Skip to main content
SpringerLink
Log in

Calculus and “Digitalization” in Finance: Change of Time Method and Stochastic Taylor Expansion with Computation of Expectation

  • Conference paper
  • First Online:
Modeling, Dynamics, Optimization and Bioeconomics I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 73))

Abstract

In this chapter, we give a review and new interpretations on the solution of stochastic differential equation by using the change of time method and the numerical solution with stochastic Taylor expansion. Firstly, a random time change is analyzed. Time change is one of the standard tools for building financial models. The process can be done by a subordinator or an absolutely continuous time change (CTM). The main results of CTM are covered in this chapter. It is applied on one of the important financial problems: Heston model, and to variance and volatility swap as well. In the second part, we focus on numerical simulation of stochastic differential equations arising from the stochastic Taylor series expansion. In order to get more accurate discrete schemes, more terms are added to the Taylor expansion. Application of Itô formula iteratively gives rise to multiple Itô integrals. As for the order of the numerical scheme, the expectations of the product of multiple Itô integrals are to be computed. We discuss the formula for the expectations of the products of Itô integrals. Throughout the chapter, we pay extra attention to the interplay between states and time, and to a “digitalization” of algebraic operations.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Institutional subscriptions

References

  1. Andreas, E.: Stochastic Analysis. Lecture Notes, University of Bonn (2012). http://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/StoAn1213/StochasticAnalysisNotes1213.pdf

  2. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  3. Burrage, P.M.: Runge-Kutta methods for stochastic differential equations. Ph.D. thesis, Department of Mathematics, University of Queensland, Australia (1999)

    Google Scholar 

  4. Burrage, K., Burrage, P.M.: High strong order methods for non-commutative stochastic ordinary differential equation system and the Magnus formula. Phys. D 133, 34–48 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal. 38, 1626–1646 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burrage, K., Burrage, P.M.: Numerical Methods for Stochastic Differential Equations. Fields Institute, Toronto (2001)

    Google Scholar 

  7. Burrage, K., Burrage, P.M., Tian, T.H.: Numerical methods for solving stochastic differential equations on parallel computers. In: Proceedings of the 5th International Conference on High-Performance Computing in the Asia-Pacific Region (2001)

    Google Scholar 

  8. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Wiley, Chichester (1987)

    MATH  Google Scholar 

  9. Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Finance 13(3), 345–382 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Demeterfi, K., Derman, E., Kamal, M., Zou, J.: A Guide to volatility and variance swaps. J. Derivatives Summer 6(4), 9–32 (1999)

    Article  Google Scholar 

  12. Doeblin, W., Yor, M.: Sur l’équation de Kolmogoroff, vol. 331. Pli Cachetè à l’Académie des Sciences, Paris (2000)

    Google Scholar 

  13. Elliott, R., Swishchuk, A.: A pricing options and volatility swaps in Markov-modulated Brownian and fractional brownian markets. In: RJE Conference, Calgary (2005)

    Google Scholar 

  14. Jacod, J.: Calcul Stochastique et Problemés de Martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)

    Google Scholar 

  15. Johnson, H.E., Shanno, D.: Option pricing when the variance is changing. J. Financ. Quant. Anal. 22, 143–151 (1987)

    Article  Google Scholar 

  16. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (2010)

    Google Scholar 

  17. Kilic, E., Karimov, A. Weber, G.-W.: Applications of stochastic hybrid systems in portfolio optimization. In: Thomaidis, N., Dash, G.H., Jr. (eds.) Recent Advances in Computational Finance, Chap. 5, pp. 75–98. Nova Science Publishers, Inc., New York (2013)

    Google Scholar 

  18. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  19. Kobayashi, K.: Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J. Theor. Probab. 24(3), 789–820 (2010)

    Article  Google Scholar 

  20. Komori, Y., Sugiura, H., Mitsui, T.: Rooted Tree Analysis of the Order Conditions of ROW-Type Schemes for Stochastic Differential Equations, Report No. 7, Mathematical Sciences, Nagoya University, Nagoya (1994)

    Google Scholar 

  21. Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol 24. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  22. Mele A.: Lectures on Financial Mathematics. Lecture Notes, University of Lugano, Swiss-Finance-Institute (2012). http://www.antoniomele.org/files/fin_eco.pdf

  23. Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht/Boston/London (1995)

    Book  Google Scholar 

  24. Movellan, J.R.: Tutorial on Stochastic Differential Equations. MPLab Tutorials Version 06.1 (2011)

    Google Scholar 

  25. Platen, E.: An introduction to numerical methods for stochastic differential equations. Acta Numerica 8, 197–246 (1999)

    Article  MathSciNet  Google Scholar 

  26. Platen, E., Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, vol. 24. North Holland Publ. Co., Amsterdam, 480 S., Dfl. 175 (1982)

    Google Scholar 

  27. Revuz, D., Yor, M.: Continuous Martingale and Brownian Motion. Springer, Berlin (2005)

    Google Scholar 

  28. Rümelin, W.: Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19, 604–613 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  29. Stele, J.M.: Stochastic Calculus and Financial Applications. Application of Mathematics, vol. 45. Springer, New York (2001)

    Google Scholar 

  30. Swishchuk, A.: Change of Time Method: Application to Mathematical Finance l, Math & Comp. Finance Lab, Department of Mathematics and Statistic, U of C “Lunch at the Lab” Talk (2005)

    Google Scholar 

Download references

Acknowledgements

The authors of this chapter would like to express their gratitude to the editors of that book, Prof. Dr. D. Zilberman, Prof. Dr. A. Pinto, for giving them the opportunity to introduce their representations and reflections to the readers.

Author information

Authors and Affiliations

  1. Gazi University, Ankara, Turkey

    Fikriye Yılmaz

  2. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Hacer Öz & Gerhard-Wilhelm Weber

Authors
  1. Fikriye Yılmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hacer Öz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gerhard-Wilhelm Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fikriye Yılmaz .

Editor information

Editors and Affiliations

  1. University of Porto, Department of Mathematics, Porto, Portugal

    Alberto Adrego Pinto

  2. Univ. of California, Dept. of Agricultural and Resource Economics, Berkeley, California, USA

    David Zilberman

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Yılmaz, F., Öz, H., Weber, GW. (2014). Calculus and “Digitalization” in Finance: Change of Time Method and Stochastic Taylor Expansion with Computation of Expectation. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_42

Download citation

Publish with us

Policies and ethics

Search

Navigation