37280 - Interest Rate Models

Course Unit Page

Academic Year 2018/2019

Learning outcomes

At the end of the course the student knows how to analyze the term structure of interest rates using several models, namely factor models (Vasixek, CIR)., HJM, LIBOR and swap market models. Particular emphasis is placed on the issue of calibration of the model to market data, from monetary, futures and swap markets, as well as from cap/floor and swaption volatilities.

Course contents

Interest rate basic concepts

  • Dimensions and units in finance and other disciplines
  • Interest rates definition and conventions

Interest rate market after the credit crunch

  • Deposits
  • The money market: central banks, interbank, retail
  • Libor/Euribor/Eonia/Repo interest rates
  • How the market changed: stylized facts and overview of market data
  • Credit and liquidity components
  • Counterparty risk and collateral
  • From Libor to OIS discounting

Modern interest rate modelling

  • Notation and basic assumptions
  • Short rate, bank account, Zero Coupon Bond, probability measure
  • Feynman-Kac and Girsanov theorems
  • Replication
  • Black-Scholes-Merton, modern perspective including multiple funding sources and collateral
  • XVAs: Credit/Debit and Funding Value Adjustments

Pricing of linear interest rate derivatives

  • A simple credit model to explain multiple interest rates
  • Spot, forward and instantaneous forward rates
  • Forward Rate Agreement
  • Futures
  • Swap, forward swap measure
  • Overnight Indexed Swap
  • Basis Swap

Multiple curve framework

  • Modern multiple curve pricing & hedging market practice
  • Multiple curves construction
  • Selection of bootstrapping instruments, market data
  • Bootstrapping formulas
  • Interpolation
  • Handling negative rates
  • Exogenous bootstrapping
  • Turn of year effect
  • Multiple curves, multiple deltas, multiple hedging
  • Performance, Sanity checks
  • Lab session: yield curve bootstrapping implementation


  • Types of bonds
  • Pricing of plain vanilla bonds
  • Credit yield curves

Forward rate modelling

  • Black model
  • Volatility smile
  • Beyond the Black model
  • Stochastic volatility SABR model
  • Handling negative rates, shifted Black, shifted SABR

Pricing of interest rate volatility products

  • Cap/Floor
  • Swaption, cash vs physical settlement
  • Constant Maturity Swap
  • CMS Cap/Floor
  • CMS Spread Option

Multiple volatility cubes

  • Modern multiple curve, multiple volatility market practice
  • Caps/Floors volatility cube
  • Handling multiple rate tenors, Kienitz model
  • Swaptions volatility cube
  • Lab session: SABR implementation

Term structure modelling

  • Exotic derivatives
  • Short rate models: Vasicek and Hull-White
  • Forward rate models: Libor Market Model (LMM)
  • Multiple curves and negative rates extension
Conclusions and references


Suggested textbooks

  • D. Brigo, F. Mercurio, "Interest Rate Models - Theory and Practice - With Smile, Inflation and Credit", Springer, 2006.
  • Leif B. G. Andersen, Vladimir V. Piterbarg, “Interest Rate Modeling”, Atlantic Financial Press, 2011.
  • M. Henrard, “Interest Rate Modelling in the Multi-Curve Framework”, Palgrave McMillan, 2014.
  • P. Wilmott, “Paul Wilmott on Quantitative Finance”, 3 vols. (2nd edition), John Wiley and Sons Ltd.
  • M. Morini, "Understanding and Managing Model Risk. A practical guide for quants, traders and validators", Wiley, 2011.
  • D. Brigo, M. Morini, A. Pallavicini, "Counterparty Credit Risk, Collateral and Funding: With Pricing Cases For All Asset Classes", Wiley, 2013

by Damiano Brigo  (Author), Massimo Morini (Author), Andrea Pallavicini (Author)
by Damiano Brigo  (Author), Massimo Morini (Author), Andrea Pallavicini (Author)

Teaching methods

The course assumes basic knowledge of elementary stochastic calculus and financial modeling, but no specific knowledge of interest rates. Interest rate definitions, markets, financial instruments and models are developed from scratch, with increasing complexity, supported by interactive Lab sessions with market data, examples and exercises.

Assessment methods

The exam consists in a homework with questions and exercises related to the course program. The homework is a relevant, not accessory, part of the course. Its purpose is to enforce the comprehension of the topics included in the course program and, more important, to help the students to evaluate their knowledge of interest rate modelling acquired during the course.

The market data relevant for the exercises, carefully discussed during the course, are included in the homework.

The results of each exercise must be reported as described in the exercise itself. Some exercises require the implementation of spreadsheets and codes, to be delivered in a fully working version and adequately commented. The chosen programming language can be VBA and/or Matlab.

The homework are evaluated according to the following criteria.

  • Problems: number of problems solved.
  • Solutions: correctness of solutions and of numerical results.
  • Exposition: correct usage of the technical language and clarity of exposition.
  • Spreadsheets: order, clarity, comments.
  • Codes: correct usage of the programming language, order, clarity, comments.

The final grade is a weighted average of the grades for each single exercise.

Non-working and/or unclear spreadsheets/codes receive very low grades. The exam is passed with a final score greater than or equal to 18/30.

The homework can be done alone or in a team with 2-3 students. Team working is encouraged, such that problems, solutions, spreadsheets and codes can be shared and cross-checked by the members of the team.

The examination text and instructions will be delivered to the students subscribed in Alma Esami at the scheduled examination date. The homework results will be returned by the students 1 week (7 days) after the examination date. The time required to correct the homeworks is 2-3 weeks.

Teaching tools

  • Slides (power point/pdf)
  • Selected literature
  • Example spreadhseets
  • Skeleton Matlab code

Office hours

See the website of Marco Bianchetti