Module 3 - Equities & Currencies

In module three, we will explore the importance of the Black- Scholes theory as a theoretical and practical pricing model which is built on the principles of delta heading and no arbitrage. You will learn about the theory and results in the context of equities and currencies using different kinds of mathematics to make you familiar with techniques in current use.

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
  • 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-Kac formula
  • Extensions to Black-Scholes: dividends and time-dependent parameters
  • Black's formula for options on futures
Option Pricing Models: Connecting the Dots
  • Show that in the 1-period binomial model, the risk-neutral measure and the equivalent martingale measure are the same
  • Derive the Fundamental Asset Pricing Formula for the 1-period and multiperiod binomial model
  • Derive the Black-Scholes PDE from the 1-period binomial model
  • Establish the connection between the multiperiod binomial model yields the Black-Scholes formula from
  • Define complete markets and incomplete markets
  • Explain how the Black-Scholes model is a complete market
  • Show that, in a complete market, the no-arbitrage approach and the martingale measure approach are strictly equivalent
Intro to Numerical Methods
  • The justification for pricing by Monte Carlo simulation
  • Grids and discretization of derivatives
  • The explicit finite-difference method
Exotic Options
  • Characterisation of exotic options
  • Time dependence (Bermudian options)
  • Path dependence and embedded decisions
  • Asian options
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?
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
Advanced Volatility Modeling 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
FX Options
  • Size and importance of the FX and FX options market
  • How FX has developed into the largest global market
  • Current uses of FX options
  • Volatility surface and out of money options
  • Pricing of simple FX options and those replicated from vanilla calls and puts
  • Path-dependent FX options American and Bermudan
  • Risk management and basic delta hedging

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

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