## [PDS] Modeling Stock Returns: II

(The discussion below borrows the general idea from Wilmott (2006), Ch 3.)

While we have approximated the scaled asset returns process by draws from a standard Gaussian distribution:

this is not a model yet. Why? Because, for it to be a model that could be used to study stock returns we should be able to say something about what happens to the returns *over a time period. *There is no time dimension explicit in the above approximation. So what we need to do is suitably modify it so that we can describe asset returns over a time interval.

Given the above approximation, it is clear that the only place time interval can play a role is in the scaling of mean and standard deviation . We begin by asking the question how does the mean of returns scale over time?

Assuming that daily returns are composed of time-steps of size , the question we are asking is what can we say about return for a time interval . Let’s first write ‘a’ daily return as:

Then if a day consists of time-steps of size , we can write:

where we have assumed that returns over any time-step are ‘identical’ over time, i.e. return over time to behaves ‘similarly’ to the return over time and and so on (technically speaking, statistically stationary).

Taking expectations on both sides then gives:

i.e. mean of returns over time-step behaves as . Now to the variance/standard deviation.

Given the approximation , and assuming that returns are uncorrelated over different time-steps (again, a property of a stationary process), variance over time-step is easily derived via:

i.e. variance of returns over time-step behaves as , or that standard deviation of returns over time-step behaves as .

Now we are ready to write our **first stock returns model** for any time-step as:

or in terms of stock prices

or alternatively, in a more familiar form as:

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