1. Problem 4. Calibration of single-factor HJM models of interest rates Coordinators Miguel Carrión Álvarez - Banco Santander Gerardo Oleaga Apadula - Universidad Complutense de Madrid Participants Antonio Bueno Universidad Complutense de Madrid Javier García - Universidad Complutense de Madrid Senshan Ji - Universidad Autónoma de Barcelona Santiago López Vizcayno- Universidad Complutense de Madrid Alejandra Sánchez - Universidad Complutense de Madrid Daniel Neira Verdes-Montenegro - Universidad Complutense de Madrid Marco Caroccia - Università degli Studi di Firenze

2. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 2 01 Introduction - The time value of money 02 Concepts - Time value of money - Interest rate - Model features 03 The HJM framework - Analysis of the forward rates - Forward correlation matrix - PCA analysis of forward rates - Arbitrage free model for the synthetic forwards Index

3. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 3 Introduction The time value of money Time is money. A dollar today is better than a dollar tomorrow. And a dollar tomorrow is better than a dollar next year. Is every day worth the same or will the price of money change from time to time? The interest rate market is where the price of money is set. What does “price of money” mean? It is the cost of borrowing and lending money. It is usually quoted by means of “rates” per unit of time (1% per annum, 2% per annum). The price of money depends not only on the length of the term, but also on the moment-to-moment random fluctuations of the market. Money behaves just like a stock with a noisy price driven by a Brownian motion. 01

4. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 4 Concepts Market Zero coupon bonds We denote by Z t (T) the value on date t of one monetary unit deliverable on date T. By definition, Z T (T) = 1, Z t (T) < 1 for all t< T. 02

5. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 5 Concepts Market Zero Coupon Bonds Due to the fact Z T (T) = 1, Z t (T) can’t be a stationary process. 02

6. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 6 Concepts Synthetic Zero Coupon Bonds However, we can define a synthetic constant-maturity bond whose price is 02

7. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 7 Concepts The synthetic Zero coupon bond The evolution of.This is a stationary process but highly autocorrelated because of price continuity. 02

8. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 8 Concepts The synthetic Zero coupon bond. If we consider we obtain a stationary and not autocorrelated process. Any function of this variable is stationary and not autocorrelated too. So, we define 02

9. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 9 Theoretical Concepts Constant maturity yields Yield. Given a discount bond price Z at time t, the yield R is given by: The forward rates and can be written in terms of the bond prices as: Instantaneous forward rates 03

10. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 10 Theorical Concepts Model features We want to consider stochastic models of interest rates with the following features: They have as few underlying stochastic factors as possible. They are consistent with absence of arbitrage opportunities (“there is no free lunch”). They can potentially accommodate any observed term structure of interest rates. 03

11. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 11 The HJM framework The Heath–Jarrow–Morton theory ("HJM") is a general framework to model the evolution of interest rate curve - instantaneous forward rate curve in particular. The key to these techniques is the recognition that the drifts of the no-arbitrage evolution of the instantaneous forwards can be expressed as functions of their volatilities, no drift estimation is needed. The general parameterization of continuous stochastic evolution due to HJM is: Where W t is basic Wiener process, so that W t ~N(0;T) 03

12. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 12 The HJM framework In the risk neutral probability measure the drift changes as. Choosing the cash bond B t to discount prices, the no-arbitrage condition implies: Whereis the log-volatility of the discounted bond price 03

13. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 13 Analysis of the forward rates We cannot obtain the instantaneous forward rates from the data, but we are able to analise the forwards between two consecutive : We have a stochastic variable for each k, so we proceed to a principal component analysis of the forwards. This allows to construct a “discretised” HJM model with no arbitrage.

14. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 14 Forward correlation matrix

15. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 15 PCA analysis for forward rates

16. IV UCM Modelling Week Calibration of single-factor HJM models of interest rates 16 The arbitrage free model for the synthetic forwards This arbitrage-free model is obtained from the HJM condition imposed to our synthetic variables. In the simplest setting, the volatilities are estimated from principal component analysis. There is only one risk factor involved. For future work, a more complex model for the volatilities is needed, and more factors may be included. Thank you!