Module 1 - Building Blocks of Quantitative Finance

In module one, we will introduce you to the rules of applied Itô calculus as a modeling framework. You will build tools using both stochastic calculus and martingale theory and learn how to use simple stochastic differential equations and their associated Fokker- Planck and Kolmogorov equations.

The Random Behavior of Assets

Different types of financial analysis
Examining time-series data to model returns
Random nature of prices
The need for probabilistic models
The Wiener process, a mathematical model of randomness
The lognormal random walk- The most important model for equities, currencies, commodities and indices

Binomial Model

A simple model for an asset price random walk
Delta hedging
No arbitrage
The basics of the binomial method for valuing options
Risk neutrality

PDEs and Transition Density Functions

Taylor series
A trinomial random walk
Transition density functions
Our first stochastic differential equation
Similarity reduction to solve partial differential equations
Fokker-Planck and Kolmogorov equations

Applied Stochastic Calculus 1

Moment Generating Function
Construction of Brownian Motion/Wiener Process
Functions of a stochastic variable and Itô’s Lemma
Applied Itô calculus
Stochastic Integration
The Itô Integral
Examples of popular Stochastic Differential Equations

Applied Stochastic Calculus 2

Extensions of Itô’s Lemma
Important Cases - Equities and Interest rates
Producing standardised Normal random variables
The steady state distribution

Martingales

Binomial Model extended
The Probabilistic System: sample space, filtration, measures
Conditional and unconditional expectation
Change of measure and Radon-Nikodym derivative
Martingales and Itô calculus
A detour to explore some further Ito calculus
Exponential martingales, Girsanov and change of measure

Lecture order and content may occasionally change due to circumstances beyond our control. However, this will never affect the quality of the program.