Skip to main content
SpringerLink
Log in
Book cover

Affine Diffusions and Related Processes: Simulation, Theory and Applications pp 1–36Cite as

Real Valued Affine Diffusions

  • Chapter

Part of the book series: Bocconi & Springer Series ((BS,volume 6))

Abstract

This chapter gives a first contact with general affine diffusions by presenting the ones that take real values. We will see that these diffusions are basically of two types, and are either a Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process. Thus, the two first sections of this chapter study these processes and present their main properties. The third section defines what are affine diffusions and characterize them by the mean of the infinitesimal generator. The last section is devoted to the application of these processes for the interest rate modelling. A quick introduction is given on the financial framework, and we present the main pricing formulas that have made the use of these processes popular.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
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

References

  1. Andersen, L., Piterbarg, V.: Interest Rate Modeling. Atlantic Financial Press (2012)

    Google Scholar 

  2. Andersen, L.B.G., Piterbarg, V.V.: Moment explosions in stochastic volatility models. Financ. Stoch. 11(1), 29–50 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baldeaux, J., Platen, E.: Functionals of Multidimensional Diffusions with Applications to Finance. B&SS – Bocconi & Springer Series, vol. 5. Springer International Publishing Switzerland, Cham (2013)

    Google Scholar 

  4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    Article  MATH  Google Scholar 

  5. Brigo, D., Mercurio, F.: Interest Rate Models—Theory and Practice. With Smile, Inflation and Credit, 2nd edn. Springer Finance. Springer, Berlin (2006)

    Google Scholar 

  6. Cox, J.C.: Ingersoll, J.E. Jr., Ross, S.A.: An intertemporal general equilibrium model of asset prices. Econometrica 53(2), 363–384 (1985)

    Google Scholar 

  7. Cox, J.C.: Ingersoll, J.E. Jr., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53(2), 385–407 (1985)

    Google Scholar 

  8. Craddock, M., Lennox, K.A.: The calculation of expectations for classes of diffusion processes by Lie symmetry methods. Ann. Appl. Probab. 19(1), 127–157 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin (2006)

    MATH  Google Scholar 

  11. Doss, H., Lenglart, E.: Sur l’existence, l’unicité et le comportement asymptotique des solutions d’équations différentielles stochastiques. Ann. Inst. H. Poincaré Sect. B (N.S.) 14(2), 189–214 (1978)

    Google Scholar 

  12. Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Feller, W.: Two singular diffusion problems. Ann. Math. 54(1), 173–182 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  14. Filipović, D.: Term-Structure Models. A Graduate Course. Springer Finance. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  15. Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hurd, T.R., Kuznetsov, A.: Explicit formulas for Laplace transforms of stochastic integrals. Markov Process. Relat. Fields 14(2), 277–290 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, vol. 24, 2nd edn. North-Holland/Kodansha, Amsterdam/Tokyo (1989)

    Google Scholar 

  18. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)

    Google Scholar 

  19. Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall/CRC Financial Mathematics Series, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL (2008)

    Google Scholar 

  20. Lord, R., Kahl, C.: Why the Rotation Count Algorithm Works. SSRN eLibrary (2006)

    Google Scholar 

  21. Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973)

    Article  MathSciNet  Google Scholar 

  22. Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5(2), 177–188 (1977)

    Article  Google Scholar 

  23. Widder, D.V.: The Laplace Transform. Princeton Mathematical Series, vol. 6. Princeton University Press, Princeton, NJ (1941)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CERMICS, Ecole Nationale des Ponts et Chaussées, Champs-sur-Marne, France

    Aurélien Alfonsi

Authors
  1. Aurélien Alfonsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Alfonsi, A. (2015). Real Valued Affine Diffusions. In: Affine Diffusions and Related Processes: Simulation, Theory and Applications. Bocconi & Springer Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-05221-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation