Multi-curve HJM modelling for risk management

Abstract

We present a HJM approach to the projection of multiple yield curves developed to capture the volatility content of historical term structures for risk management purposes. Since we observe the empirical data at daily frequency and only for a finite number of time-to-maturity buckets, we propose a modelling framework which is inherently discrete. In particular, we show how to approximate the HJM continuous time description of the multi-curve dynamics by a Vector Autoregressive process of order one. The resulting dynamics lends itself to a feasible estimation of the model volatility-correlation structure and market risk-premia. Then, resorting to the Principal Component Analysis we further simplify the dynamics reducing the number of covariance components. Applying the constant volatility version of our model on a sample of curves from the Euro area, we demonstrate its forecasting ability through an out-of-sample test.

Keywords:

• HJM
• Multiple-curve
• Scenario generation
• PCA

JEL Classification:

• C32
• C5
• C51
• G01
• G10

Acknowledgements

We thank Flavia Barsotti, Andrea Bertagna, Tommaso Colozza, Fulvio Corsi, Niccolò Cottini, Lorenzo Liesch, Stefano Marmi, Aldo Nassigh, Andrea Pallavicini and Roberto Renò for many inspiring discussions. We also acknowledge Andrea Sillari for having provided the historical data.

Notes

No potential conflict of interest was reported by the authors.

1 The symbol stands for the usual scalar product.

2 As before, the drift terms appear because we parametrize the rate dynamics in terms of the time-to-maturity.

3 We drop the dependence of , and on the vector of time-to-maturity buckets for ease of notation.

4 We drop the dependence of on the vector for ease of notation.

5 Market quotes are taken from Bloomberg and Reuters.

6 .

7 For the computation of the statistical errors affecting the principal components we refer to Anderson (1963); Flury (1997).

9 We have dropped the dependence of on for ease of notation.

10 We drop the dependence of M, and on for ease of notation.

Additional information

Funding

The activity of CS is supported by the Research Agreement number [grant number 1300240/2013]; “Stabilità e corenza con le aspettative di mercato degli scenari a lungo termine impiegati nella gestione dei rischi finanziari” between UniCredit S.p.A. and Scuola Normale Superiore.