The lognormal random walk and the Black-Scholes model have been very successful in practice. Yet there is plenty of room for improvement. The benefits of new models will be discussed from theoretical, practical and commercial viewpoints. When pricing complex products it is necessary to be able to correctly value vanilla products. Modern models adopt frameworks that ensure that basic products are perfectly calibrated initially.

The models derived in earlier parts of the course are only as good as the solution. Increasingly often the problems must be solved numerically. We explain the main numerical methods, and their practical implementation.

• Volatility surfaces: Analysis and calibration, the behaviour of implied volatility.
• Stochastic volatility: Modelling and empirical evidence, pricing and hedging, mean-variance analysis.
• Jump diffusion: Discontinuous price paths, the Merton model, jump distributions, expectations and worst-case analysis.
• Exotic options: Common OTC contracts and their mathematical analysis.
• Monte Carlo simulations: Use for option pricing, speculation and scenario analysis, differences
  between equity/currency/commodity and the fixed-income worlds, accuracy, variance reduction,
  bootstrapping.
• Quasi-Monte Carlo methods: Low-discrepancy series for numerical quadrature, Halton, Sobol, Faure
  and Haselgrove methods.
• Finite-difference methods: Crank-Nicolson, and Douglas multi-time level methods, convergence,
  accuracy and stability.
• Non-probabilistic models: Uncertainty in parameter values versus randomness in variables, non-
  Brownian processes, nonlinear diffusion equations.
• Static hedging: Hedging exotic target contracts with exchange-traded vanilla contracts, optimal static
  hedging.
• Brace, Gatarek and Musiela: The evolution of forward rates continued, the discrete-maturity case.

Lecture 6.1

• Characterizing exotic options
• Simulations and partial differential equations
• Examples

Lecture 6.2

• The Poisson process for modelling jumps
• Hedging in the presence of jumps
• How to price derivatives when the path of the underlying can be discontinuous
• Modeling volatility as a stochastic variable
• How to price contracts when volatility is stochastic
• The market price of volatility risk

Lecture 6.3

• 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

Lecture 6.4

• Advanced Monte Carlo techniques
• Variance reduction
• Building spreadsheets for pricing and default timing
• Low-discrepancy sequences

Lecture 6.5

• Implicit finite-difference methods including Crank-Nicolson
• Douglas schemes
• Richardson extrapolation
• American-style exercise and exotic options
• About the explicit finite-difference method for two-factor models
• About the ADI and Hopscotch methods

Lecture 6.6

• Brace, Gatarek, Musiela, the LIBOR market model
• Discrete interest rates

Lecture 6.7

• The effect of hedging at discrete times
• Hedging error
• The real distribution of profit and loss
• How to allow for transaction costs in option prices

Preparatory reading:

• P. Wilmott, Paul Wilmott On Quantitative Finance, second edition, 2006, John Wiley.
  Chapters 22-29, 37, 45-48, 50, 51-53, 57, 73, 76-79, 82
• Jaeckel, Monte Carlo Methods in Finance, 2002, John Wiley. Chapters 1—14

Further reading:

• 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

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

• How to Hedge
• Inverse Problems in Finance
• Advanced Brace, Gatarek and Musiela
• Transform Methods