Itô Stochastic Calculus

Abstract

One of the first applications of the Wiener process was proposed by Bachelier, who around 1900 wrote a ground-breaking paper on the modelling of asset prices at the Paris Stock Exchange. Of course Bachelier could not have called it the Wiener process, but he used what in modern terminology amounts to W(t) as a description of the market fluctuations affecting the price X(t) of an asset. Namely, he assumed that infinitesimal price increments dX(t) are proportional to the increments dW(t) of the Wiener process, dX(t) = σdW(t), where σ is a positive constant. As a result, an asset with initial price X(0) = x would be worth X(t) = x + σW(t) at time t. This approach was ahead of Bachelier’s time, but it suffered from one serious flaw: for any t > 0 the price X(t) can be negative with non-zero probability. Nevertheless, for short times it works well enough, since the probability is negligible. But as t increases, so does the probability that X(t) < 0, and the model departs from reality.