Historical Vignette
Introduction
There are legions of famous mathematicians, economists, and finance academics whose names appear throughout the literature and the author's texts. How can any example do justice to this history? Indeed, it can not. Depending on one's field and training, different names may come to mind, or may be ranked in different ways. However, we chose one example because it highlights well the interconnection between research in theoretical mathematics, and a foundational application to finance.
Kiyosi Itô (1915–2008)
Kiyosi Itô was a Japanese mathematician in the field of probability theory, who in the 1940s invented the fields of stochastic calculus, also known as Itô calculus, and stochastic differential equations. Grounded in the measure-theoretic field of stochastic processes, it would not be an exaggeration to say that these studies were abstract and firmly embedded in the field of theoretical mathematics, apparently far from anticipated applications.
However, the name Itô is familiar today to every student of quantitative finance, at least for Itô's lemma, and for the more advanced students, Itô processes and Itô calculus as well. While built upon a mountain of theoretical mathematical results in measure theory, stochastic processes can be described intuitively, and then the results of Itô stated and applied, again intuitively, to solve a problem of central importance to quantitative finance, the pricing of European put and call options.
Fischer Black (1938–1995) & Myron Scholes (b. 1941)
The European put and call option pricing problem was solved in a 1973 paper by economics Fischer Black (1938 – 1995) and Myron Scholes (b. 1941), obtaining the now famous Black-Scholes option pricing formula. This formula resulted from their deriving and solving a parabolic partial differential equation for the price of such an option. Once derived, such equations could be solved by methods derived in the field of physics, where the foundational parabolic partial differential equation is known as the heat equation, and approaches to solving such equations date back to the early 1800s.
This partial differential equation was in turn derived by Black and Scholes insight that such an option can be "replicated" by a continuously rebalanced portfolio of risk free assets and the asset underlying the option. Such replication of the option is now known as delta hedging, and provides a fundamental insight in risk management applications today.
Robert C Merton (b. 1944)
The put and call option pricing problem was also solved in another 1973 paper by Robert Merton. Explicitly framing the price of the asset underlying the option as a stochastic process known as geometric Brownian motion, he again derived the parabolic partial differential equation of Black and Scholes for the price of such an option. This derivation again reflected replication of the option with a portfolio of risk free and underlying assets, but now the final differential equation was derived with an application of Itô's lemma.
It is common to today see the Black-Scholes option pricing formula referenced as Black-Scholes-Merton.
Postscript
It is not an overstatement to assert that the replication argument of Black and Scholes, the stochastic process modeling of Merton, and the stochastic calculus results of Itô, have become the foundational tools in many problems in quantitative finance today.
in 1997, Scholes and Merton were awarded the Nobel Memorial Prize in Economic Sciences for this work. Though cited in the press release and award ceremony speech, Black was not formally awarded since such prizes are not given posthumously.