Advertisement

Notions of Mathematical Finance

  • Norbert Hilber
  • Oleg Reichmann
  • Christoph Schwab
  • Christoph Winter
Part of the Springer Finance book series (FINANCE)

Abstract

The present notes deal with topics of computational finance, with focus on the analysis and implementation of numerical schemes for pricing derivative contracts. There are two broad groups of numerical schemes for pricing: stochastic (Monte Carlo) type methods and deterministic methods based on the numerical solution of the Fokker–Planck (or Kolmogorov) partial integro-differential equations for the price process. Here, we focus on the latter class of methods and address finite difference and finite element methods for the most basic types of contracts for a number of stochastic models for the log returns of risky assets. We cover both, models with (almost surely) continuous sample paths as well as models which are based on price processes with jumps. Even though emphasis will be placed on the (partial integro)differential equation approach, some background information on the market models and on the derivation of these models will be useful particularly for readers with a background in numerical analysis.

Keywords

Stochastic Differential Equation Option Price Risky Asset Price Process Geometric Brownian Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 4.
    L. Bachelier. Théorie de la spéculation. Ann. Sci. Éc. Norm. Super., 17:21–86, 1900. MathSciNetzbMATHGoogle Scholar
  2. 40.
    R. Cont and P. Tankov. Financial modelling with jump processes. Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, 2004. zbMATHGoogle Scholar
  3. 53.
    F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of asset pricing. Math. Ann., 300(3):463–520, 1994. MathSciNetzbMATHCrossRefGoogle Scholar
  4. 54.
    F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann., 312(2):215–250, 1998. MathSciNetzbMATHCrossRefGoogle Scholar
  5. 55.
    F. Delbaen and W. Schachermayer. The mathematics of arbitrage. Springer Finance, 2nd edition. Springer, Heidelberg, 2008. Google Scholar
  6. 71.
    I.I. Gihman and A.V. Skorohod. The theory of stochastic processes I. Grundlehren Math. Wiss. Springer, New York, 1974. zbMATHCrossRefGoogle Scholar
  7. 72.
    I.I. Gihman and A.V. Skorohod. The theory of stochastic processes II. Grundlehren Math. Wiss. Springer, New York, 1975. zbMATHCrossRefGoogle Scholar
  8. 73.
    I.I. Gihman and A.V. Skorohod. The theory of stochastic processes III. Grundlehren Math. Wiss. Springer, New York, 1979. zbMATHCrossRefGoogle Scholar
  9. 97.
    J. Jacod and A. Shiryaev. Limit theorems for stochastic processes, 2nd edition. Springer, Heidelberg, 2003. zbMATHCrossRefGoogle Scholar
  10. 109.
    D. Lamberton and B. Lapeyre. Introduction to stochastic calculus applied to finance. Chapman & Hall/CRC Financial Mathematics Series, 2nd edition. Chapman & Hall/CRC, Boca Raton, 2008. zbMATHGoogle Scholar
  11. 120.
    X. Mao. Stochastic differential equations and applications, 2nd edition. Horwood Publishing, Chichester, 2007. zbMATHGoogle Scholar
  12. 131.
    B. Øksendal. Stochastic differential equations: an introduction with applications. Universitext, 6th edition. Springer, Berlin, 2003. Google Scholar
  13. 132.
    P.E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability, 2nd edition. Springer, Berlin, 2005. CrossRefGoogle Scholar
  14. 152.
    A.N. Shiryaev. Essentials of stochastic finance: facts, models, theory. World Scientific, Singapore, 2003. Google Scholar
  15. 161.
    P. Wilmott, S. Howison, and J. Dewynne. The mathematics of financial derivatives: a student introduction. Cambridge University Press, Cambridge, 1993. Google Scholar
  16. 162.
    P. Wilmott, S. Howison, and J. Dewynne. Option pricing: mathematical models and computation. Oxford Financial Press, Oxford, 1995. zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Norbert Hilber
    • 1
  • Oleg Reichmann
    • 2
  • Christoph Schwab
    • 2
  • Christoph Winter
    • 3
  1. 1.Dept. for Banking, Finance, Insurance, School of Management and LawZurich University of Applied SciencesWinterthurSwitzerland
  2. 2.Seminar for Applied MathematicsSwiss Federal Institute of Technology (ETH)ZurichSwitzerland
  3. 3.Allianz Deutschland AGMunichGermany

Personalised recommendations