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School of Mathematics
School of Mathematics         The diagram illustrates the local accuracy of the tangent line approximation to a smooth curve, or--otherwise stated--the closeness of the differential of a function to the difference of function values due to a small increment of the independent variable.  
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Course Sequence Syllabus

 

Syllabus for FM 5001/FM 5002, Preparation for Financial Mathematics

This syllabus is currently under development.

Syllabus for FM 5011/FM5012, Mathematical Background for Finance

This syllabus is currently under development.

Syllabus for FM 5011

  1. Discrete Models of Financial Markets
    • Information Structures, Trading Strategies
    • Completeness of Markets
    • Adapted Processes, Conditional Expectations, Martingales
    • Discrete Versions of the Stochastic Integral, Itô's Lemma, Girsanov's Theorem
    • Application to Option Pricing Models
  2. Introduction to Continuous Stochastic Processes
    • General Probability Spaces and Information Structures
    • The Wiener Process
    • Stochastic Processes as Solutions of Stochastic Differential Equations
    • Itô Processes, Itô's Lemma as an Analogue of the Chain Rule
  3. Application to the Black-Scholes Model
  4. Derivation of the Black-Scholes Partial Differential Equation
  5. Continuous Stochastic Processes
    • The Stochastic Integral, Properties of Itô's Integral
    • Itô's Formula for the Wiener Process
    • Representation of Martingales as Stochastic Integrals
    • Stochastic Exponential, Girsanov's Theorem
    • The Radon-Nikodym Theorem and Radon-Nikodym Derivative
    • The Black-Scholes Revisited
  6. Local Martingales and Semi-Martingales
    • Maximal and Quadratic Spaces, Orthogonal Local Martingales
    • Stopping Time and Optional Stopping
    • Doob's Inequality, Predictable Processes
    • The Doob-Meyer Decomposition
    • Quadratic Variation Processes
  7. General Stochastic Calculus
    • Lebesgue-Stieltjes Integral
    • Stochastic Integrals
    • Stable Spaces of Martingales
  8. Statistics
    • Regression and Multiple Regression Analysis
    • Statistical Distributions
    • Statistical Tests
    • Student t-Test
    • F-Test
    • Chi-Squared Test
    • Hypothesis Testing
    • Confidence Intervals
    • Error Analysis
    • Sources of Errors
    • Estimating the Size of Errors
    • Propagation of Errors
    • Working with Time Series

Syllabus for FM 5012

  1. Trees
    • Approximation of Diffusions by Binomial and Trinomial Trees
    • Option Pricing: Path Independent, Path Dependent, Early Exercise
  2. Numerical PDE Solutions by Finite Difference Methods,
  3. Including Explicit, Fully Implicit and Crank-Nicholson
    • Convergence Analysis: Accuracy and Stability
    • Solution Methods for Linear Systems:
      • Direct and Iterative PSOR for Early-Exercise
      • Linear Complementarity Problems
  4. Monte Carlo Methods
    • Approximation of Expectations:
      • Use of Pseudo-Random Numbers, Error Analysis
    • Discretization Methods for SDE Simulation
    • Early Exercise Analysis via Regression Methods
  5. Transform Methods: Use of Discrete Fourier Transforms in Option Pricing

Syllabus for FM 5021/FM 5022, Mathematical Theory Applied to Finance

This syllabus is currently under development.
  1. Capital Markets & Derivatives
    1. Time Value of Money
      • Compounding, Conversion between Rates and Industry Practice
      • Zero Coupon Bonds
      • Coupon Bonds
      • Perpetuities
      • Annuities
      • Accumulation and Annuitization
      • Dividend discount models
    2. The Elements of the Basic Derivatives Tool Box
      • Forwards & Futures
      • Vanilla Interest Rate Fixed for Floating Swaps
      • Short Dated European Equity Options
      • Short Dated American Equity Options
      • Short Dated Interest Rate European Options
    3. Market Data Structures
      • Bootstrapping a Yield Curve
      • Cash Market Curve
      • Treasury Zero Curve
      • US Dollar LIBOR
      • Bootstrapping ATM Equity Volatility
      • Modeling ATM Interest Rate Volatility
      • Modeling Credit (Probability of Default)
      • Prepayments
      • Currencies
    4. Deeper Look at the Basic Tool Box using the Market Data Structures
      • Futures
      • Equity & Equity Index Futures
      • Repo Rate
      • Dividends
      • Bond Futures
      • Cheapest To Deliver
      • Eurodollar Futures
      • The Convexity Adjustment
      • Swaps
      • How the ISDA Agreement and Confirmation Affects Valuation
      • Day Count Conventions and Calendars
      • Payment Adjusted / Unadjusted
      • Other Swaps
      • Fixed to Fixed / Floating to Floating
      • USD Currency Swap / Currency to Currency Swap
      • Asset Swap
      • Short Dated European Equity Options
      • Risk Neutral Pricing
      • Black Scholes Merton Model
      • 1st Derivation Using a Matching Portfolio of Bonds and Stocks
      • Discrete & Continuous Dividends
      • Other Financing Considerations
      • Implications of Underlying Assumptions
      • Equity Volatility is Key to Pricing
      • Historical Volatility
      • Implied Volatility
      • Realized Volatility
      • Short Dated American Equity Options
      • Modeling a Path Dependent Option with a Binomial Model
      • Short Dated Interest Rate European Options
    5. Deeper Look at Market Data Structures
      • Introduction to Monte Carlo Simulation
      • Modeling Equity Prices with Brownian Motion
      • Modeling Interest Rates with Mean Reversion
      • Volatility
      • Skew
      • Regime Switching
      • Long Dated Implied Volatility
      • Relationship Between Equities, Rates and Cash
      • Historical Correlation & Covariance
      • Implied Correlation & Covariance
      • Realized Correlation & Covariance
      • Cholesky Decomposition
      • Closer Look at Actual Capital Market Instrument Distributions
      • Equity Returns
      • Index Returns
      • Mortgage Backed Security Returns
      • Prepayment Model Factors and Path Dependence
      • Credit Curves
      • Forecasting Market Data Structures
      • Forecasting Interest Rate Implied Volatility
      • Forecasting Equity Implied Volatility
      • Long Dated Volatility Includes Rates
      • Forecasting Correlations
      • Methods for Finding the Closest Positive Definite Matrix
    6. Complex Derivatives
      • Forward Start Equity Options
      • Swaptions
      • Long Dated Puts (Particularly the Spot-at-the-Money Put)
      • Caps & Floors
      • Digital / Barrier / Knock-Out &-In options
      • Asians (Strike Based on an Average Value of the Underlying)
      • Look-Backs (Striking Based on a Previous Value of the Underlying)
      • Credit Default Swaps
    7. Compound Derivatives
      • Rainbows (Striking of any of a Set of Options)
      • Collars
      • Straddles, Strangles & Spreads
      • Butterflies and Condors
      • Callable Bonds
      • Mortgage Backed Securities
      • Equity Linked Notes
    8. Measuring Risk
      • BSM Greeks
      • Delta
      • Rho
      • Gamma
      • Convexity
      • Vega
      • Taylor Series Expansion
      • A Key Cross-Term for Long Dated Options: dValue''/dDelta*dRho
      • Closer Look at Rho
      • Term Structure and Key Rates
      • Cost of Carry and Dividends
      • Estimating Greeks Through Simulation
    9. Using Risk Measures
      • Hedging
      • Delta Hedge
      • Duration Hedging (Key Rate Rhos)
      • Delta-Rho Hedging
      • Delta-Gamma
      • Delta-Rho-Gamma-Vega
      • Risk Isolation
      • VAR (Value at Risk)
      • Parametric For portfolio with Linear Deltas (Gamma & Convexity $=0$)
      • Simulation for Portfolios with Gamma &/or Convexity $\ne0$
      • Using Cholesky Decomposition to Create Correlated Time Series
  2. Insurance Liabilities
    1. Understanding Life Products
      • Mortality
      • Lapse
      • Interest Rates
      • Expenses
      • Single Pay
      • Fee Based Account Value, Accumulation Value, etc.
      • Policy Loans
    2. Understanding Annuity Products
      • Income Annuities
      • Immediate
      • Deferred
      • Period Certain, Life-only, Period Certain and Life
      • Joint and Last to Die
      • Indexed (This is a Call Spread with Fees)
      • Accumulation
      • Riders
      • GMDB - Guaranteed Minimum Death Benefit
      • GMAB - Guaranteed Minimum Accumulation Benefit
      • GMIB - Guaranteed Minimum Income Benefit
      • GMWB - Guaranteed Minimum Withdrawal Benefit
    3. Modeling Liabilities
      • SPDA Contracts
      • Variable Annuity Living & Death Benefits
      • Minimum Rate Guarantees On Fixed Annuities And Universal Life
      • Potentially Secondary Guarantees On Life Contracts
    4. Asset Liability Management
      • Topics to be Decided
  3. Portfolio Construction
    1. Asset Allocation
      • CAPM (Capital-Asset Pricing Model)
      • APT (Arbitrage-Free Pricing Theory)
      • Markowitz Mean Variance Optimization, MVO, Using Historical Data
      • Historical MVO with Constraints
      • MVO Using Expected Values
      • Expected Value MVO with Constraints
    2. Risk Appetite and the Utility Function
  4. Performance Attribution
    1. Measuring Performance
    2. Benchmarks
    3. Tracking Error
    4. Decomposing Returns by Factors
    5. Curve
    6. Benchmark
    7. Asset Classes
    8. Style
    9. Alpha
  5. Predictive Mutli-Factor Modeling
    1. Fundamental Models
    2. Econometric Model
    3. Principal Component Analysis
    4. Challenges with Multifactor Models
    5. Multi-Colinearity
    6. Homoskedascity, heteroskedacity and GARCH models
    7. Autocorrelation
    8. Survivor Bias
    9. Look-Ahead Bias
    10. Having Too Many Variables
    11. Working with Large Sparse Matrices

Description of FM 5031/FM 5032, A Practitioner's Course in Finance

This syllabus is currently under development.
This course will be taught by industry professionals. We may have two professionals, each teaching a one-semester course, but likely more. Six different instructors, each teaching for five weeks is our initial target. Recruitment for these courses will not begin until Fall 2006, one year before the expected launch of the program itself.

In-depth training in valuation of derivatives and portfolio management will the foundation for our program, but this course sequence will provide a forum to discuss some of the finer issues. Education on the various scenario generators used, the situations in which they are used, and recommendations for model usage, calibration, and scenario reduction techniques are all topics for consideration.

Some tentative decisions for the fall semester:
       John Dodson (RiverSource Investments) will teach the first seven weeks
       William Barr (RiverSource Investments) will teach the next four weeks
       Ryan Williams (Kenwood Capital) will teach the last four weeks

Some tentative decisions for the spring semester:
       John Dodson (RiverSource Investments) will teach the first seven weeks
       William Barr (RiverSource Investments) will teach the next four weeks
       Gary Hatfield (Securian) will teach the last four weeks

Syllabi are under preparation.

Syllabus for FM 5091/FM 5092, Programming and Presentation in Finance

This syllabus is currently under development.

 

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