The Black-Scholes theory, built on the principles of delta-hedging and no arbitrage, has been very successful and fruitful as a theoretical model and in practice. The theory and results are explained using different kinds of mathematics to make the student familiar with techniques in current use.

• The Black-Scholes model: A stochastic differential equation for an asset price, the delta-hedged
• portfolio and self-financing repliction, no arbitrage, the pricing partial differential equation and simple solutions.
• The greeks: delta, gamma, theta, vega and rho and their uses in hedging.
• Risk-neutrality: Fair value of an option as an expectation with respect to a risk-neutral density function.
• Early exercise: American options, elimination of arbitrage, modifying the binomial method, gradient conditions, formulation as a free boundary problem.
• Elementary numerical analysis: Monte Carlo simulation and the explicit finite-difference method.
• Value at Risk: Portfolios of derivatives.
• Martingales: The probabilistic mathematics used in derivatives theory

Lecture 3.1

• 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

Lecture 3.2

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

Lecture 3.3

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

Lecture 3.4

• The many types of volatility
• What the market prices of options tells us about volatility
• The term structure of volatility
• Volatility skews and smiles
• Exploiting your volatility models
• Should you hedge using implied or actual volatility?

Lecture 3.5

• Martingale theory and its relevance to pricing
• Its role in practice
• Examples

Preparatory reading:

• P. Wilmott, Paul Wilmott Introduces Quantitative Finance, second edition, 2007, John Wiley. Chapters 6, 8, 27-30
• M. Jackson and M. Staunton, Advanced Modelling in Finance Using Excel and VBA, 2001, John Wiley.
  Chapters 9, 11—12
• E.G. Haug, The Complete Guide to Option Pricing Formulas, second edition, 2007, McGraw-Hill Professional.
  Chapters 1, 2, 7, 8, 12
• P. Wilmott, Paul Wilmott On Quantitative Finance, second edition, 2006, John Wiley. Chapter 12, 49, 50

Further reading:

• N. Taleb, Dynamic Hedging, 1996, John Wiley
• J.C. Hull, Options, Futures and Other Derivatives (5th Edition), 2002, Prentice-Hall
• K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 1994,
  Cambridge University Press
• G.D. Smith, Numerical Solution of Partial Differential Equations, 1985, Oxford University Press
• S.N. Neftci, An Introduction to the Mathematics of Financial Derivatives, 1996, Academic Press

Follow-up recording(s), extra lecture(s):

• American Options
• Infinite Variance