The Core Modules cover the mathematical foundations (probability, statistics, PDEs) stochastic calculus and martingale theory, portfolio theory, the Black-Scholes model and extensions, numerical methods (finite differences and Monte Carlo), interest rate modeling, stochastic optimization, exotic derivatives and stochastic volatility. Matlab is taught as a practical computing language.  

The Core Modules are assessed formatively.  (Feedback will be given with an indicative mark in order to assist students in improving their future performance.)  Attendance at the Core Modules is compulsory.  Two two-hour written examinations held in September cover the material of the Core Modules.  

Please note that the content of the Modules is subject to slight variation.  

Core Modules 2013

Module 1: Mathematical and Technical Prerequisites

7-11 January 2013

Assignment due: 12 noon, 11 February 2013

Syllabus

• Probability: basics, review of discrete and continuous random variables, properties,important distributions, measure theory, change of measure, convergence of random variables, limit theorems
• Statistics: review of sampling and estimation, parameter estimation, regression techniques, tests for normality, QQ plots, Bayesian techniques, elementary principal components analysis
• PDEs: parabolic partial differential equations; heat equation, link to random walks, similarity solutions, Fourier transform; qualitative properties of solutions, maximum principles, smoothness
• Introduction to Matlab: basics, plotting, implementation of elementary numerical concepts applied to finance
• Binomial trees, discrete martingales: one-period and multi-period binomial stock price models, arbitrage pricing of options on trees
• Portfolio theory, utility: expected returns, variance and covariances, benefits of diversification, the opportunity set, efficient frontiers and the Sharpe ratio, utility, risk aversion, optimal investment, convex duality
• Stochastic calculus: Brownian motion, constructions, non-differentiability, quadratic variation, stochastic integration, construction of Ito integral and properties, the Ito formula
• SDEs: random walks in continuous time, strong and weak solutions, expectations of solutions

Timetable

Course Materials

Module 2: Black-Scholes Theory

11-15 March 2013

Assignment due: 12 noon, 15 April 2013

Syllabus

• The Black-Scholes model: perfect replication, risk-neutral valuation, the Black-Scholes PDE and solutions
• Extensions of Black-Scholes: discrete and continuous dividend payments; time-dependent volatility, dividends and interest rates
• Introduction to Monte Carlo: uniform randome number generators, sampling non-uniform distributions, implementation of MC methods, simple variance reduction techniques, workshop
• Basic exotic options: general payoffs, options on futures, pay-later options; multi-stage options, forward-start options, ratchets, compound options, choosers
• Multi-factor problems, quantos, FX and basket options
• Elementary stochastic differential equations, transition density functions, Feynman-Kac formula, Kolmogorov equations, exit times and hitting probabilities, Girsanov's theorem, maximum/minimum of Brownian motion
• Multi-factor problems, quantos, FX and basket options
• Elementary stochastic differential equations, transition density functions, Feynman-Kac formula, Kolmogorov equations, exit times and hitting probabilities, Girsanov's theorem, maximum/minimum of Brownian motion
• Martingale methods in discrete and continuous time, martingale representation theorem
• Explicit and implicit finite difference schemes, implementation, accuracy and stability, Greeks and smoothing schemes; workshop
• Introduction to the term structure of interest rates, bond price equilibria, duration and convexity, caps, floors, swaps
• Overview of instruments in rates markets, interpolation of yield curves and volatility term structures, bootstrapping

Timetable

Course Materials

Module 3: Extensions of the Black-Scholes Framework

22-26 April 2013

Assignment due: 12 noon, 27 May 2013

Syllabus

• Hedging, Greeks: delta, gamma, theta; vega, rho as out-of-model hedges; less common sensitivities
• American options: early exercise, linear complementarity problem, perpetual put, free boundary formulation, smooth pasting
• Finite differences for American options: explicit methods, projected iterations, a penalty method; workshop
• Stochastic optimization: dynamic programming, continuous time stochastic control, Merton problem
• Asset pricing: equivalent measures, martingale measures, numeraires, risk-neutral pricing, market price of risk, arbitrage pricing in binomial models, completeness, trinomial trees, fundamental theorems of asset pricing in discrete and continuous time, stopping times and American options
• Interest rate trees, the role of numeraires for interest rate derivatives
• Models for the short rate, use and calibration: Vasicek, CIR, Hull-White

Timetable

Course Materials

Module 4: Exotic Options and Advanced Modelling Techniques

24-28 June 2013

Assignment will not be marked but solutions will be place on the website on or after 29 July 2013

Syllabus

• Barrier options: down-and-out and other barrier contracts, intermittent sampling, American digitals, reflection principle
• Further exotic options: Asians, rate and strike options, similarity reductions, discretely sampled options and jump conditions; lookbacks, continuous and discrete sampling, probabilistic methods via reflection
• Monte Carlo for exotic options: continuity corrections for discretely sampled paths for barriers and lookbacks; correlation and basket options; variance reduction; workshop
• Implied and local volatility, Shimko's method and Dupire's formula; stochastic volatility models: stylised facts, econometric models, complete stochastic volatility models; two-factor models, hedging and market completion, special cases
• Jump diffusion and Levy processes: Poisson and Cox process, Ito with jumps, Merton's model; Levy processes, hedging and pricing jump-risk, characteristic functions
• Yield curve modelling, market models (HJM, LMM)
• Practicalities of pricing and hedging interest rate products; SABR; running a swaps desk; managing exotics
• Practicalities of FX and equity derivatives pricing, calibration and implementation

Timetable

Course Materials from 2012

Written Examinations

23 September 2013

Two two-hour written examinations cover the material of modules 1-4.  The examination papers will consist of the following sections.  Each section predominantly covers one of the components of the core module material (see Course Handbook for details of these components) though may draw on other components of the core module material.

 Paper 1

• Section A “Mathematical Techniques” (two questions)
• Section B “Portfolio Theory, Asset Pricing” (two questions)
• Section C “Interest Rates” (two questions)

 Paper 2

• Section D “Derivative Pricing” (four questions)
• Section E “Numerical Methods” (two questions)

Past examination papers

Calculators are not allowed in the examination room.

For further details on assessment see Examination Conventions.