 # Interest rate curve models

#### HILLAIRET Caroline

Department: Finance

### Objective

The objective of this course is to present the methods of valuation and hedging of interest rate products – in particular derivatives - as implemented today in fixed income trading rooms. The first part, which is more applied, goes back over standard instruments (Libor rates, bonds, interest rate swaps, caps, floors, swaptions, etc.) and the stripping of the yield curve, before tackling some more complex derivatives which will motivate and give full meaning to the models introduced in the second part. The second part of the course, which is theoretical, aims to detail the different approaches to modeling the yield curve, with particular emphasis on the HJM framework, which is now unavoidable.

At the end of this course, students should be able to :
- a zero-coupon yield curve from the quotation of market instruments (bonds or Euribor derivatives)
- Valuing the main standard interest rate products on the basis of a zero-coupon curve and, where appropriate, a volatility cube (FRAs and futures, interest rate swaps, caps & floors, swaptions)
- Master the convexity adjustment techniques used in the valuation of many structured interest rate products.
- Know the main curve models used on interest rate exotics and know how to evaluate and hedge an interest rate product using these models (by direct calculation and by EDP).
- Master the concept of forward neutral measurement and know how to use it to value and hedge interest rate derivatives.

### Planning

Part I - Antonin Chaix

1. Underlying interest rate instruments and yield curve stripping LIBOR / EURIBOR / FRA, bonds, interest rate swaps: how to value them from zero-coupons? What is a yield curve? How to build it from the quotation of standard instruments?
2. Valuation of vanilla interest rate options: caps / floors and swaptions. How to value assets in the presence of stochastic rates? Reminder on neutral forward measures. Modeling a LIBOR rate or a swap rate under the associated measure. Valuation of a cap/floor and a swaption in the Black model. Volatility model for interest rate options.
3. Introduction to the valuation of interest rate exotics. Structuring of interest rate products. Architecture of a pricing platform for exotics and in particular representation of market volatilities: volatility cube and SABR model. Convexity adjustment concept: LIBOR in arrears and CMS. Panorama of 1st generation exotics: digital, corridors, quantos, spread options...
4. Complex exotics or the need for a stochastic model of the yield curve. Qualitative feedback on curve models: short rate models, HJM framework, Gaussian models vs. market models. Focus on the Hull & White 1 factor model and application to the valuation of a Bermuda swaption. Which approach for more complex multi-callable or path-dependent products?

Part II - Caroline Hillairet

1. Vasicek model, Hull and White model
2. Cox-Ingersoll-Ross Model (CIR)
3. Pricing and coverage by EDP
4. Heath, Jarrow and Morton's approach and factor models
5. Neutral forward measurement, cash change
6. Valuation and hedging of interest rate derivatives

### References

• GRBAC  Z. and RUNGGALDIER  W. J.  (2016). Interest Rate Modeling: Post-Crisis Challenges and Approaches.SpringerBriefs in Quantitative Finance, Springer.
• HULL J. (1999) : Options, futures and other derivatives, Prentice Hall.
• MARTELLINI L. et PRIAULET P. (2000) : Produits de taux d’intérêt : méthodes dynamiques d’évaluation et de couverture, Economica.
• MARTELLINI L., PRIAULET P. and PRIAULET S. (2003) : Fixed-Income Securities : Valuation, Risk
• Management and Portfolio Strategies, Wiley.
• MUSIELA M. and RUTKOWSKY M. (2005) : Martingale Methods in Financial Modelling, Springer.
• REBONATO R. (1998) : Interest Rate Option Models, Wiley.