The Black-Scholes Equation

The most important application of the Itô calculus, derived from the Itô lemma, in financial mathematics is the pricing of options. The most famous result in this area is the Black-Scholes formulae for pricing European vanilla call and put options. As a consequence of the formulae, both in theoretical and practical applications, Robert Merton and Myron Scholes were awarded the Nobel Prize for Economics in 1997 to honour their contributions to option pricing. Unfortunately, Fischer Black, who has also given his name and contributions, had passed away two years before. In their famous work, in 1973, Black and Scholes transformed the option pricing problem into the task of solving a (parabolic) partial differential equation (PDE) with a final condition. The main conceptual idea of Black and Scholes lies in the construction of a riskless portfolio taking positions in bonds (cash), option, and the underlying stock. Such an approach strengthens the use of the no-arbitrage principle as well. Derivation of a closed-form solution to the Black-Scholes equation depends on the fundamental solution of the heat equation. Hence, it is important, at this point, to transform the Black-Scholes equation to the heat equation by change of variables. Having found the closed-form solution to the heat equation, it is possible to transform it back to find the corresponding solution of the Black-Scholes PDE. The connection between an initial and/or boundary value problem for differential equations, the so-called a Cauchy problem, and the computation of the expected value of a functional of a solution of an SDE is covered by the Feynman-Kac representation theorem. However, we leave it to interested readers, but apply the celebrated closed-form solutions to various examples. Indeed, an important consequence of these closed-form solutions is the use of the Greeks: the partial derivatives of the value of an option with