Monte Carlo Simulation and Finance

1. Introduction.
2. Some Basic Theory of Finance.
- Introduction to Pricing: Single PeriodModels.
- MultiperiodModels.
- Determining the Process Bt.
- Minimum Variance Portfolios and the Capital Asset Pricing Model.
- Entropy: choosing a Q measure.
- Models in Continuous Time.
- Problems.
3. Basic Monte Carlo Methods.
- Simulation andMonte CarloMethods.
- UniformRandomNumber Generation.
- Apparent Randomness of Pseudo-RandomNumber Generators.
- Generating Random Numbers from Non-Uniform Continuous Distributions.
- Generating RandomNumbers fromDiscrete Distributions.
- RandomSamples Associated withMarkov Chains.
- Simulating Stochastic Partial Differential Equations.
- Problems.
4. Variance Reduction Techniques.
- Introduction.
- Variance reduction for one-dimensional Monte-Carlo Integration.
- Problems.
5. Simulating the value of Options.
- Asian Options.
- Pricing a Call option under stochastic interest rates.
- Simulating Barrier and lookback options.
- Survivorship Bias.
- Problems.
6. Quasi- Monte Carlo Multiple Integration.
- Introduction.
- Theory of Low discrepancy sequences.
- Examples of low discrepancy sequences.
- Problems.
7. Estimation and Calibration.
- Introduction.
- Finding a Root.
- Maximization of Functions.
- MaximumLikelihood Estimation.
- Using Historical Data to estimate the parameters in Diffusion Models.
- Estimating Volatility.
- Estimating Hedge ratios and Correlation Coefficients.
- Problems.
8. Sensitivity Analysis, Estimating Derivatives and the Greeks.
- Estimating Derivatives.
- Infinitesimal Perturbation Analysis: Pathwise differentiation.
- Calibrating aModel using simulations.
- Problems.
9. Other Directions and Conclusions.
- Neural Nets, Chaos, Non-linear Dynamics.
- Arch and GarchModels.
- Monte Carlo Tomorrow.
- References.

DON L. McLEISH is Professor of Statistics and Actuarial Science at the University of Waterloo. His research has focused on probability, statistical methods and models in general, and their application to financial data, including wide-tail alternatives to the normal distribution and the consequences for derivatives and asset pricing. He has contributed to the application of Monte Carlo techniques, variance reduction, and stochastic calculus to problems in finance, and is cofounder of the University of Waterloo's Center for Advance Studies in Finance. McLeish is also coauthor, with C.G. Small, of The Theory and Application of Statistical Inference Functions and Hilbert Space Methods in Probability and Statistical Inference (Wiley).