FM 5031/2 - Financial Mathematics Practitioner Sequence

Module - Risk & Asset Allocation

Instructor - John Dodson

Fall 1 - Introduction & Random Variables

• Introduction
• Summary
• Random Variables
• Characterizations
• Transformations
• Expectation
• Law of Large Numbers
• Monte Carlo
• Summary Statistics
• Central Moments
• Normal Distribution
• Central Limit Theorem
• Change of Measure
• Mixtures


• exercise: Jensen's inequality

Fall 2 - Common Characterizations

• Topological Taxonomy
• Finite Support
• Countable Support
• Interval Support
• Half-line Support
• Unbounded Support
• Common Transforms
• Common Mixtures
• Non-Parametric


• exercise: simulating and working with samples of random variates

Fall 3 - Dependence

• Independence
• Conditioning & Margining
• Bivariate Normal
• Dependence
• Copulae
• Concordance
• Bayes' Rule
• Covariance
• Cholesky Decomposition
• Spectral Decomposition
• Mahalanobis Distance
• Multivariate Distributions


• case: regression as conditional expectation
• exercise: non-linear dependence from a non-normal copula

Fall 4 - Market Prices as Random Variables

• Modeling the Market
• Securities Markets
• Market Conventions
• Investment Horizon
• Quest for Invariance
• Identifying Invariants
• Projecting Invariants
• Mapping Invariants


• case: scaling between the sampling period and the investment period
• exercise: graphical tests for invariants

Fall 5 - Estimators

• Motivation
• Sample
• Sufficient Statistic
• Estimator
• Loss
• Maximum Likelihood Estimator
• Standard Error
• Admissibility


• exercise: estimating drift and volatility

Fall 6 - Heteroskedasticity

• Specification
• Estimation
• Variance Targeting
• Forecasting


• case: comparing MLE's
• exercise: estimating ARCH parameters

Fall 7 - Real-World Data

• Robustness
• M-Estimators
• Missing Data
• Overlapping Data
• Other Techniques
• Implied Characterization


• case: option-implied standard deviation
• assignment: estimating a copula from de-volatized residuals