A model for interest rates with clustering effects. (English) Zbl 1400.91630
Summary: We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for the characteristic function for interest rates and their integral. We introduce a class of equivalent measures under which the features of the process are preserved. We infer the prices of bonds and their dynamics under a risk-neutral measure. The question of derivatives pricing is developed under a forward measure, and a numerical algorithm is proposed to evaluate caplets and floorlets. The model is fitted to EONIA rates from 2004 to 2014 using a peaks-over-threshold procedure. From observation of swap curves over the same period, we filter the evolution of risk premiums for Brownian and jump components. Finally, we analyse the sensitivity of implied caplet volatility to parameters defining the level of self-excitation.
MSC:
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
References:
| [1] | Ait-Sahalia, Y., Cacho-Diaz, J. and Laeven, R.J.A., Modeling financial contagion using mutually exciting jump processes. J. Financ. Econ., 2015, 117, 585-606. |
| [2] | Ait-Sahalia, Y., Laeven, R.J.A. and Pelizzon, L., Mutual excitation in Eurozone sovereign CDS. J. Econometrics, 2014, 183, 151-167. · Zbl 1312.91089 |
| [3] | Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.F., Modelling microstructure noise with mutually exciting point processes. Quant. Finance, 2013, 13(1), 65-77. · Zbl 1280.91073 |
| [4] | Bauwens, L. and Hautsch, N., Handbook of Financial Time Series: Modelling Financial High Frequency Data Using Point Processes, 2009 (Springer: Berlin). · Zbl 1178.91218 |
| [5] | Barndorff-Nielsen, O.E. and Shephard, N., Power and bipower variation with stochastic volatility and jumps (with discussion). J. Financ. Econ., 2004, 2, 1-48. |
| [6] | Barndorff-Nielsen, O.E. and Shephard, N., Econometrics of testing for jumps in financial economics using bipower variation. J. Financ. Econ., 2006, 4, 1-30. |
| [7] | Brigo D. and Mercurio F., Interest Rate Models - Theory and Practice, 2007. (Springer: Berlin). · Zbl 1038.91040 |
| [8] | Chavez-Demoulin, V., Davison, A.C. and McNeil, A.J., A point process approach to value-at-risk estimation. Quant. Finance, 2005, 5(2), 227-234. · Zbl 1118.91353 |
| [9] | Chavez-Demoulin, V. and McGill, J.A., High-frequency financial data modeling using Hawkes processes. J. Bank. Financ., 2012, 36, 3415-3426. |
| [10] | Chen K. and Poon S.H., Variance swap premium under stochastic volatility and self-exciting jumps. SSRN Working Paper, SSRN-id2200172, 2013. |
| [11] | Cox, J.C., Ingersoll, J.E. and Ross, S.A., A theory of the term structure of interest rates. Econometrica, 1985, 53, 385-407. · Zbl 1274.91447 |
| [12] | Dai, Q. and Singleton, K.J., Specification analysis of affine term structure models. J. Financ., 2000, 55, 1943-1978. |
| [13] | Dassios, A. and Jang, J., A double shot noise process and its application in insurance. J. Math. Syst. Sci., 2012, 2, 82-93. |
| [14] | Duffie, D. and Kan, R., A yield-factor model of interest rates. Math. Financ., 1996, 6, 379-406. · Zbl 0915.90014 |
| [15] | Eberlein, E. and Kluge, W., Exact pricing formulae for caps and swaptions in a Levy term structure model. J. Comput. Financ., 2006, 9(2), 99-125. |
| [16] | Embrechts P., Liniger T. and Lu L., Multivariate Hawkes processes: An application to financial data. J. Appl. Proba.2011, 48, A367-A378. · Zbl 1242.62093 |
| [17] | Errais, E., Giesecke, K. and Goldberg, L.R., Affine point processes and portfolio credit risk. SIAM J. Financ. Math., 2010, 1, 642-665. · Zbl 1200.91296 |
| [18] | Filipović, D. and Tappe, S., Existence of Levy term structure models Fin. Stoch., 2008, 12(1), 83-115. · Zbl 1150.91017 |
| [19] | Giot, P., Market risk models for intraday data. Euro. J. Finanac., 2005, 11(4), 309-324. |
| [20] | Hainaut, D. and MacGilchrist, R., An interest rate tree driven by a L\’{e}vy process. J. Deriv., 2010, 18(2), 33-45. |
| [21] | Hawkes, A., Point sprectra of some mutually exciting point processes. J. R. Stat. Soc. B, 1971a, 33, 438-443. · Zbl 0238.60094 |
| [22] | Hawkes, A., Spectra of some self-exciting and mutually exciting point processes. Biometrika, 1971b, 58, 83-90. · Zbl 0219.60029 |
| [23] | Hawkes, A. and Oakes, D., A cluster representation of a self-exciting process. J. Appl. Proba., 1974, 11, 493-503. · Zbl 0305.60021 |
| [24] | Hull J. and White A., Pricing interest rate derivatives. Rev. Financ. Stud.1990, 3, 4573-592. · Zbl 1386.91152 |
| [25] | Mancini, C., Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat., 2009, 36, 270-296. · Zbl 1198.62079 |
| [26] | Salmon M. and Tham W.W., Time deformation and term structure of interest rates. SSRN Working Paper, SSRN-id999841, 2007. |
| [27] | Vasicek, O., An equilibrium characterization of the term structure. J. Financ. Econ., 1977, 5, 177-188. · Zbl 1372.91113 |
| [28] | Zhang X.W., Glynn P.W. and Giesecke K., Rare event simulation for a generalized Hawkes process. Proceedings of the 2009 Winter Simulation Conference, Austin, Texas. |
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