In this section, we provide an introduction to what is known as stochastic calculus. Our goal is not to achieve a working knowledge in the subject, but rather to provide context for some of the terminology and the formulas encountered in the literature on modeling asset prices with random walks.
So far, we discussed random walks for which every step is taken at a specific discrete point in time. When the time increments are very small, almost zero in length, the equation of a random walk describes a stochastic process in continuous time. In this context, the arithmetic random walk model is known as a generalized Wiener process or Brownian motion (BM). The geometric random walk is referred to as geometric Brownian motion (GBM), and the arithmetic mean-reverting walk is the Ornstein-Uhlenbeck process mentioned earlier.
Special notation is used to denote stochastic processes in continuous time. Increments are denoted by d
or Δ. (For example, (St+1
) is denoted dSt
, meaning a change in St
over an infinitely small interval.) The equations describing the process, however, have a very similar form to the equations we introduced earlier in this section:
Equations involving small changes (“differences”) in variables are referred to as differential equations. In words, the equation above reads: “The change in the price St over a small time period dt equals the average drift µ multiplied by the small time change plus a random term equal to ...