Simplified Stochastic Calculus With Applications in Economics and Finance

European Journal of Operational Research

30 Pages Posted: 23 Dec 2019 Last revised: 4 Jan 2021

See all articles by Aleš Černý

Aleš Černý

Bayes Business School, City, University of London

Johannes Ruf

London School of Economics & Political Science (LSE) - London School of Economics

Date Written: December 8, 2019

Abstract

The paper introduces a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically present). The calculus is fail-safe in that, under minimal assumptions, all informal calculations yield mathematically well-defined stochastic processes. The calculus is also intuitive as it allows the user to pretend all jumps are of compound Poisson type. The new calculus is very effective when it comes to computing drifts and expected values that possibly involve a change of measure. Such drift calculations yield, for example, partial integro-differential equations, Hamilton–Jacobi–Bellman equations, Feynman–Kac formulae, or exponential moments needed in numerous applications. We provide several illustrations of the new technique, among them a novel result on the Margrabe option to exchange one defaultable asset for another.

Keywords: drift, Émery formula, Girsanov's theorem, simplified stochastic calculus

JEL Classification: C60, G11, G13

Suggested Citation

Černý, Aleš and Ruf, Johannes, Simplified Stochastic Calculus With Applications in Economics and Finance (December 8, 2019). European Journal of Operational Research , Available at SSRN: https://ssrn.com/abstract=3500384 or http://dx.doi.org/10.2139/ssrn.3500384

Aleš Černý (Contact Author)

Bayes Business School, City, University of London

Northampton Square
London, EC1V 0HB
United Kingdom

Johannes Ruf

London School of Economics & Political Science (LSE) - London School of Economics ( email )

United Kingdom

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