# Modules

Module 1: Basic mathematics – Calculus

(by Professor Peter Wood)

In this module, we will review the essential concepts from calculus used later in this course. Topics include the derivative of single and multivariable functions; linear and higher-order approximations to differentiable functions; and a brief introduction to integration.

Module 2: Basic mathematics – Matrix Algebra

(by Professor Thomas F. Coleman)

This module covers some of the basic matrix concepts and ideas that are commonly used in the solution of computational problems in financial risk management. Included are a review of fundamental matrix operations and methods to solve linear equations: the LU-factorization and the Cholesky factorization are highlighted.  In addition, there is a review of the basic approaches to solving overdetermined linear regression problems such as the method of normal equations and the QR-factorization.

Module 3: Basic mathematics – Numerical Methods

(by Professor Thomas F. Coleman)

This module covers some of the basic numerical methods (excluding those for linear problems) that commonly arise in the computational solution of problems in financial risk management. Included are a review of methods for one-dimension zero-finding and minimization problems: i.e., Newton-like and bisection approaches. One-dimensional problems are followed by multi-dimensional nonlinear systems solutions and nonlinear optimization, including linearly constrained problems. Finally there is a brief discussion on the finite-difference solution approach to solving partial differential equations (PDEs).

Module 4: Basic mathematics – Probability

(by Professor Surya Banerjee)

This module covers the basic concepts of probability. The classical definition and its various properties are discussed. The rules of probability, conditional probability and Bayes’ Rule is also discussed. In addition there is an introduction to random variables, and discrete and continuous distributions.

Module 5: Basic mathematics – Statistics

(by Professor Surya Banerjee)

This module covers the various methods of descriptive statistics that are commonly used in data analysis. Algebraic data summaries—Central Tendency, Variability, Skewness are discussed. In addition, common graphical data summaries like histograms, box-plots, empirical c.d.f. are also explained.

Module 6: Basic mathematics – Financial Mathematics & Bonds

(by Professor Peter Wood)

In this module, we will cover some of the basics of financial mathematics, including the calculation of the present and future value of an investment given an interest rate; a review of the money markets; simple mortgage calculations and a brief discussion of the yield curve.

For Bond, we will cover the important principles of bond valuation and bond price sensitivities. Topics include the yield to maturity of a bond; the clean and dirty prices; and sensitivity measures including duration and convexity. We will also introduce swaps, interest rate caps, floors and collars.

Module 7: Portfolio Theory – Utility Theory

(by Professor Ken Seng Tan)

This module provides a basic introduction to utility theory and individual's aversion to risk. We also relate this theory to decision making under uncertainty and to insurance.

Module 8 & 9: Portfolio Theory – Capital Asset Pricing Model & Mean Variance

(by Professor David Saunders)

These modules cover the fundamental trade-off between risk and return when investing in financial markets. The mean-variance model, in which investors maximizing expected return while minimizing the variance of returns, is presented in detail. This is followed by a discussion of systematic and idiosyncratic risks in financial markets, and the capital asset pricing model.

Module 10: Derivatives 1 – Introduction

(by Professor Carole Bernard)

This is an introduction to the most basic derivatives: we define forward, futures and options. We then explain how they can be useful in hedging and risk management as well as for trading. In particular we review the most popular strategies involving options.

Module 11: Derivatives 2 – Arbitrage

(by Professor Carole Bernard)

We define the important concept of "arbitrage" and explain how to use it to find the price of forwards and futures. We show that arbitrage arguments can be used to derive relationships among prices of options including the famous put call parity and convexity property of options.

Module 12: Derivatives 3 – Binomial Model

(by Professor Carole Bernard)

We start by explaining how to price options in the most basic model in finance: the one period binomial tree. We derive a pricing formula as well as a replication strategy for any option. We extend the presentation to a slightly more realistic model, i.e. the multiperiod model.

Module 13: Derivatives 4 – Black Scholes Model

(by Professor Carole Bernard)

We present the Black Scholes model, the price of call and put options in the binomial model. We also introduce the "implied volatility", an important concept in practice. Finally, we study the sensitivity of the price of calls and puts with respect of the different market parameters, called Greeks. We briefly explain how to extend the hedging principles seen in the multiperiod binomial model in continuous time.

Module 14: Derivatives 5 – Advanced Concepts

(by Professor Carole Bernard)

We study more advanced concepts that extend Part 3 and 4. We start with a review of the most common exotic options. We briefly explain what American options are and how to price them in a multiperiod binomial model. We then show how to price in the Black Scholes model a large number of exotic options. We end with the limitations of the Binomial and Black-Scholes model and explain with an example that markets in practice are incomplete.

Module 15: Advanced Risk Management – Credit Risk

(by Professor David Saunders)

This module presents the basics of credit risk, one of the oldest and most important forms of financial risk.  In particular, we will study mathematical models for credit risk losses, both at the single instrument, and the portfolio level.

Module 16: Advanced Risk Management – Economic Capital

(by Professor David Saunders)

This module introduces the concept of economic capital, which represents the amount of capital an institution estimates that it needs to maintain to operate as a going concern and avoid insolvency with a high degree of confidence. We will also discuss the use of economic capital as a management tool, considering applications to performance measurement and the estimation of return on risk adjusted capital.

Module 17: Advanced Risk Management – Liquidity Risk

(by Professor David Saunders)

This module presents an introduction to liquidity risk. Basic liquidity risk management concepts are introduced, and some cases in which the failure of liquidity risk management caused serious problems for financial institutions are discussed. Regulatory attempts to limit and control liquidity risk are also discussed.

Module 18: Advanced Risk Management – Risk Measures

(by Professor David Saunders)

This module concerns risk measures and their use in financial institutions, with a focus on practical issues in the use of risk measures. Topics covered include Value-at-Risk (VaR) and its alternatives, estimating risk measures and evaluating their accuracy through historical backtests, and assessing the risk contributions of portfolio constituents.

Module 19: Advanced Risk Management – Stress Testing

(by Professor David Saunders)

This module presents an introduction to stress tests, which are tools employed by financial institutions to understand the impact of extreme events on their portfolios. Topics covered include types of stress tests, methods of defining stress scenarios, stress testing frameworks, and issues that arise in stress testing in practice.

Module 20: Case Study – China Aviation Oil

(by Professor Ranjini Jha)

The case focuses on the importance of risk management for the survival of an organization. The case shows the importance of the development of risk management procedures and their effective implementation and compliance with the policies. It also shows the importance of risk measurement and its implications for internal and external financial reporting.