## Elective modules

### Derivatives Pricing Stream

Stochastic Calculus for Finance (Thomas Cass)

This module will introduce and develop the basic elements of stochastic calculus and illustrate how these mathematical ideas can be applied to study problems arising in finance. The module will begin by reviewing the core results from the theory of stochastic integration and stochastic differential equations, including the Cameron Martin Theorem, strong existence and uniqueness of stochastic differential equations (SDEs) and the martingale representation theorem. We then give a brief overview of the tools of the so-called stochastic calculus of variations (or Malliavin calculus), including results on the derivatives of SDE solutions with respect to their parameters. The final part of the course will consider applications in finance, in particular to the efficient computation of sensitivities of option prices under diffusion models.

Topics in Derivatives Pricing (Alex Tse)

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).

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.

Numerical Methods in Finance (Alex Tse)

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.

### 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.

Advances in Machine Learning (Paul Bilokon)

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.

Advanced topics in Data Science: Signatures and Rough Paths in Machine Learning (Thomas Cass)

Rough path theory was developed in the 1990s in order to understand the response of a nonlinear system to highly oscillatory input signal. A key element of this theory so-called signature transform which gives an economical way to represent and extract information from high dimensional ordered data, such as a complex financial time series. Over the last decade it has been used to achieve state-of-the art outcomes in several data science challenges. This short module will give an overview of the mathematical properties of the signature, explain how it can be used as a feature set in machine learning application with a particular emphasis on problems inspired by finance. Topics covered will include:

• Key mathematical properties of the signature transform
• The use of the signatures a feature set in machines learning. Two examples will be developed in detail to illustrate this: (a) learning a solution to a stochastic differential equation, and (b) learning a high-frequency trading strategy. Computational methods. Other examples will be explored in the coursework and as time permits.
• Recovering information about a data stream from the signature, the asymptotic analysis of the signature.
• Signatures and kernel methods. Using signatures in neural networks.

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)

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 (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.