lognormal process
A lognormal process is a diffusion of a quantity Q (stock price, fx rate or something else) that starts at time t = 0 with some known initial known value Q₀ (referred to as spot price) and thereafter evolves according to the SDE (Stochastic Differential Equation):
dQ = μ(t)Qdt + σ(t)Qdw
where:
μ(t) is a deterministic function of time t known as drift (or more properly as log drift to account for the presence of the multiplier Q).
σ(t) is a deterministic function of time t known as vol (or more properly as log vol to account for the presence of the multiplier Q).
dt is a formal mathematical symbol pertaining to the determistic independent time variable of the SDE, but it may be regarded in a heuristic sense to represent an infinitesimal (very small) increment of the time t
dw is a formal mathematical symbol pertaining to the stochastic Brownian variable of the SDE, but it may be regarded in a heuristic sense to represent a random noise amount generated during the time lapse dt
The word random means that for any given time lapse dt, the corresponding dw is a random number and therefore not uniquely determined.
As an example, consider a simulation of the evolution of a stock price Q described by an SDE where μ(t) = 0.1, σ(t) = 0.2 and Q₀ = 100
Assume the simulation moves in time steps of one day each, i.e. dt = 1/365 in annual units.
By definition, at time t = 0, the stock price Q equals Q₀ = 100
At the next simulation time t = dt, the stock price Q will equal Q₀ + dQ = 100 + dQ
with
dQ = μ(t)Qdt + σ(t)Qdw = 0.1 * 100 * 1/365 + 0.2 * 100 * dw
As mentioned, dw is random and - more precisely - is a normally distributed random variable with expectation 0 and variance equal to dt = 1/365, i.e. with standard deviation equal to (1/365)¹ᐟ² ≈ 0.052
It turns out that there is not a single quantity dQ that needs to be calculated, but rather several - in theory infinite many - such quantities corresponding to all possible dw values.
While dw may attain any possible value, most likely are the values around 0 and within the standard deviation of 0.052, with the usual bell curve describing the exact likelihood of occurrence.
The universe of dw values lead to a corresponding universe of dQ values.
For example, the most likely value dw = 0 leads to dQ = 0.1 * 100 * 1/365 + 0.2 * 100 * 0 = 0.1 * 100 * 1/365 ≈ 0.027 and therefore to Q ≈ 100 + 0.027 = 100.027
The less likely but still frequently occurring value dw = 0.052 leads to dQ = 0.1 * 100 * 1/365 + 0.2 * 100 * 0.052 ≈ 0.027 + 1.04 ≈ 1.067 and therefore to Q ≈ 100 + 1.067 = 101.067
Proceeding similarly, a computer algorithm can easily generate a chart with several possible Q values along their occurrence frequencies, i.e. a probability distribution of Q as of t = 1/365