Efficient solution of structural default models with correlated jumps and mutual obligations

References

• L. Ballotta and E. Bonfiglioli, Multivariate asset models using Lévy processes and applications, SSRN: 1695527, 2014.
   Google Scholar
• L. Ballotta and I. Kyriakou, Monte Carlo simulation of the CGMY process and option pricing, J. Futures Markets 34(12) (2014), pp. 1095–1121. doi: 10.1002/fut.21647
   Web of Science ®Google Scholar
• M. Baxter, Lévy simple structural models, Int. J. Theor. Appl. Financ. 10 (2007), pp. 607–631. doi: 10.1142/S021902490700438X
   Google Scholar
• T. Bielecki, S. Crépey, and A. Herbertsson, Markov chain models of portfolio credit risk, in The Oxford Handbook of Credit Risk, A. Lipton and A. Rennie, eds., Oxford University Press, Oxford, 2011, pp. 327–382.
   Google Scholar
• F. Black and J.C. Cox, Valuing corporate securities: Some effects of bond indenture provisions, J. Finance 31(2) (1976), pp. 351–367. doi: 10.1111/j.1540-6261.1976.tb01891.x
   Web of Science ®Google Scholar
• S.S. Clift and P.A. Forsyth, Numerical solution of two asset jump diffusion models for option valuation, Appl. Numer. Math. 58 (2008), pp. 743–782. doi: 10.1016/j.apnum.2007.02.005
   Web of Science ®Google Scholar
• R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Matematics Series, Chapman & Hall/CRCl, Boca Raton, 2004.
   Google Scholar
• G. Deelstra and A. Petkovic, How they can jump together: Multivariate Lévy processes and option pricing, Belg. Actuar. Bull. 9(1) (2009–2010), pp. 29–42.
   Google Scholar
• Y. d'Halluin, P.A. Forsyth, and K.R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal. 25 (2005), pp. 87–112. doi: 10.1093/imanum/drh011
   Web of Science ®Google Scholar
• E. Eberlein, Jump-type Lévy processes, in Handbook of Financial Time Series, T.G. Andersen, R.A. Davis, J.-P. Kreiß, and T. Mikosch, eds., Springer Verlag, Berlin, 2009, pp. 439–455.
   Google Scholar
• E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli 1 (1995), pp. 281–299. doi: 10.2307/3318481
   Google Scholar
• L. Eisenberg and T.H. Noe, Systemic risk in financial systems, Manage. Sci. 47(2) (2001), pp. 236–249. doi: 10.1287/mnsc.47.2.236.9835
   Web of Science ®Google Scholar
• A. Elhashash and D.B. Szyld, Generalizations of M-matrices which may not have a nonnegative inverse, Linear Algebra Appl. 429 (2008), pp. 2435–2450. doi: 10.1016/j.laa.2008.01.006
   Web of Science ®Google Scholar
• H. Elsinger, A. Lehar, and M. Summer, Using market information for banking system risk assessment, Int. J. Central Banking 2(1) (2006), pp. 137–166.
   Google Scholar
• J. Garcia, S. Goossens, V. Masol, and W. Schoutens, Lévy based correlation, Wilmott J. 1(2) (2009), pp. 95–100. doi: 10.1002/wilj.6
   Google Scholar
• G. Gauthier, A. Lehar, and M. Souissi, Macroprudential regulation and systemic capital requirements, Technical Report 2010-4, Bank of Canada, 2010.
   Google Scholar
• F. Guillaume, The αVG model for multivariate asset pricing: Calibration and extension, Rev. Deriv. Res. 16(1) (2013), pp. 25–52. doi: 10.1007/s11147-012-9080-2
   Web of Science ®Google Scholar
• T. Haentjens and K.J. In't Hout, Alternating direction implicit finite difference schemes for the Heston–Hull–White partial differential equation, J. Comput. Financ. 16 (2012), pp. 83–110.
   Web of Science ®Google Scholar
• N. Hilber, O. Reichmann, C. Winter, and C. Schwab, Computational Methods for Qantitative Finance, Springer, Heidelberg, 2013.
   Google Scholar
• K.J. In't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Model. 7(2) (2010), pp. 303–320.
   Web of Science ®Google Scholar
• K.J. In't Hout and C. Mishra, Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms, Appl. Numer. Math. 74 (2013), pp. 83–94. doi: 10.1016/j.apnum.2013.07.003
   Web of Science ®Google Scholar
• K.J. In't Hout and B.D. Welfert, Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms, Appl. Numer. Math. 57 (2007), pp. 19–35. doi: 10.1016/j.apnum.2005.11.011
   Web of Science ®Google Scholar
• A. Itkin, Efficient solution of backward jump-diffusion PIDEs with splitting and matrix exponentials, J. Comput. Financ. (2014a), forthcoming. Available at http://arxiv.org/abs/1304.3159.
   Web of Science ®Google Scholar
• A. Itkin, High-order splitting methods for forward PDEs and PIDEs, preprint (2014b). Available at http://arxiv.org/abs/1403.1804.
   Google Scholar
• A. Itkin, Splitting and matrix exponential approach for jump-diffusion models with inverse normal Gaussian, hyperbolic and Meixner jumps, Algorithmic Financ. (2014c), forthcoming. Available at http://arxiv.org/abs/1405.6111.
   Google Scholar
• A. Itkin and P. Carr, Jumps without tears: A new splitting technology for barrier options, Int. J. Numer. Anal. Model. 8(4) (2011), pp. 667–704.
   Web of Science ®Google Scholar
• A. Itkin and P. Carr, Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models, Comput. Econ. 40(1) (2012), pp. 63–104. doi: 10.1007/s10614-011-9269-8
   Web of Science ®Google Scholar
• S.G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Manag. Sci. 50(9) (2004), pp. 1178–1192. doi: 10.1287/mnsc.1030.0163
   Web of Science ®Google Scholar
• O.L. de Lange and R.E. Raab, Operator Methods in Quantum Mechanics, Oxford Science Publications, Oxford, Chapter 3, 1992.
   Google Scholar
• A. Lipton, The vol smile problem, Risk (2002), pp. 61–65.
   Google Scholar
• A. Lipton and I. Savescu, Pricing credit default swaps with bilateral value adjustments, Quant. Finance 14(1) (2014), pp. 171–188. doi: 10.1080/14697688.2013.828239
   Web of Science ®Google Scholar
• A. Lipton and A. Sepp, Credit value adjustment for credit default swaps via the structural default model, J. Credit Risk 5(2) (2009), pp. 123–146.
   Web of Science ®Google Scholar
• A. Lipton and A. Sepp, Credit Value Adjustment in the Extended Structural Default Model, The Oxford Handbook of Credit Derivatives, Oxford University, Oxford, 2011, 406–463.
   Google Scholar
• E. Luciano and P. Semeraro, Multivariate time changes for Lévy asset models: Characterization and calibration, J. Comput. Appl. Math. 233 (2010), pp. 1937–1953. doi: 10.1016/j.cam.2009.08.119
   Web of Science ®Google Scholar
• J. Mai, M. Scherer, and T. Schulz, Sequential modeling of dependent jump processes, Wilmott Mag. 70 (2014), pp. 54–63. doi: 10.1002/wilm.10309
   Google Scholar
• A. Marshall and I. Olkin, A multivariate exponential distribution, J. Amer. Statist. Assoc. 2 (1967), pp. 84–98.
   Google Scholar
• J.M. McDonough, Lectures on Computational Numerical Analysis of Partial Differential Equations, University of Kentucky, 2008. Available at http://www.engr.uky.edu/acfd/me690-lctr-nts.pdf.
   Google Scholar
• R. Merton, On the pricing of corporate debt: The risk structure of interest rates, J. Finance 29 (1974), pp. 449–470.
   Web of Science ®Google Scholar
• T. Moosbrucker, Copulas from infinitely divisible distributions: Applications to credit value at risk, Technical report, Department of Banking – University of Cologne, 2006. Available at http://gloria-mundi.com/Library_Journal_View.asp?Journal_id=7547.
   Google Scholar
• P. Roach, Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, 1976.
   Google Scholar
• W. Schoutens, Meixner processes in finance, Technical report, K.U.Leuven–Eurandom, 2001.
   Google Scholar
• G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), pp. 509–517. doi: 10.1137/0705041
   Web of Science ®Google Scholar
• Y. Sun, R. Mendoza-Arriaga, and V. Linetsky, Valuation of collateralized debt obligations in a multivariate subordinator model. In Proceedings of the 2011 Winter Simulation Conference (WSC), S. Jain, R.R. Creasey, J. Himmelspach, K.P. White, and M. Fu, eds., IEEE, Phoenix, AZ, 2011, pp. 3742–3754.
   Google Scholar
• O. Vasicek, Limiting loan loss probability distribution, Technical report, KMV Co., 1987.
   Google Scholar
• O. Vasicek, Loan portfolio value, Risk 15(12) (2002), pp. 160–162.
   Google Scholar
• T. von Petersdorff and C. Schwab, Numerical solution of parabolic equations in high dimensions, Math. Model. Numer. Anal. 38(1) (2004), pp. 93–127. doi: 10.1051/m2an:2004005
   Web of Science ®Google Scholar
• L. Webber and M. Willison, Systemic capital requirements, Technical Report 436, Bank of England, 2011. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1945654.
   Google Scholar
• C. Winter, Wavelet Galerkin schemes for option pricing in multidimensional Lévy models, Ph.D. thesis, Eidgenössische Technische Hochschule ETH Zürich, 2009.
   Google Scholar
• C. Yang, R. Duraiswami, N.A. Gumerov, and L. Davis, Improved fast Gauss transform and efficient kernel density estimation, In IEEE International Conference on Computer Vision, Nice, 2003, pp. 464–471.
   Google Scholar
• F. Yu, Correlated defaults in intensity-based models, Math. Financ. 17 (2007), pp. 155–173. doi: 10.1111/j.1467-9965.2007.00298.x
   Web of Science ®Google Scholar
• C. Zhou, The term structure of credit spreads with jump risk, J. Bank. Financ. 25 (2001), pp. 2015–2040. doi: 10.1016/S0378-4266(00)00168-0
   Web of Science ®Google Scholar