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Introduction to Interest Rate Curves and the LIBOR Market Model

Theory, Modeling and Implementation.

07.03.2013 – 15.03.2013


Interest rate modeling and the valuation of related financial contracts continue to be important research topics in financial mathematics and risk management. This course has the intention to give a step-by-step introduction to theory and applications of mathematical models for the term structure of interest rates.

The focus of the course will be a comprehensive discussion of the well-known LIBOR Market Model. In particular, we detail the arbitrage-free modeling of LIBOR rates, the pricing of interest rate derivatives in the LMM and address the calibration of the model. Some more advanced topics, such as hybrid models and multi-curve extensions will also be addressed. 

The course will complement theoretical parts with practical applications and software implementations and will also include hands-on exercises sessions.


  • Thursday, March 7th, 2013. 14:00-17:30
  • Friday, March 8th, 2013. 08:30-17:30
  • Thursday, March 14th, 2013. 14:00-17:30
  • Friday, March 15th, 2013. 08:30-17:30


The workshop takes place at

quantLab - Room B 121
LMU Institute of Mathematics
Theresienstr. 39
80333 Munich

A detailed location plan can be found here.


Part 1: Foundations, Single-Curve and Multi-Curve Interest Rate Theory

  1. Risk Neutral Valuation: A Review
    • Foundations from Probability Theory
    • Stochastic Processes
    • Brownian Motion
    • Geometric Brownian Motion
    • Ito Calculus
    • Replication
    • Change of Measure, Risk Neutral Measure
    • Black-Scholes Model and Monte-Carlo Simulation
  2. Interest Rates (Single Curve Interest Rates Theory)
    • Zero Coupon Bonds
    • Forward Rates
  3. Simple Interest Rates Products: Linear Products (Single Curve Interest Rates Theory)
    • Swaps
    • Swap Rates
  4. Simple Interest Rates Products: European Options (Single Curve Interest Rates Theory)
    • Caplets, Cap
    • Swaptions
  5. Collateralization, Funding and Basis-Spreads (Multi-Curve Interest Rates Theory)
    • Collateralization and Funding
    • Cross-Currency Analogy to Collateralization
  6. Curve Calibration (with object oriented implementation)
    • Discount Factors and Forwards
    • Swaptions

The resources for this session (spreadsheet, source code) are available at

Part 2: LIBOR Market Model: Theory and Implementation

  1. LIBOR Market Model: Definition, Drift, Model Parameters (Single Curve Interest Rate Theory)
    • Motivation of the model.
    • LMM dynamic under spot and terminal measure.
    • Model parameters (intuition).
  2. LIBOR Market Model: Calibration (Single Curve Interest Rate Theory)
    • Calibration to forward rate curve.
    • Calibration to caplets.
    • Calibration to swaptions.
  3. Cross-Currency and Hybrid LIBOR Market Model
    • Motivation.
    • CCY LMM dynamic under spot and terminal measure.
    • Equity Hybrid LMM dynamic under spot and terminal measure.
    • Multi-Curve LMM.
    • Calibration.
  4. Discretization and Monte-Carlo Simulation
    • Monte-Carlo Simulation
    • Euler-Scheme
  5. Object Oriented Implementation
    • Object oriented implementation of a Monte-Carlo Simulation of the LMM
    • Calibration example (calibration to swaptions)
  6. Valuation of Bermudan Option
    • Object oriented implementation of a Monte-Carlo Simulation of the LMM
    • Calibration example (calibration to swaptions)

Prof. Dr. Christian Fries

Christian Fries is head of model development at DZ Bank’s risk control and Professor for Applied Mathematical Finance at Department of Mathematics, LMU Munich.

His current research interests are hybrid interest rate models, Monte Carlo methods, and valuation under funding and counterparty risk. His papers and lecture notes may be downloaded from

He is the author of “Mathematical Finance: Theory, Modeling, Implementation”, Wiley, 2007 and runs

Dr. Alessandro Gnoatto

Alessandro Gnoatto is a post-doctoral researcher with the Workgroup Financial Mathematics of LMU Munich. His current research interests include advanced asset pricing models based on matrix-valued affine processes and their various applications, e.g. to multiple-curve interest rate models.

He is an experienced lecturer on a broad range of topics from mathematical finance such as interest rate models, Lévy processes and Monte Carlo methods and is a contributor to finmath. Alessandro holds a Ph.D. in Computational Mathematics.

To register for the workshop a workshop fee is requested:

  • 400 Euro a single week (part 1 or part 2).
  • 700 Euro for two weeks (part 1 and part 2).

For registration please contact Christina Broza via

Note: The workshop is offered at a competitive price and will not include any catering (coffee, tea, lunch or dinner). However, located in Munich's "Maxvorstadt", corresponding locations are in short walking distance from the venue.