The constant volatility random walk with normal innovations

A random walk, with constant volatility is given by

x (t + δt) = x (t) + r[δt](t + δt) r [δt](t + δt) = σ ∞ ϵ(t)
where x is the logarithm of the price. The random innovations ϵ(t) (or the residues) are random variables with an independent and identical distribuion (iid), and with E[ϵ] = 0, E[ϵ2] = 1.

The most common choice for p(ϵ) is a Gaussian distribution, since this choice allows to build a continuum model for δt → 0. This is the model initiated by Bachelier Bachelier [1900], and used extensively in finance. In particular, in the option pricing theory of Black and Scholes [1973] and Merton [1973], this model is used to describe the dynamics of the underlying asset.

The parameter for the simulations is:

The simulation time corresponds to 200 years with a time increment δt = 3 minutes.

References

   Louis Bachelier. Théorie de la spéculation. Annales de l’Ecole Normale Supérieure, 17, 1900.

   Fischer Black and Myron Scholes. The pricing of option and corporate liabilities. Journal of Political Economy, 81:637–659, 5 1973.

   Robert C. Merton. Theory of rational option pricing. Bell Journal of Economics and Management Science, 4:141–183, 4 1973.