37280 - Interest Rate Models

Learning outcomes

The course covers in detail the interest rate market and its evolution across the financial crisis. The modern interest rate modelling approach is developed to include funding and collateral, and applied to the most important interest rate derivatives. Market quotations for plain vanilla linear derivatives and options are used to bootstrap multiple-yield curves and volatility cubes. Interest rate models are introduced step by step with increasing complexity, including extensions to multiple-curves, negative rates, convexity adjustments and correlations.

At the end of the course the students will know how the most important interest rates are fixed on the market, and how the modern interest modelling approach is extended to include funding and collateral. They will also understand the most important interest rate derivatives and their market quotations, and how to build the corresponding interest rate yield curves and volatility cubes. They will be able to model the term structure of interest rates using several approaches with increasing complexity, from the simplest Black model and its extension to stochastic volatility (SABR), to factor models for short rates (Heath-Jarrow-Morton, Vasicek, Hull-White) and forward rates (Libor Market Model), dealing with multiple-curves, negative rates and convexity adjustments. Practical examples and case studies will be illustrated using spreadsheet and codes.

Course contents

Interest rate basics 

• Dimensions and units in finance and other disciplines
• Interest rates definition and conventions
• Multiple types of interest rates

Interest rate market

• 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
• Symmetry breaking and market segmentation after the credit crunch
• 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
• Multiple funding sources
• Funding and funding value adjustment (FVA)
• Collateral: perfect, partial, hedge collateral
• Stochastic funding rates, multiple currencies

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
• Overnight Indexed Swap
• Swap, forward swap measure
• Basis Swap
• Bond

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

Forward rate modelling: single rates

• Black’s model
• Beyond the Black’s 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
• Market quotations

Multiple volatility cubes

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

Convexity adjustment and Constant Maturity Swaps

• IRS convexity adjustment
• Constant Maturity Swaps
• CMS convexity adjustment
• SABR calibration to Swaptions and CMSs 

• CMS Options
• CMS Spread Options
• Bootstrapping implied correlations

Short rate modelling and forward rate modelling

• Instantaneous forward rates: the HJM model
• Short rate models, Vasicek and Hull-White models
• Libor Market Model (LMM)
• Dealing with multiple curves and negative rates

Conclusions and references

Readings/Bibliography

• D. Brigo, F. Mercurio, "Interest Rate Models - Theory and Practice", Springer, 2006.
• L.B.G. Andersen, V. V. Piterbarg, “Interest Rate Models”, 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.

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.

• Number of problems solved with respect to the minimum number required
• Correctness of the solutions and numerical results
• Accuracy of the technical language, clarity of exposition.
• Correctness, clarity, elegance and comments of spreadsheets and codes.

The final grade is the average of the grades for each single exercise. Non-working and/or unclear spreadsheets and/or codes receive very low grades. Possible non-original homework receive a zero grade (copying is discouraged). 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 by the course tutor to the students at the examination date. The examination results will be delivered by the students to the course tutor within 1 week (7 days) after the examination date. The time required to correct the homework is 1-2 weeks.

Teaching tools

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

Office hours

See the website of Marco Bianchetti