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Course 2021-2022 a.y.


Department of Finance

Course taught in English

Go to class group/s: 31

DSBA (6 credits - I sem. - OP  |  SECS-S/06)
Course Director:

Classes: 31 (I sem.)

Suggested background knowledge

It is suggested to have a good preliminary knowledge of: financial calculus (compounding & discounting and the different conventions; bond payment schedule; yield to maturity; term structure of spot rates and discount factors; present value of a stream of cash flows); basic derivatives (forwards and options) and their pricing (cash-and-carry & Black-Scholes formula); stochastic calculus (martingale, Brownian motion and its properties, Ito's lemma, arithmetic & geometric Brownian motion ); no-arbitrage and risk-neutral pricing.

Mission & Content Summary

The purpose of the course is to present the latest achievements in the term structure modeling for pricing and hedging interest rate derivatives. Emphasis is devoted to the theoretical and practical implementation of the models, and the suitability of different models in complex valuation and hedging problems of interest rate options, equity-linked fixed income securities and structured products commonly used in the industry. Particular emphasis is also devoted to credit risk issues, such as assessment of counterparty credit risk and to the new approaches in interest rate modelling over the last decade. On completing the course the participants have a clear and thorough understanding of the different methodologies in the pricing and hedging of interest rate options. The course is quantitatively oriented, but financial and practical issues are greatly discussed. On successful completion of this module, you are expected to be able to: 1. To provide a foundation in a crucial area of financial markets and quantitative finance. 2. To complement the general derivatives course with specific instruction in a key derivatives area. 3. To acquaint the student with the main modelling streams in fixed income securities. 4. To enable students to use models in this area in practical applications. 5. To transmit the student with fundamental mathematical modelling techniques underpinning the subject.


The main topics covered in the course are detailed here below:

  • Introduction to Fixed Income World: Examples of Structured products and Financial Engineering. Review of Basic elements of financial math and Interest Rate Conventions.
  • Building Blocks: LIBOR, FRA, Eurofutures, Swap.
  • Yield Curve Stripping: the Bootstrapping Procedure. Interpolating the yield curve: parametric and non-parametric methods.
  • Pricing Floating Rate Notes.
  • Interest Rate Options: Caps, Swaptions and Bond Options.
  • Use and transformations of the central Black Formula. The Black Model and the volatility surface
  • Allowing for negative rates: Bachelier and Shifted Black model.
  • Application to pricing structured products and their use for hedging interest rate risk.
  • The foundations of risk-adjusted pricing in Fixed Income. The change of numeraire and pricing of interest rate derivatives.
  • Understanding the different approaches to modelling: from short rates to HJM to the Libor Market Model. Short rate models. Merton, Vasicek, CIR. The Heath-Jarrow-Morton (HJM) Model. Gaussian HJM Models. Multivariate HJM models. The Libor Market Model: theory and fundamental formulas and techniques for pricing. Comparing different Term Structure Models: Market Models, Short Rate Models, HJM.
  • Practical application, calibration and management of volatility and correlation.
  • Fixed Income in the last decade: the multicurve market. Double discounting for collateralized Derivatives. Overnight Indexed Swaps. Bootstrapping the curves. Understanding how it emerges from credit and funding risk. Multicurve short rate models and multicurve Libor Market Model. The interaction of credit and interest rate risk - The XVAs - Collateral and OIS discounting.

Intended Learning Outcomes (ILO)
At the end of the course student will be able to...
  • Have a foundation in a crucial area of financial markets and quantitative finance.
  • Complement the general derivatives course with specific instruction in a key derivatives area.
  • Be acquainted with the main modelling streams in fixed income securities.
  • Be able to use models in this area in practical applications.
  • Be able to deal with fundamental mathematical modelling techniques underpinning the subject.
At the end of the course student will be able to...

Knowledge and understanding:

  • Show knowledge of the some of the main models used in the mathematical modelling of fixed income.
  • Understand how models are applied in practice.
  • Understand the key differences between different modelling approaches.


  • Performing basic fixed income computations.
  • Building the term structure of interest rates.
  • Valuing interest rate swaps.
  • Setting up hedges for fixed income portfolios.
  • Implementing  the main models and using them to value fixed income securities and fixed income derivatives.

Teaching methods
  • Face-to-face lectures
  • Online lectures
  • Exercises (exercises, database, software etc.)
  • Case studies /Incidents (traditional, online)
  • Group assignments

This module is taught primarily through lectures and laboratories, making use of numerical and analytic examples with the support of case studies.

Assessment methods
  Continuous assessment Partial exams General exam
  • Written individual exam (traditional/online)
  •   x x
  • Group assignment (report, exercise, presentation, project work etc.)
  •   x x

    Written exam with numerical and theoretical questions, according to two possibilities:

    1. Two partial examinations (closed books) are planned. The first follows the initial six lectures and is based on the first part of the course. The second examination occurs at the end of the course and is based on the second part of the course. Grades are assigned as follows: 55% for the part having highest mark and 45% for the part having lowest score.
    2. A Final examination (closed books) at the end of the course is planned, and is based on the entire course. 


    In addition, a not-compulsory group coursework (maximum 3 persons per group) on pricing a structured product is possible.  The usual deadline for the take home examination is around mid-January. The use of Excel VBA/Matlab/R/Pyhton will be required. The coursework gives the possibility of adding a maximum of 3 points to the written mark. 

    Teaching materials

    Textbook and course material:

    1. Lecture Slides provided by the teachers.
    2. Chapters from the following sources:
    •  D. BRIGO, F. MERCURIO, Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit, Springer Finance, 2007, 2nd ed.
    • P. VERONESI, Fixed Income Securities: Valuation, Risk, and Risk Management, Wiley, 2009.
    • B. TUCKMAN, A.SERRAT,  textit{Fixed Income Securities: Tools for Today's Markets}, 3rd Edition.
    • L. BALLOTTA, FUSAI, G. and MARENA, M., A Gentle Introduction to Default Risk and Counterparty Credit Modelling, available at SSRN http://ssrn.com/abstract=2816355.
    • A complete suggested reading list is distributed at the beginning of the course.
    Last change 05/08/2021 16:03