Module 1: Building Blocks of Finance

The Random Behaviour 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

PDE's 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

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

Discrete Martingales

  • Binomial Model extended
  • The Probabilistic System: sample space, filtration, measures
  • Conditional and unconditional expectation
  • Change of measure and Radon-Nikodym derivative

Continuous Martingales

  • What is a continuous time martingale?
  • Martingales and Itô calculus
  • A detour to explore some further Ito calculus
  • Exponential martingales, Girsanov and change of measure
  • The martingale zoology in a hurry: martingale, local martingale, supermartingale, submartingales, semimartingales
Lecture order and content may occasionally change due to circumstances beyond our control; however this will never affect the quality of the program.