Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model
Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning
World Scientific Publishing
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.
Chalamandaris, G and Malliaris, A. (Tassos) G.. Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model. Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning, 2, : , 2020. Retrieved from Loyola eCommons, School of Business: Faculty Publications and Other Works, http://dx.doi.org/10.1142/11335
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