Ito calculus and the Black-Scholes equation

After having discussed at length the peculiarities of the continuous time Gaussian limit in the previous chapters, it is interesting to describe how option pricing theory is usually introduced using Ito's stochastic differential calculus. This framework is extremely powerful and allows one to obtain rather quickly some of the results that were painfully derived in the previous chapters, such as the optimal hedge or the independence of the price on the drift. Once the limitations of the Black-Scholes model and the subtleties of the continuous limit are fully understood, the ‘blind’ use of Ito calculus to obtain a first approximation to the price of derivatives becomes justified. However, we are convinced that in order to go beyond Black-Scholes and study more realistic models, one should be prepared to abandon Ito calculus and the world of partial differential equations on which most of mathematical finance relies.

The Gaussian Bachelier model

We first assume that the price itself (and not its logarithm) follows a continuous time random walk, with drift m and diffusion constant D. We also assume that the interest rate r is equal to zero.