94424 - Advanced Interest Rate Models And Market

Academic Year 2021/2022

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Quantitative Finance (cod. 8854)

Learning outcomes

At the end of the course the students will know about the most recent developments in interest rate market. Particular emphasis is placed on how interest rate market data, derivatives, bonds and pricing models are used in practice in a real financial institution for trading and risk management purposes.

Course contents


1) Introduction

  • Why interest rates are important
  • Classic vs modern pricing framework

2) Interest rate basic concepts

  • Dimensions and units
  • Interest rate definition and conventions
  • Types of interest rates
  • Lab session: basic concepts (Excel)

3) The market after the credit crunch

  • How the market changed: stylized facts
  • The money market: Deposits and Repos
  • The money market: central banks, interbank, retail markets
  • The age of negative rates
  • Interest rate benchmarks: IBORs, overnight rates, and their reform
  • Credit and liquidity risks
  • Counterparty risk and collateral
  • From IBOR to OIS discounting
  • From EONIA to €STR discounting
  • Valuation adjustments (XVA)
  • Lab session: basic concepts (Excel)

4) Modern interest rate modelling

  • Short rate, Bank Account, Zero Coupon Bond
  • Risk neutral and forward measures, change of measure
  • Feynman-Kac and Girsanov theorems
  • Pricing by replication
  • Black-Scholes-Merton from a modern perspective
  • Multiple funding sources
  • Pricing including multiple funding sources and collateral
  • Funding Value Adjustment (FVA)
  • Solving the discounting puzzle



5) Linear interest rate products

  • A simple credit model
  • Interest rate instruments
  • Forward rates
  • Forward Rate Agreements (FRA)
  • Futures
  • Swaps (IRS), forward swap measure
  • Overnight Indexed Swaps (OIS)
  • Basis Swaps
  • Lab session: market data, pricing linear derivatives (Excel)

6) Multiple curve framework

  • Modern multiple curve pricing & hedging market practice
  • Yield curve construction
  • Bootstrapping
  • Interpolation
  • Negative rates
  • Yield curve jumps
  • Sensitivities
  • Performance, sanity checks
  • Lab session: yield curve bootstrapping implementation

7) Bonds

  • Bond: payoff and types
  • Pricing of plain vanilla bonds
  • Credit spread measures
  • Bond credit spread vs CDS spread
  • Credit spread curve construction
  • Sensitivities and duration
  • Negative yields
  • Lab session: market data, pricing bonds (Excel)



8) Forward rate modelling

  • Black model
  • Beyond the Black model
  • Shifted Black model
  • Shifted SABR model

9) Interest rate volatility products

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

10) Multiple volatility cubes

  • Introduction
  • Swaptions volatility cube
  • Caps/Floors volatility cube
  • SABR usage
  • Lab session: SABR implementation and calibration

11) Term structure modelling

  • Exotic derivatives
  • Term structure modelling
  • Short rate models: Vasicek and Hull-White, multi-curve extension
  • Forward rate models: Libor Market Model (LMM)
  • Lab session: Hull-White (Excel)


Mandatory materials

  • Slides used during the course
  • Excel spreadsheets used during Lab sessions

Accessory textbooks

Interest rate modelling:

  • M. Henrard, “Interest Rate Modelling in the Multi-Curve Framework”, Palgrave McMillan, 2014.
  • Jorg Kienitz, "Interest Rate Derivatives Explained: Volume 1: Products and Markets", Palgrave Macmillan, 2014.
  • Jorg Kienitz, Peter Caspers, "Interest Rate Derivatives Explained: Volume 2: Term Structure and Volatility Modelling", Palgrave Macmillan, 2017.
  • 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.
  • Paul Wilmott, “Paul Wilmott on Quantitative Finance”, 3 vols., John Wiley and Sons Ltd., 2nd edition, 2006.
  • Marco Bianchetti, Massimo Morini, editors, “Interest Rate Modelling after the Financial Crisis”, Risk Books, 2013.

Model risk:

  • M. Morini, "Understanding and Managing Model Risk. A practical guide for quants, traders and validators", Wiley, 2011.

Counterparty and liquidity risk (XVAs):

  • D. Brigo, M. Morini, A. Pallavicini, "Counterparty Credit Risk, Collateral and Funding: With Pricing Cases For All Asset Classes", Wiley, 2013.
  • Andrew Green, XVA, “Credit, Funding and Capital Valuation Adjustments”, Wiley, 2015.
    Gregory J., “The xVA Challenge: Counterparty Risk, Funding, Collateral, Capital and Initial Margin”, Wiley, 4th Edition, 2020.

Teaching methods

The course assumes basic knowledge of elementary stochastic calculus and financial modeling, but no specific knowledge of interest rates.

Interest rate basics, markets, financial instruments and pricing models are developed from scratch, with increasing complexity.

Theoretical lessons are accompanied and completed by interactive Lab sessions with real market data, examples and exercises.

Lessons are based on slides and Excel exercises, delivered online in advance (https://virtuale.unibo.it).

Students are strongly encouraged to form and consolidate teams of 2-3 people (3 preferred) since the very beginning of the course, to study together and face the final examination (see section "assessment methods" below).

Students are strongly encouraged to attend all the lessons in presence or online, to study the material during the course, and to face the examination at the first date available. Experience from the past courses shows that this is the best approach to successfully benefit the course and obtain a good grade.

Assessment methods

Examination: 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 purposes are the following:

  • enforce the comprehension of the topics discussed during the course,
  • help students to assess their knowledge of the topics,
  • encourage and test team-working skills.

Attendance to the course lessons is essential for a successful examination.

Teams: the exam should be taken by teams of 2-3 students as described in section "Teaching methods". During the examination, members of the same team are encouraged to work together, sharing and cross-checking questions, solutions, spreadsheets and codes. The interaction is allowed only among members of the same team. Cross-team interaction is strictly forbidden and will be considered as cheating, penalizing the whole group.

Schedule: the examination text and instructions are delivered to the students subscribed in Alma Esami (no exceptions) at the scheduled examination date. The homework results are returned by the students 1 week (7 days) after the examination date. Afterwards, each team illustrate the homework in a dedicate meeting with the Professor. The time required to correct the homeworks is 2-3 weeks.

Exercises: theoretical questions typically require simple mathematical proofs beyond those reported in the slides. Practical questions typically 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/Matlab/Python. 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.

Valuation: the homeworks are evaluated according to the following criteria.

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

Grades: the final grade is a weighted average of the grades for each single exercise. Wrong, non-working, confused, non-commented spreadsheets/codes are penalyzed. Clearly copied exercises are strongly penalyzed. The exam is passed with a final grade greater than or equal to 18/30.

Teaching tools

  • Slides (power point/pdf)
  • Selected literature
  • Example spreadhseets (Excel)

Office hours

See the website of Marco Bianchetti


Quality education Decent work and economic growth

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.