# Courses

## Mathematics 16:643:626 Fixed income Securities and Derivative Modeling

### Schedule

The course is normally offered during the Spring semester.
• Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
• Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

### Course Abstract

This course covers the Theory and principles behind Fixed Income Securities and Fixed Income Derivatives modeling. Fixed Income Derivatives modeling is a triumph of Mathematical Finance, since all major institutions use the theory for pricing, hedging, and management risk,is by far the largest instrument class.

### Pre-requisites and Co-requisites

Prerequisites: Math 16:643:621, 16:643:573 Co-requisites: Math 16:643:622

### Primary Textbooks

"Interest Rate Modeling" by Andersen and Piterbarg (3 volumes)

• Volume I: foundations and Vanilla Models
• Volume II: Term Structure Models
• volume III: Products and Risk Management

### Sakai

All course content – lecture notes, homework assignments and solutions, exam solutions, supplementary articles, and computer programs – are posted on Sakai and available to registered students.

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### Class Policies

Please see the MSMF common class policies.

### Weekly Lecturing Agenda and Readings

This will be provided on Sakai.
WeekTopics
Instruments
• Instruments. Flow: Treasury bonds, money market, swaps, forward rate agreements, futures. Vanillas: caps, oors, swaptions. Quasi- vanillas: constant maturity swap options, spread options, range ac- cruals. Exotics: Bermudans, callable range accruals, barriers
• How do you price them? Flow: closed form, no model needed. Vanil- las and quasi-vanillas: closed form + semi-analytical, models needed. erences, Monte Carlo, models needed
• Why do you need models? For pricing, hedging and risk management. Short description of the life of an option
• Introduction to Python and Subversion.
2 Zero curves
• Zero coupon bonds, forward ZCB's, forward rates, short rate, forward instantaneous rate, formulas
• Zero curve
• How to price a swap
• How to bootstrap a curve
• Basis: what it is, why it exists. Example: index basis, 3M-1M basis, FX basis.
• How to price a swap with basis
• How to bootstrap the basis.
3 Caps, oors, swaptions
• Definitions
• Black pricing, normal, lognormal
• Implied vol, normal lognormal. Vlo units
• Vega
• Smile
• Liquid strikes
• The 1-q (shifted log-normal) model
• Smile dynamics, normal and log-normal backbone
• 1-q parameterization
• Multi-q
4 Linear Gaussian Model (a.k.a. Vasicek or Hull-White)
• The SDE for the Vasicek model. Solution. Interpretation of the mean reversion.
• Zero coupon bond by integration
• Zero coupon bond by PDE
• Forward rates (animation)
• Hull-White
• SDE for forward rates.
5 More involved static models
• CMS pricing - convexity adjustment
• Implied marginal distributions of swap rates
• Binary options. Range accruals
• Quanto CMS
• Relationship between forwards and futures.
6 Linear Gaussian Model (continued)
• Zero coupon bond options
• Pricing of caps and oors
• Jamshidian's trick for swap options
• Implementation of cap/ oor/swaption pricers.
• Analysis of the smile very close to normal
• Approximate formula for implied bp vol - empirical determination.
7 Linear Gaussian Model (continued)
• MLE for Vasicek using historical data
• Calibration of Hull-White using market data
• Implement Vasicek in a tree
• Implement Vasicek in a Monte Carlo
• Moment matching.
8 Linear Gaussian Model (continued)
• Verification of the CMS convexity adjustment formula
• CMS hedging
• Multi-factor Vasicek
• Cross-curency Vasicek
• Quanto CMS pricing
9 Mortgages
• Prepayment and default
• Securitization
• Hedging
10 LGM with local volatility
• 1-q and multi-q local volatility examples
• Zero Coupon Bond formula
• Implementation in tree and MC
• Smile
11 HJM, BGM
• HJM - drift condition
• Implementation in Monte Carlo
• Markovian HJM
• Explosion
• BGM - drift condition
• Correlations
• Markovian BGM.
12 LGM with stochastic volatility
• Heston and SABR SDE's
• Fourier method for Heston
• Asymptotic methods for Heston and SABR
• Which one is better?
• 1-factor QGM. Formula for bond
• Implementation in tree
• How to control skew
• 2-factor QGM. Stochastic volatility interpretation
• Skew and smile for 2-factor QGM.
14 Calibration and parametrization of models
• Approximate formula for the implied vol in the LGM model
• Iterative procedure for calibration of the LGM model
• Parametrization of a stoch vol LGM model

### Library Reserves

All textbooks referenced on this page should be on reserve in the Hill Center Mathematical Sciences Library (1st floor). Please contact the instructor if reserve copies are insufficient or unavailable.

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