# Module 3: Equities & Currencies

### Black-Scholes Model

- The assumptions that go into the Black-Scholes equation
- Foundations of options theory: delta hedging and no arbitrage
- The Black-Scholes partial differential equation
- Modifying the equation for commodity and currency options
- The Black-Scholes formulae for calls, puts and simple digitals
- The meaning and importance of the Greeks, delta, gamma, theta, vega and rho
- American options and early exercise
- Relationship between option values and expectations

### Martingale Theory – Applications to Option Pricing

- The Greeks in detail
- Delta, gamma, theta, vega and rho
- Higher-order Greeks
- How traders use the Greeks

### Martingales and PDEs: Which, When and Why

- Computing the price of a derivative as an expectation
- Girsanov's theorem and change of measures
- The fundamental asset pricing formula
- The Black-Scholes Formula
- The Feynman-K_ac formula
- Extensions to Black-Scholes: dividends and time-dependent parameters
- Black's formula for options on futures

### Understanding Volatility

- The many types of volatility
- The market prices of options tells us about volatility
- The term structure of volatility
- Volatility skews and smiles
- Volatility arbitrage: Should you hedge using implied or actual volatility?

### Introduction to Numerical Methods

- The justification for pricing by Monte Carlo simulation
- Grids and discretization of derivatives
- The explicit finite-difference method

### Further Numerical Methods

- Implicit finite-difference methods including Crank-Nicolson schemes
- Douglas schemes
- Richardson extrapolation
- American-style exercise
- Explicit finite-difference method for two-factor models
- ADI and Hopscotch methods

### Exotic Options

- Characterisation of exotic options
- Time dependence (Bermudian options)
- Path dependence and embedded decisions
- Asian options

### Derivatives Market Practice

- Option traders now and then
- Put-Call Parity in early 1900
- Options Arbitrage Between London and New York (Nelson 1904)
- Delta Hedging
- Arbitrage in early 1900
- Fat-Tails in Price Data
- Some of the Big Ideas in Finance
- Dynamic Delta Hedging
- Bates Jump-Diffusion

### Advanced Greeks

- The names and contract details for basic types of exotic options
- How to classify exotic options according to important features
- How to compare and contrast different contracts
- Pricing exotics using Monte Carlo simulation
- Pricing exotics via partial differential equations and then finite difference methods

### Advanced Volatility Modelling in Complete Markets

- The relationship between implied volatility and actual volatility in a deterministic world
- The difference between 'random' and 'uncertain'
- How to price contracts when volatility, interest rate and dividend are uncertain
- Non-linear pricing equations
- Optimal static hedging with traded options
- How non-linear equations make a mockery of calibration

### Market-Based Valuation of Equity Index Options

- Stylized Facts of Equity & Options Markets
- Numerically efficient valuation of equity index options
- Calibration of option pricing models to market data
- Simulation of option pricing models for European & American options
- Canonical Example
- Euro Stoxx 50 index and options
- Merton (1976) jump-diffusion model
- Python implementation

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

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