Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

Document Type

Book Chapter

Publication Date

9-2020

Publication Title

Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning

Volume

2

Publisher Name

World Scientific Publishing

Abstract

The purpose of this chapter is to develop certain relatively recent mathematical discoveries known generally as stochastic calculus (or more specifically as Itô’s Calculus) and to illustrate their application in the pricing of options. The topic is motivated by a desire to provide an intuitive understanding of certain probabilistic methods that have found significant use in financial economics. A rigorous presentation of the same ideas is presented briefly in Malliaris and Brock (1982) and more recently, Chalamandaris and Malliaris (2011). Itô’s Calculus was prompted by purely mathematical questions originating in Wiener’s work in 1923 on stochastic integrals and was developed by the Japanese probabilist Kiyosi Itô during 1944–1951. Two decades later economists such as Merton (1973) and Black and Scholes (1973) started using Itô’s stochastic differential equation to describe the behavior of asset prices. Because stochastic calculus is now used regularly by financial economists, some attention must be given to its mathematical meaning, its appropriateness in economic modeling, and its applications in economic modeling, and to finance.

Identifier

ISBN: 978-981-120-239-1

Comments

Electronic version of an article published as Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning Volume 2, 10.1142/11335 data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNgYAAAAAMAASsJTYQAAAAASUVORK5CYII= © World Scientific Publishing Company, 2020, https://doi.org/10.1142/11335

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

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