## The Annals of Applied Probability

### Time-changed CIR default intensities with two-sided mean-reverting jumps

#### Abstract

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state variable $X$ driving default intensity and a default indicator process $D$ and time change it with a Lévy subordinator $\mathcal{T}$. We characterize the time-changed process $(X^{\phi}_{t},D^{\phi}_{t})=(X(\mathcal{T}_{t}),D(\mathcal{T}_{t}))$ as a Markovian–Itô semimartingale and show from the Doob–Meyer decomposition of $D^{\phi}$ that the default time in the time-changed model has a jump-diffusion or a pure jump intensity. When $X$ is a CIR diffusion with mean-reverting drift, the default intensity of the subordinate model (SubCIR) is a jump-diffusion or a pure jump process with mean-reverting jumps in both directions that stays nonnegative. The SubCIR default intensity model is analytically tractable by means of explicitly computed eigenfunction expansions of relevant semigroups, yielding closed-form pricing of credit-sensitive securities.

#### Article information

Source
Ann. Appl. Probab. Volume 24, Number 2 (2014), 811-856.

Dates
First available in Project Euclid: 10 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1394465372

Digital Object Identifier
doi:10.1214/13-AAP936

Mathematical Reviews number (MathSciNet)
MR3178498

Zentralblatt MATH identifier
1291.91225

Subjects
Primary: 91G40: Credit risk
Secondary: 60J75: Jump processes 91G30: Interest rates (stochastic models) 60G55: Point processes

#### Citation

Mendoza-Arriaga, Rafael; Linetsky, Vadim. Time-changed CIR default intensities with two-sided mean-reverting jumps. Ann. Appl. Probab. 24 (2014), no. 2, 811--856. doi:10.1214/13-AAP936. https://projecteuclid.org/euclid.aoap/1394465372

#### References

• Ahn, D.-H. and Gao, B. (1999). A parametric nonlinear model of term structure dynamics. The Review of Financial Studies 12 721–762.
• Andreasen, J. (2001). Credit explosives. Bank of America Fixed Income Research Working Paper.
• Barlow, M. T. (2002). A diffusion model for electricity prices. Math. Finance 12 287–298.
• Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2 41–68.
• Beaglehole, D. and Tenney, M. (1992). Corrections and additions to “A nonlinear equilibrium model of the term structure of interest rates.” Math. Finance 32 345–353.
• Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
• Bertoin, J. (1999). Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717. Springer, Berlin.
• Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2011). Hedging of a credit default swaption in the CIR default intensity model. Finance Stoch. 15 541–572.
• Bielecki, T. R. and Rutkowski, M. (2004). Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin.
• Bielecki, T. R., Crépey, S., Jeanblanc, M. and Rutkowski, M. (2008). Defaultable options in a Markovian intensity model of credit risk. Math. Finance 18 493–518.
• Bielecki, T. R., Cousin, A., Crépey, S. and Herbertsson, A. (2013). Dynamic modeling of portfolio credit risk with common shocks. J. Optim. Theory Appl. To appear.
• Bielecki, T. R., Crépey, S., Jeanblanc, M. and Zargari, B. (2012). Valuation and hedging of CDS counterparty exposure in a Markov copula model. Int. J. Theor. Appl. Finance 15 1250004, 39.
• Bochner, S. (1949). Diffusion equation and stochastic processes. Proc. Natl. Acad. Sci. USA 35 368–370.
• Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
• Boyarchenko, N. and Levendorskiĭ, S. (2007). The eigenfunction expansion method in multi-factor quadratic term structure models. Math. Finance 17 503–539.
• Brigo, D. and Alfonsi, A. (2005). Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance Stoch. 9 29–42.
• Brigo, D. and El-Bachir, N. (2006). Credit derivatives pricing with a smile-extended jump stochastic intensity model. Working paper.
• Brigo, D. and El-Bachir, N. (2010). An exact formula for default swaptions’ pricing in the SSRJD stochastic intensity model. Math. Finance 20 365–382.
• Carr, P. and Linetsky, V. (2006). A jump to default extended CEV model: An application of Bessel processes. Finance Stoch. 10 303–330.
• Çinlar, E. and Jacod, J. (1981a). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981) (E. Cinlar, K. L. Chung and R. K. Getoor, eds.). Progr. Prob. Statist. 1 159–242. Birkhäuser, Boston, MA.
• Çinlar, E. and Jacod, J. (1981b). Semimartingales defined on Markov processes. In Stochastic Differential Systems (Visegrád, 1980) (M. Arató, D. Vermes and A. V. Balakrishnan, eds.). Lecture Notes in Control and Information Sci. 36 13–24. Springer, Berlin.
• Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54 161–219.
• Chen, Z.-Q. and Fukushima, M. (2011). Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton Univ. Press, Princeton, NJ.
• Cox, J. C., Ingersoll, J. E. Jr. andRoss, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–407.
• Cuchiero, C., Filipović, D., Mayerhofer, E. and Teichmann, J. (2011a). Affine processes on positive semidefinite matrices. Ann. Appl. Probab. 21 397–463.
• Cuchiero, C., Keller-Ressel, M., Mayerhofer, E. and Teichmann, J. (2011b). Affine processes on symmetric cones. Unpublished manuscript.
• Davies, E. B. (2007). Linear Operators and Their Spectra. Cambridge Studies in Advanced Mathematics 106. Cambridge Univ. Press, Cambridge.
• Davydov, D. and Linetsky, V. (2003). Pricing options on scalar diffusions: An eigenfunction expansion approach. Oper. Res. 51 185–209.
• Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984–1053.
• Duffie, D. and Garleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analysts Journal 57 41–59.
• Duffie, D. and Kan, R. (1996). A yield-factor model of interest rates. Math. Finance 6 379–406.
• Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68 1343–1376.
• Duffie, D. and Singleton, K. J. (1999). Modeling term structures of defaultable bonds. The Review of Financial Studies 12 687–720.
• Duffie, D. and Singleton, K. J. (2003). Credit Risk: Pricing, Measurement, and Management. Princeton Univ. Press, Princeton, NJ.
• Elkamhi, R., Jacobs, K., Langlois, H. and Ornthanalai, C. (2012). Accounting information releases and CDS spreads. Working paper.
• Erdelyi, A. (1953). Higher Transcendental Functions II. McGraw-Hill, New York.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• Feller, W. (1951). Two singular diffusion problems. Ann. of Math. (2) 54 173–182.
• Filipović, D. (2001). A general characterization of one factor affine term structure models. Finance Stoch. 5 389–412.
• Fukushima, M., Oshima, Y. and Takeda, M. (2011). Dirichlet Forms and Symmetric Markov Processes, extended ed. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin.
• Geman, H. and Roncoroni, A. (2006). Understanding the fine structure of electricity prices. Journal of Business 79 1225–1261.
• Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 313–349.
• Gorovoi, V. and Linetsky, V. (2004). Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates. Math. Finance 14 49–78.
• Itô, K. and McKean, H. P. (1974). Diffusion Processes and Their Sample Paths, corrected 2nd ed. Springer, Berlin.
• Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
• Jacod, J. and Protter, P. (2011). Discretization of Processes. Springer, Berlin.
• Jacod, J. and Shiryaev, A. N. (2002). Limit Theorems for Stochastic Processes, 2nd. ed. Comprenhensive Studies in Mathematics 288. Springer, Berlin.
• Jamshidian, F. (1996). Bond, futures and option evaluation in the quadratic interest rate model. Appl. Math. Finance 3 93–115.
• Janson, S., M’Baye, S. and Protter, P. (2011). Absolutely continuous compensators. Int. J. Theor. Appl. Finance 14 335–351.
• Jarrow, R. A., Lando, D. and Turnbull, S. M. (1997). A Markov model for the term structure of credit risk spreads. The Review of Financial Studies 10 481–523.
• Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer, London.
• Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 36–54.
• Keller-Ressel, M., Schachermayer, W. and Teichmann, J. (2011). Affine processes are regular. Probab. Theory Related Fields 151 591–611.
• Kita, A. (2012). CDS spreads explained with credit spread volatility and jump risk of individual firms. Working paper.
• Lewis, A. L. (1994). Three expansion regimes for interest rate term structure models. Analytic Investment Management. Available at http://optioncity.net/.
• Li, L. and Linetsky, V. (2013a). Optimal stopping and early exercise: An eigenfunction expansion approach. Oper. Res. 61 625–644.
• Li, L. and Linetsky, V. (2013b). Time-changed Ornstein–Uhlenbeck processes and their applications in commodity derivative models. Math. Finance. To appear.
• Lim, D., Li, L. and Linetsky, V. (2012). Evaluating callable and putable bonds: An eigenfunction expansion approach. J. Econom. Dynam. Control 36 1888–1908.
• Linetsky, V. (2004). The spectral decomposition of the option value. Int. J. Theor. Appl. Finance 7 337–384.
• Linetsky, V. (2006). Pricing equity derivatives subject to bankruptcy. Math. Finance 16 255–282.
• Linetsky, V. (2008). Spectral methods in derivatives pricing. In Handbooks in Operations Research and Management Science: Financial Engineering 15 223–300. Elsevier/North-Holland, Amsterdam.
• Lorig, M., Lozano-Carbassé, O. and Mendoza-Arriaga, R. (2013). Variance swaps on defaultable assets and market implied time-changes. Unpublished manuscript.
• Madan, D. B., Carr, P. and Chang, E. C. (1998). The variance gamma process and option pricing. European Finance Review 2 79–105.
• McKean, H. P. Jr. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82 519–548.
• Mendoza-Arriaga, R. (2012). Credit default swap options under the subordinate diffusion framework. Working paper.
• Mendoza-Arriaga, R., Carr, P. and Linetsky, V. (2010). Time-changed Markov processes in unified credit-equity modeling. Math. Finance 20 527–569.
• Mendoza-Arriaga, R. and Linetsky, V. (2013). Multivariate subordination of Markov processes with financial applications. Math. Finance. To appear.
• Meyer-Brandis, T. and Tankov, P. (2008). Multi-factor jump-diffusion models of electricity prices. Int. J. Theor. Appl. Finance 11 503–528.
• Nikiforov, A. F. and Uvarov, V. B. (1988). Special Functions of Mathematical Physics. Birkhäuser, Basel.
• Nowak, A. and Stempak, K. (2010). On $L^{p}$-contractivity of Laguerre semigroups. Working paper.
• Phillips, R. S. (1952). On the generation of semigroups of linear operators. Pacific J. Math. 2 343–369.
• Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425–457.
• Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals Series: Special Functions. Integrals and Series II. Gordon & Breach, New York.
• Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics I: Functional Analysis, revised ed. Academic Press, San Diego, CA.
• Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
• Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
• Schilling, R. L., Song, R. and Vondraček, Z. (2010). Bernstein Functions: Theory and Applications. de Gruyter Studies in Mathematics 37. de Gruyter, Berlin.
• Zhang, B. Y., Zhou, H. and Zhu, H. (2009). Explaining credit default swap spreads with the equity volatility and jump risks of individual firms. Review of Financial Studies 22 5099–5131.