Daily Returns and the Black Scholes Model

The Daily Return:

Let Sn be the stock index at the end of the n-th trading day. Then the
daily return of the stock is defined by

i.e. the relative change. If we take the log-return instead

we get an even easier to handle quantity since the log return over k days
is simply the sum over the daily returns

The Desired Properties For Our Model:

Under the assumptions that the log returns of disjunct equidistant time intervals
are independently and identically distributed we can use the central limit
theorem of probability theory to state:

Log-returns seen as a sum of a large number of independent identically
distributed random variables with finite variance are approximately normally

Therefore we are looking for a stochastic market model, defined in
continuous time where the log-returns are normally distributed for arbitrary
time intervals. Let St be the stock price at time t. Bachelier modelled the stock price
with a Brownian motion with the disadvantage that the stock price within
this model can be negative. Samuelson modified the model using the geometric
Brownian motion for the dynamics of the stock price.

We denote μ  as the expected return rate (drift) and σ as the
volatility. The process described above is a special case of the Ito process with general μ and σ.