# 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

### 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

### 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