37280 - Interest Rate Models

Learning outcomes

At the end of the course the student will know about the most recent developments on interest rate markets and products, yield curve and volatility surface construction, and the most important models used to price interest rate instruments. Particular emphasis is placed on how market data, products and models are used in practice in a real finanzial institution.

Course contents

Interest rate basic concepts

• Why interest rates are so important
• 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 markets
• Interest rate benchmarks: IBOR rates, overnight rates, and their reform 
• How the market changed: stylized facts and overview of market data
• Credit and liquidity risks
• Counterparty risk and collateral
• From Libor to OIS discounting
• XVAs overview

Modern interest rate modelling

• Notation and basic assumptions
• Short rate, bank account, Zero Coupon Bond, probability measure, change of measure
• Feynman-Kac and Girsanov theorems
• Pricing by replication
• Black-Scholes-Merton, classic vs modern perspective including multiple funding sources and collateral
• Funding Value Adjustment (FVA)

Linear interest rate derivatives

• A simple credit model to explain multiple interest rates
• Spot, forward and instantaneous forward rates
• Forward Rate Agreements
• Futures
• Swaps, forward swap measure
• Overnight Indexed Swaps (OIS)
• Basis Swaps
• Lab session: pricing linear derivatives in excel with real market data

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

Bonds

• Types of bonds
• Pricing of plain vanilla bonds
• Credit spread measures
• Credit spread curve construction
• Sensitivities, duration
• Negative yields
• Lab session: pricing bonds in excel with real market data

Forward rate modelling

• Black model and its limitations
• Volatility smile
• Beyond the Black model
• Shifted Black model
• Stochastic volatility: shifted SABR model
  

Pricing of interest rate volatility products

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

Multiple volatility cubes

• Modern multi-curve, multi-volatility market practice
• Swaptions volatility cube
• Caps/Floors volatility cube
  
• Lab session: SABR implementation and calibration

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 in excel

Conclusions and references

Readings/Bibliography

Mandatory materials

• Course slides and spreadsheets. 

 

Accessory textbooks

For 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.

For model risk:

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

For advanced pricing of 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 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 the assessment methods described 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 included in the course program,
• help students to self-assess their knowledge of interest rate modelling acquired during the course,
• encourage and test team-working skills.

Teams: the homework should be taken by teams of 2-3 students. Students are encouraged to form and consolidate such teams since the very beginning of the course. During the examination, members of the same team are encouraged to work together, sharing and cross-checking solutions, spreadsheets and codes. Cross-team interaction is strictly forbidden and will be considered as cheating.

Schedule: 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.

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 receive very low grades. Clearly copied exercises receive zero grade. 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
  

Office hours

See the website of Marco Bianchetti