Actuarial Finance

15Introduction to stochastic calculus***

Usually, in most textbooks and research papers, the evolution of the stock price in the Black-Scholes-Merton (BSM) model is given by a so-called stochastic differential equation:

(see also equation (17.1.8) in Chapter 17). This last equation has an equivalent stochastic integral form:

In other words, in the BSM framework, the dynamic of the stock price S is the sum of two random integrals: the integral of the process {μS_t, t ⩾ 0} with respect to the time variable and an integral of the process {σS_t, t ⩾ 0} with respect to Brownian motion.

Before making sense of these integrals, recall that we mentioned in Chapter 14 that the stock price in the BSM framework is represented by a geometric Brownian motion. Therefore, the latter two representations, in terms of a stochastic differential equation and of the sum of random integrals, should be two equivalent representations of the same stochastic process, namely a GBM. To better understand these concepts, we will provide below a heuristic introduction to stochastic calculus.

Stochastic calculus is a set of tools, in the field of probability theory, to work with continuous-time stochastic processes, just like we do with functions in classical differential and integral calculus. One of the main ...