Skip to main content

Validation and Applications

  • Chapter
  • First Online:
  • 1237 Accesses

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 77))

Abstract

In this chapter, we present various numerical experiments with different applications from finance. We study the performance of the different numerical quadrature methods from Chapter 3 and Chapter 4 and investigate the impact of the different approaches for dimension reduction and smoothing from Chapter 5. Parts of this chapter are taken from [58].

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albizzati, M., Geman, H.: Interest rate risk management and valuation of the surrender option in life insurance policies. J. Risk and Insurance 61(4), 616–637 (1994)

    Article  Google Scholar 

  2. Bacinello, A.: Fair pricing of life insurance participating contracts with a minimum interest rate guaranteed. Astin Bulletin 31(2), 275–297 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bacinello, A.: Pricing guaranteed life insurance participating policies with annual premiums and surrender option. Astin Bulletin 7(3), 1–17 (2003)

    MATH  MathSciNet  Google Scholar 

  4. Ballotta, L., Haberman, S., Wang, N.: Guarantees in with-profit and unitized with-profit life insurance contracts: Fair valuation problem in presence of the default option. Insurance: Mathematics and Economics 73(1), 97–121 (2006)

    Google Scholar 

  5. Bernsten, J., Espelid, T., Genz, A.: Algorithm 698: DCUHRE – an adaptive multidimensional integration routine for a vector of integrals. ACM Transactions on Mathematical Software 17, 452–456 (1991)

    Article  Google Scholar 

  6. Brigo, D., Mercurio, F.: Interest Rate Models - Theory and Practice. Springer (2001)

    Google Scholar 

  7. Briys, E., Varenne, F.: On the risk of life insurance liabilities: Debunking some common pitfalls. J. Risk and Insurance 64, 37–57 (1997)

    Google Scholar 

  8. Caflisch, R., Morokoff, W., Owen, A.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Comp. Finance 1(1), 27–46 (1997)

    Google Scholar 

  9. Cottin, C., Kurz, A.: Asset–Liability–Management in der Lebensversicherung. Preprint 16, Fachhochschule Bielefeld (2003)

    Google Scholar 

  10. Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)

    Article  MathSciNet  Google Scholar 

  11. DAV: Abschlussbericht der DAV-Arbeitsgruppe Market Consistent Embedded Value (2010). To appear.

    Google Scholar 

  12. De Felice, M., Moriconi, F.: Market based tools for managing the life insurance company. Astin Bulletin 1(35), 79–111 (2005)

    Article  Google Scholar 

  13. Diers, D.: Interne Unternehmensmodelle in der Schaden- und Unfallversicherung. ifa-Schriftenreihe, Ulm (2007)

    Google Scholar 

  14. Drezner, Z., Wesolowsky, G.: On the computation of the bivariate normal integral. J. Statistical Computation and Simulation 35, 101–107 (1990)

    Article  MathSciNet  Google Scholar 

  15. Egloff, D., Leippold, M.: The valuation of American options with stochastic stopping time constraints. Appl. Math. Finance 16(3), 287–305 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fischer, T., May, A., Walther, B.: Anpassung eines CIR–1–Modells zur Simulation der Zinsstrukturkurve. Blätter der DGVFM 2(XXVI), 193–206 (2003)

    Article  Google Scholar 

  17. Genz, A.: Numerical computation of multivariate normal probabilities. J. Comput. Graph. Statist. 1, 141–150 (1992)

    Article  Google Scholar 

  18. Genz, A.: Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Statistics and Computing 14, 151–160 (2004)

    Article  MathSciNet  Google Scholar 

  19. Genz, A., Malik, A.: An adaptive algorithm for numerical integration over an n-dimensional rectangular region. J. Comp. Appl. Math. 6, 295–302 (1980)

    Article  MATH  Google Scholar 

  20. Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numerical Algorithms 18, 209–232 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gerstner, T., Griebel, M.: Dimension–adaptive tensor–product quadrature. Computing 71(1), 65–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gerstner, T., Griebel, M., Holtz, M.: The Effective Dimension of Asset-Liability Management Problems in Life Insurance. In: C. Fernandes, H. Schmidli, N. Kolev (eds.) Proc. Third Brazilian Conference on Statistical Modelling in Insurance and Finance, pp. 148–153 (2007)

    Google Scholar 

  23. Gerstner, T., Griebel, M., Holtz, M.: Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance. Insurance: Mathematics and Economics 44, 434–446 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gerstner, T., Griebel, M., Holtz, M., Goschnick, R., Haep, M.: A general asset-liability management model for the efficient simulation of portfolios of life insurance policies. Insurance: Mathematics and Economics 42(2), 704–716 (2008)

    Article  MATH  Google Scholar 

  25. Gerstner, T., Griebel, M., Holtz, M., Goschnick, R., Haep, M.: Numerical simulation for asset-liability management in life insurance. In: H.J. Krebs, W. Jäger (eds.) Mathematics – Key Technology for the Future, Part 6, pp. 319–341. Springer (2008)

    Google Scholar 

  26. Gerstner, T., Holtz, M.: Geometric tools for the valuation of performance-dependent options. In: M. Costantino, C. Brebbia (eds.) Computational Finance and its Applications II, pp. 161–170. WIT Press, London (2006)

    Chapter  Google Scholar 

  27. Gerstner, T., Holtz, M.: The orthant decomposition of hyperplane arrangements. Working paper, University Bonn (2006)

    Google Scholar 

  28. Gerstner, T., Holtz, M.: Valuation of performance-dependent options. Appl. Math. Finance 15(1), 1–20 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Gerstner, T., Holtz, M., Korn, R.: Valuation of performance-dependent options in a Black-Scholes framework. In: J. Appleby, D. Edelman, J. Miller (eds.) Numerical Methods for Finance, Financial Mathematics Series, vol. 8, pp. 203–214. Chapman & Hall/CRC (2007)

    Google Scholar 

  30. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer (2003)

    Google Scholar 

  31. Goecke, O.: Über die Fähigkeit eines Lebensversicherers Kapitalmarktrisiken zu transfor-mieren. Blätter der DGVFM 2, 207–227 (2003)

    Article  Google Scholar 

  32. Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complexity (2010). Appeared online.

    Google Scholar 

  33. Grosen, A., Jorgensen, P.: Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options and bonus policies. Insurance: Mathematics and Economics 26(1), 37–57 (2000)

    Article  MATH  Google Scholar 

  34. Grosen, A., Jorgensen, P.: Life insurance liabilities at market value: An analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework. J. Risk and Insurance 69(1), 63–91 (2002)

    Article  Google Scholar 

  35. Harrison, J., Pliska, S.: Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11, 215–260 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hull, J.: Options, Futures and other Derivative Securities. Prentice Hall, Upper Saddle River (2000)

    Google Scholar 

  37. Hürlimann, W.: Fair pricing using deflators and decrement copulas: The unit linked endowment approach. Blätter der DGVFM 3(XXVI), 421–437 (2004)

    Article  Google Scholar 

  38. Imai, J., Tan, K.: A general dimension reduction technique for derivative pricing. J. Comp. Finance 10(2), 129–155 (2006)

    Google Scholar 

  39. Jaquemod, R.: Abschlussbericht der DAV-Arbeitsgruppe Stochastisches Unternehmensmo-dell für deutsche Lebensversicherungen. Schriftenreihe angewandte Versicherungsmathematik.Versicherungswirtschaft (2005)

    Google Scholar 

  40. Jorgensen, P.: On accounting standards and fair valuation of life insurance and pension liabilities. Scand. Actuarial J. 5, 372–394 (2004)

    Google Scholar 

  41. Karatzas, I.: Lectures on the Mathematics of Finance, CRM Monograph Series, vol. 8. American Mathematical Society (1997)

    Google Scholar 

  42. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998)

    MATH  Google Scholar 

  43. Kling, A., Richter, A., Russ, J.: The interaction of guarantees, surplus distribution, and asset allocation in with-profit life insurance policies. Insurance: Mathematics and Economics 40(1), 164–178 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Korn, R.: A valuation approach for tailored options (1996). Technical Report, University Kaiserslautern

    Google Scholar 

  45. Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. American Mathematical Society (2001)

    Google Scholar 

  46. Leobacher, G.: Stratified sampling and quasi-Monte Carlo simulation of Lévy processes. Monte Carlo Methods and Applications 12(3-4), 231–238 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Linetsky, V.: The path integral approach to financial modeling and options pricing. Computational Economics 11, 129–163 (1998)

    Article  MATH  Google Scholar 

  48. Miltersen, K., Persson, S.: Guaranteed investment contracts: Distributed and undistributed excess returns. Scand. Actuarial J. 23, 257–279 (2003)

    Article  MathSciNet  Google Scholar 

  49. Moller, T., Steffensen, M.: Market-Valuation Methods in Life and Pension Insurance. International Series on Actual Science. Cambridge University Press (2007)

    Google Scholar 

  50. Moro, B.: The full Monte. RISK 8(2) (1995)

    Google Scholar 

  51. Morokoff, W.: Generating quasi-random paths for stochastic processes. SIAM Review 40, 765–788 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  52. Nahm, T.: Error estimation and index refinement for dimension-adaptive sparse grid quadrature with applications to the computation of path integrals. Master’s thesis, University Bonn (2005)

    Google Scholar 

  53. Ninomiya, S., Tezuka, S.: Toward real-time pricing of complex financial derivatives. Appl. Math. Finance 3, 1–20 (1996)

    Article  MATH  Google Scholar 

  54. Paskov, S., Traub, J.: Faster valuation of financial derivatives. J. Portfolio Management 22, 113–120 (1995)

    Article  Google Scholar 

  55. Petras, K.: Smolyak cubature of given polynomial degree with few nodes for increasing dimension. Numer. Math. 93(4), 729–753 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  56. Redfern, D.: Low discrepancy numbers and their use within the ESG. Research Report, Barrie & Hibbert (2010)

    Google Scholar 

  57. Reisinger, C.: Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2004)

    Google Scholar 

  58. Rodrick, S.: Incentive Compensation and Employee Ownership. National Center for Employee Ownership (2004)

    Google Scholar 

  59. Schervish, M.: Multivariate normal probabilities with error bound. Applied Statistics 33, 81–87 (1984)

    Article  MATH  Google Scholar 

  60. Sloan, I., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14, 1–33 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  61. Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40, 1650 – 1665 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  62. Sullivan, M.: Discrete-time continuous-state interest rate models. J. Economic Dynamics and Control 25(6-7), 1001–1017 (2001)

    Article  MATH  Google Scholar 

  63. Tanskanen, A., Lukkarinen, J.: Fair valuation of path-dependent participating life insurance contracts. Insurance: Mathematics and Economics 33(3), 595–609 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  64. Wang, X.: On the effects of dimension reduction techniques on some high-dimensional problems in finance. Operations Research 54(6), 1063–1078 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  65. Wang, X., Fang, K.T.: The effective dimension and quasi-Monte Carlo integration. J. Complexity 19(2), 101 – 124 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  66. Wang, X., Sloan, I.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput. 27(1), 159 – 183 (2005)

    MATH  MathSciNet  Google Scholar 

  67. Wüthrich, M., Bühlmann, H., Furrer, H.: Market-Consistent Actuarial Valuation. EAA Lecture Notes. Springer (2008)

    Google Scholar 

  68. Zhang, P.: Exotic Options. World Scientific Publishing (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Markus Holtz .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Berlin Heidelberg

About this chapter

Cite this chapter

Holtz, M. (2011). Validation and Applications. In: Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance. Lecture Notes in Computational Science and Engineering, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16004-2_6

Download citation

Publish with us

Policies and ethics