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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Bachelier. Théorie de la spéculation. Ann. Sci. Éc. Norm. Super., 17:21–86, 1900.
R. Cont and P. Tankov. Financial modelling with jump processes. Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, 2004.
F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of asset pricing. Math. Ann., 300(3):463–520, 1994.
F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann., 312(2):215–250, 1998.
F. Delbaen and W. Schachermayer. The mathematics of arbitrage. Springer Finance, 2nd edition. Springer, Heidelberg, 2008.
I.I. Gihman and A.V. Skorohod. The theory of stochastic processes I. Grundlehren Math. Wiss. Springer, New York, 1974.
I.I. Gihman and A.V. Skorohod. The theory of stochastic processes II. Grundlehren Math. Wiss. Springer, New York, 1975.
I.I. Gihman and A.V. Skorohod. The theory of stochastic processes III. Grundlehren Math. Wiss. Springer, New York, 1979.
J. Jacod and A. Shiryaev. Limit theorems for stochastic processes, 2nd edition. Springer, Heidelberg, 2003.
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.
X. Mao. Stochastic differential equations and applications, 2nd edition. Horwood Publishing, Chichester, 2007.
B. Øksendal. Stochastic differential equations: an introduction with applications. Universitext, 6th edition. Springer, Berlin, 2003.
P.E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability, 2nd edition. Springer, Berlin, 2005.
A.N. Shiryaev. Essentials of stochastic finance: facts, models, theory. World Scientific, Singapore, 2003.
P. Wilmott, S. Howison, and J. Dewynne. The mathematics of financial derivatives: a student introduction. Cambridge University Press, Cambridge, 1993.
P. Wilmott, S. Howison, and J. Dewynne. Option pricing: mathematical models and computation. Oxford Financial Press, Oxford, 1995.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hilber, N., Reichmann, O., Schwab, C., Winter, C. (2013). Notions of Mathematical Finance. In: Computational Methods for Quantitative Finance. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35401-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-35401-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35400-7
Online ISBN: 978-3-642-35401-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)