Elective modules

Derivatives Pricing Stream

Malliavin Calculus in Finance (Thomas Cass)

Malliavin Calculus is an extremely powerful tool in stochastic analysis, extending the classical notion of derivative to the space of stochastic processes. A certain number of results arising from this theory turn out to provide the right framework to analysis several problems in mathematical finance. The module will be divided into two parts: the first one will concentrate on developing the theoretical tools of Malliavin Calculus, including analysis on Wiener space, the Wiener chaos decomposition, the Ornstein-Uhlenbeck semigroup and hypercontractivity, the Malliavin derivative operator and the divergence operator, and Sobolev spaces and equivalence of norms. The second part of the module will focus on understanding how these tools come in handy in order to price and hedge financial derivatives, and to compute their sensitivities.

Topics in Derivatives Pricing (Alex Tse)

Derivatives pricing is at the core of trading and model validation, in so far as traders and quantitative analysts rely on stochastic models to build their trades and monitor their risks. The goal of this module is to introduce the technical tools needed to understand the specificities of these models and their inherent risks.

Selected Topics in Quantitative Finance (Vladimir Lucic)

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. In order to do so, the module will constantly strike a fair balance between the mathematical framework and specific tools (stochastic analysis, Fourier methods, fractional calculus), the numerical aspects (actual implementation of the models, optimization routines) and the data (calibration on real data, backward testing of hedging strategies)

Numerical Methods in Finance (Alex Tse)

Numerical Methods are at the very core of quantitative modelling, no matter which area one considers. 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 case, we will start with a theoretical framework, explaining how the option pricing problem can be turn, depending on the model used, either into a PDE problem, or into a Fourier integration issue. We shall then focus our attention to the actual numerical methods needed to implement these two approaches, and test them on real models and real data.

Machine Learning in Finance Stream

Algorithmic Trading and Machine Learning (Giuseppe Di Graziano)

The aim of the course is to present in some detail a series of models/techniques used in the algorithmic trading space. For each topic, we shall emphasize both theoretical aspects as well as practical applications. The course consists of two main blocks: 1) optimal execution theory and 2) machine learning for finance.

Optimal execution techniques are typically used by quantitative brokers to buy/sell large numbers of securities. Machine learning algorithms are often used by hedge fund and trading desks to generate trading signals, quote on exchange and hedge complex portfolios. The basic optimal execution problem consists of an agent (e.g. a bank or a broker) who needs to buy or sell a pre-specified number of units of a given asset within a fixed time frame (e.g. an hour, a day, etc.). Assuming that the purchase or sale of the asset will have an adverse impact on its price, what is the execution policy which minimizes market impact? This problem can be formulated as a trade-off between the expected execution cost and the price risk due to exogenous factors. We shall solve the optimization problem using different types of impact models (temporary, transient, permanent) and risk functions (variance, VaR).

Machine learning techniques are becoming increasingly popular in the financial industry. For example, they are used to help predict asset prices, improve the hedging and pricing of complex portfolios. In the lectures we shall analyse in detail some of the most popular supervised learning algorithms such as LASSO/Ridge regression, logistic regression and support vector machines. We shall also introduce unsupervised learning techniques such as clustering and PCA. We will talk about issues related to model selection, overfitting and explore ways to deal with other problems such as selection bias. Trading applications will be presented during the course. Students will be requested to implement some of the models presented in the lectures in Python.

Data Analysis and Machine Learning (Paul Bilokon)

The course 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 Path Theory in Machine Learning (Thomas Cass)

Rough path theory was developed in the 1990s in order to understand the structure and information content of a given path (be it a financial time series, a hand-drawn character or the route taken by a vehicle). It turned out to be one of the key developments in stochastic analysis over the past 20 years, and has allowed for a better understanding (and new proofs) to many problems in this field. The goal of this module is to provide students with a flavour of this powerful theory and to understand how it can efficiently be applied in machine learning, one of the fast-developing techniques in the financial industry nowadays. One of the key elements in this exploration is the so-called signature of a path, of which we shall study the algebraic properties, the faithfulness, as well as the inversion and asymptotic properties. We shall further see how this signature is in fact a feature set in machine learning, and illustrate these results in mathematical finance (in particular to predict financial time series), as well as in other areas (handwriting recognition, computer vision, classification problems in medical data).

Deep Learning (Mikko Pakkanen)

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.

Market Microstructure Stream

Convex Optimisation (Andreas Sojmark)

Many optimisation problems can be reformulated as convex optimisation problems (COP). These have many highly desirable features, in particular it is possible to numerically solve them efficiently and reliably. Morever, COP have a rich, and beautiful theory behind them: convex analysis, i.e. the theory of convex functions and sets, which leads to duality techniques and optimality and complimentary conditions crucial to solving a COP. Moreover, COP also admit several important subclasses of problems with additional nice properties: Linear, Quadratic and Conical optimisation problems, to just mention a few.

This module is devoted to explaining how to identify and solve several classes of COP. To better do this, we cover some of the underlying convex analysis, e.g. we talk about subgradients and Fenchel-Moreau conjugates. In particular, we cover in some detail the geometry of linear programming, discussing for example Farkas' lemma, the Bipolar theorem and the Minkowski-Weyl characterisation of polyhedra. Moreover, as COP occur naturally throughout mathematical finance, participants of this course will learn a spectrum of COP applications through a series of examples of practical relevance; in particular, we will cover cash flow matching, mean-variance portfolio optimization (in the way of Markowitz), robust portfolio optimisation etc.

In this module we will at times use Python and CVXOPT to numerically solve some COP. We will discuss the simplex algorithm, and its theoretical relevance to the study of Linear optimisation problems. We will also briefly talk of numerical methods for COP in more generality.

Stochastic Control in Finance (Eyal Neuman)

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.

Algorithmic and High-Frequency Trading(Eyal Neuman)

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 (Johannes Muhle-Karbe)

The goal of the module is to develop thorough understanding of how 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. The remaining part of the module focuses on statistical analysis of market microstructure, concentrating on statistical modelling of tick-by-tick data, measurement of price impact and volatility estimation using high-frequency data.

Portfolio Management(Johannes Muhle-Karbe)

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.