It will be necessary to bring all students up to the same technical level. Most students will be familiar with the contents of this first module, but any gaps in a student’s background will be identified. We introduce the rules of applied Itô calculus as a modelling framework. Simple stochastic differential equations and their associated Fokker-Planck and Kolmogorov equations are introduced.

• Important mathematical tools and results
• Taylor series
• Ordinary differential equations
• Probabilistic concepts
• Gaussian, Poisson, Cauchy, Binomial, etc.
• Central Limit Theorem
• The random behaviour of asset prices
• Stochastic calculus and Itô’s Lemma
• Transition density functions
• Partial differential equations
• Martingale theory
• Change of numéraire                                                                                   
• The Radon-Nikodym derivative                                                                  

Lecture 1.1

• Notation commonly used in mathematical finance                                                  
• How to examine time-series data to model returns
• The random nature of prices
• Unpredictability
• The need for probabilistic models
• The Wiener process, a mathematical model of randomness
• A simple model for equities, currencies, commodities and indices

Lecture 1.2

• Taylor series
• A trinomial random walk
• Transition density functions
• Our first differential equation
• Similarity solutions

Lecture 1.3

• The Central Limit Theorem
• The meaning of Markov and martingale
• Brownian motion
• Stochastic differential equations
• Itô’s lemma

Lecture 1.4

• Continuous-time stochastic differential equations as discrete-time processes
• Simple ways of generating random numbers in Excel
• Correlated random walks
• Using Itô’s lemma to manipulate stochastic differential equations

Lecture 1.5

• Visual Basic workshop
• Hands-on implementation, an introduction

Lecture 1.6: Methods for Quantitative Finance: I

• Double Integration & Applications
• Probability Distributions

Preparatory reading:

• P. Wilmott, Paul Wilmott Introduces Quantitative Finance, second edition, 2007, John Wiley. Chapters 4, 5, 7
• M. Jackson and M. Staunton, Advanced Modelling in Finance Using Excel and VBA, 2001, John Wiley. Chapters 1 - 4

Further reading:

• G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes, 1997, Oxford University Press
• J.D. Hamilton, Time Series Analysis, 1994, Princeton University Press
• J.A. Rice, Mathematical Statistics and Data Analysis, 1988, Wadsworth-BrooksCole
• S.N., An Introduction to the Mathematics of Financial Derivatives, 1996, Academic Press

Follow-up recording(s), extra lecture(s):

• Linear Algebras
• Stochastic Calculus
• Differential Equations
• Fundamentals of Optimization