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.