to develop such algorithms in this high-frequency framework. The module will start with a short review of stochastic optimal control, which forms the mathematical background. We shall then move on to study optimal execution, namely how and when to place buy/sell orders in this market, both assuming continuous trading and in the context of limit and market orders. The last part of the module will be dedicated to the concept of market making and statistical arbitrage in high-frequency settings. 

Market Microstructure: MATH70125 (Dr M. Rosenbaum)

The goal of the module is to develop thorough understanding of how form, information is aggregated, and trades occur in financial markets. The main market types will be described as well as traders’ main motives for why they trade. Market manipulation and high-frequency trading strategies have received a lot of attention in the press recently, so the module will illustrate them and examine recent developments in regulations that aim to limit them. Liquidity is a key theme in market microstructure, and the students will learn how to measure it and to recognise the recent increase in liquidity fragmentation and hidden, “dark” liquidity. The Flash Crash of 6 May 2010 will be analysed as a case study of sudden loss of liquidity.

Portfolio Management: MATH70129 (Dr Y. Shadmi)

This module gives students a foundation for quantitative portfolio management and for understanding market price determination. Key concepts include risk measurement, risk-reward trade-offs, portfolio optimization, benchmarking, equilibrium asset pricing, market efficiency, and pricing anomalies. Specific portfolio management tools include mean-variance optimization, CAPM and APT asset pricing, factor models (e.g., Fama-French), momentum strategies, and performance evaluation. The course will present essential theories and formulas and will also review important institutional and empirical facts about equity, bond, and commodity markets.