Skip to main content
SpringerLink
Log in

Multidimensional Structural Default Models and Correlated Jumps

  • Chapter
  • First Online:
Pricing Derivatives Under Lévy Models

Part of the book series: Pseudo-Differential Operators ((PDO,volume 12))

  • 1098 Accesses

Abstract

In this chapter, we extend the MPsDO to the multidimensional case. To make our description more transparent, we use a concrete example, first considered in Itkin and Lipton (Int. J. Comput. Math. 92(12):2380–2405, 2015). In that paper, the structural default model of Lipton and Sepp (J. Credit Risk 5(2):123–146, 2009) is generalized to a set of banks with mutual interbank liabilities whose assets are driven by correlated Lévy processes with idiosyncratic and common components. Below we show how efficient FD schemes can be constructed using the MPsDO under this model in two- and three-dimensional cases. Also, the effects of mutual liabilities are discussed, and numerical examples are given to illustrate these effects.

Extra dimensional theories are sometimes considered science fiction with equations. I think that’s a wrong attitude. I think extra dimensions are with us, they are with us to stay, and they entered physics a long time ago. They are not going to go away.

Leonard Susskind.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It should be emphasized that the Vasicek model considers a single-period setting, whereas Lévy models have to be analyzed in continuous time. In addition, Lévy models use infinitely divisible distributions, rather than standard Gaussian random variables.

  2. 2.

    The expression given below assumes that the bank assets are allowed to be below its liabilities up to some value determined by the recovery rate. In this case, there is no default if such a breach is observed at some time before the maturity T. In this setup, the default boundary has a kink at t = T.

  3. 3.

    In order to better fit the market data, we can replace σ i with the local volatility function σ i (t, A i, t ).

  4. 4.

    Since we use splitting on financial processes, pure jump models are naturally covered by the same method. In the latter case, there is no diffusion at the first and third steps of the method, so one has to solve a pure convection equation. This can be achieved by applying various methods known in the fluid mechanics literature; see, e.g., [41].

  5. 5.

    By definition of A 2 B, the matrix M 2 is a lower triangular matrix with three nonzero diagonals. The main and the first lower diagonals are positive, and the second lower diagonal is negative. However, the former two dominate the latter.

References

  1. L. Ballotta, E. Bonfiglioli, Multivariate asset models using Lévy processes and applications. Eur. J. Finance. (DOI:10.1080/1351847X.2013.870917), April 2014

    Google Scholar 

  2. H. Bateman, A. Erdélyi (eds.), Tables of Integral Transforms (McGraw-Hill, New York, 1954)

    Google Scholar 

  3. M. Baxter, Lévy simple structural models. Int. J. Theor. Appl. Finance 10, 607–631 (2007)

    Article  MATH  Google Scholar 

  4. T.R. Bielecki, S. Crépey, A. Herbertsson, Markov chain models of portfolio credit risk, in The Oxford Handbook of Credit Risk, ed. by A. Lipton, A. Rennie (Oxford University Press, Oxford, 2011), pp. 327–382

    Google Scholar 

  5. F. Black, J.C. Cox, Valuing corporate securities: Some effects of bond indenture provisions. J. Finance 31 (2), 351–367 (1976)

    Article  Google Scholar 

  6. S.S. Clift, P. Forsyth, Numerical solution of two asset jump diffusion models for option valuation. Appl. Numer. Math. 58, 743–782 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Cont, P. Tankov, Financial Modelling with Jump Processes. Financial Mathematics Series (Chapman & Hall /CRCl, London, 2004)

    Google Scholar 

  8. J. Dash, Quantitative Finance and Risk Management: A Physicist’s Approach (World Scientific, Singapore, 2004)

    Book  MATH  Google Scholar 

  9. G. Deelstra, A. Petkovic, How they can jump together: Multivariate Lévy processes and option pricing. Belgian Actuarial Bull. 9 (1), 29–42 (2009–2010)

    Google Scholar 

  10. Y. d’Halluin, P.A. Forsyth, K.R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. Anal. 25, 87–112 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Eberlein, U. Keller, Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)

    Article  MATH  Google Scholar 

  12. L. Eisenberg, T.H. Noe, Systemic risk in financial systems. Manag. Sci. 47 (2), 236–249 (2001)

    Article  MATH  Google Scholar 

  13. H. Elsinger, A. Lehar, M. Summer, Using market information for banking system risk assessment. Int. J. Central Bank. 2 (1), 137–166 (2006)

    MATH  Google Scholar 

  14. J. Garcia, S. Goossens, V. Masol, W. Schoutens, Lévy based correlation. Wilmott J. 1 (2), 95–100 (2009)

    Article  Google Scholar 

  15. G. Gauthier, A Lehar, M. Souissi, Macroprudential regulation and systemic capital requirements. Technical Report 2010-4, Bank of Canada, 2010

    Google Scholar 

  16. F. Guillaume, The αVG model for multivariate asset pricing: calibration and extension. Rev. Deriv. Res. 16 (1), 25–52 (2013)

    Article  MATH  Google Scholar 

  17. T. Haentjens, K.J. In’t Hout, Alternating direction implicit finite difference schemes for the Heston–Hull–White partial differential equation. J. Comput. Finance 16, 83–110 (2012)

    Article  Google Scholar 

  18. N. Hilber, O. Reichmann, C. Winter, C. Schwab, Computational Methods for Quantitative Finance (Springer, New York, 2013)

    Book  MATH  Google Scholar 

  19. S. Howison, Barrier options, 1995. Available at https://people.maths.ox.ac.uk/howison/barriers.pdf

    Google Scholar 

  20. K.J. In’t Hout, S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7 (2), 303–320 (2010)

    MathSciNet  Google Scholar 

  21. K.J. In’t Hout, C. Mishra, Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms. Appl. Numer. Math. 74, 83–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. K.J. In’t Hout, B.D. Welfert, Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms. Appl. Numer. Math. 57, 19–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Itkin, Splitting and matrix exponential approach for jump–diffusion models with inverse normal Gaussian, hyperbolic, and Meixner jumps. Algorithmic Finance 3, 233–250 (2014)

    MathSciNet  Google Scholar 

  24. A. Itkin, High-Order Splitting Methods for Forward PDEs and PIDEs. Int. J. Theor. Appl. Finance 18 (5), 1550031–1—1550031–24 (2015)

    Google Scholar 

  25. A. Itkin, Efficient solution of backward jump–diffusion PIDEs with splitting and matrix exponentials. J. Comput. Finance 19, 29–70 (2016)

    Article  Google Scholar 

  26. A. Itkin, P. Carr, Jumps without tears: A new splitting technology for barrier options. Int. J. Numer. Anal. Model. 8 (4), 667–704 (2011)

    MathSciNet  MATH  Google Scholar 

  27. A. Itkin, P. Carr, Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models. Comput. Econ. 40 (1), 63–104 (2012)

    Article  MATH  Google Scholar 

  28. A. Itkin, A. Lipton, Efficient solution of structural default models with correlated jumps and mutual obligations. Int. J. Comput. Math. 92 (12), 2380–2405 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Kuznetsov, A.E. Kyprianou, J.C. Pardo, Meromorphic Lévy processes and their fluctuation identities, 2011. Available at http://arxiv.org/pdf/1004.4671.pdf

    MATH  Google Scholar 

  30. A.L. Lewis, Option Valuation under Stochastic Volatility (Finance Press, Newport Beach, 2000)

    MATH  Google Scholar 

  31. A. Lipton, Assets with jumps. RISK, 149–153 (2002)

    Google Scholar 

  32. A. Lipton, The vol smile problem. RISK, 61–65 (2002)

    Google Scholar 

  33. A. Lipton, A. Sepp, Credit value adjustment for credit default swaps via the structural default model. J. Credit Risk 5 (2), 123–146 (2009)

    Article  Google Scholar 

  34. A. Lipton, A. Sepp, Credit value adjustment in the extended structural default model, in The Oxford Handbook of Credit Derivatives, pp. 406–463 (Oxford University, Oxford, 2011)

    Google Scholar 

  35. E. Luciano, P. Semeraro, Multivariate time changes for Lévy asset models: characterization and calibration. J. Comput. Appl. Math. 233, 1937–1953 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. J.F. Mai, M. Scherer, T. Schulz, Sequential modeling of dependent jump processes. Wilmott Mag. 70, 54–63 (2014)

    Article  Google Scholar 

  37. A.W. Marshall, I. Olkin, A multivariate exponential distribution. J. Am. Stat. Assoc. 2, 84–98 (1967)

    MathSciNet  MATH  Google Scholar 

  38. 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 

  39. R. Merton, On the pricing of corporate debt: The risk structure of interest rates. J. Finance 29, 449–470 (1974)

    Google Scholar 

  40. 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 

  41. P.J. Roach, Computational Fluid Dynamics (Hermosa Publishers, Albuquerque, 1976)

    Google Scholar 

  42. W. Schoutens, Meixner processes in finance. Technical report, K.U. Leuven–Eurandom, 2001

    Google Scholar 

  43. G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 509–517 (1968)

    Article  MathSciNet  Google Scholar 

  44. Y. Sun, R. Mendoza-Arriaga, V. Linetsky, Valuation of collateralized debt obligations in a multivariate subordinator model, in Proceedings of the 2011 Winter Simulation Conference (WSC), ed. by S. Jain, R.R. Creasey, J. Himmelspach, K.P. White, M. Fu (IEEE, Phoenix, AZ, 2011), pp. 3742–3754

    Chapter  Google Scholar 

  45. O. Vasicek, Limiting loan loss probability distribution. Technical report, KMV Co., 1987

    Google Scholar 

  46. O. Vasicek, Loan portfolio value. RISK 15 (12), 160–162 (2002)

    Google Scholar 

  47. T. von Petersdorff, C. Schwab, Numerical solution of parabolic equations in high dimensions. Math. Modell. Numer. Anal. 38 (1), 93–127 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. L. Webber, M. Willison, Systemic capital requirements. Technical Report 436, Bank of England, 2011. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1945654

  49. C. Winter, Wavelet Galerkin schemes for option pricing in multidimensional Lévy models, PhD thesis, Eidgenössische Technische Hochschule ETH Zürich, 2009

    Google Scholar 

  50. C. Yang, R. Duraiswami, N.A. Gumerov, L. Davis, Improved fast Gauss transform and efficient kernel density estimation, in EEE International Conference on Computer Vision, pp. 464–471 (2003)

    Google Scholar 

  51. F. Yu, Correlated defaults in intensity-based models. Math. Finance 17, 155–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. C. Zhou, An analysis of default correlations and multiple defaults. Rev. Financ. Stud. 14 (2), 555–576 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Finance and Risk Engineering, NYU Tandon School of Engineering, Brooklyn, NY, USA

    Andrey Itkin

Authors
  1. Andrey Itkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Science+Business Media LLC

About this chapter

Cite this chapter

Itkin, A. (2017). Multidimensional Structural Default Models and Correlated Jumps. In: Pricing Derivatives Under Lévy Models . Pseudo-Differential Operators, vol 12. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-6792-6_8

Download citation

Publish with us

Policies and ethics

Search

Navigation