# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [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:

$R_t = \mu + \sigma \epsilon$

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 $(\mu)$ and standard deviation $(\sigma)$. We begin by asking the question how does the mean of returns scale over time?

Assuming that daily returns are composed of $n$ time-steps of size $\Delta t$,  the question we are asking is what can we say about return for a time interval $\Delta t$. Let’s first write ‘a’ daily return as:

$\displaystyle R_{1} = ln \frac{P_t}{P_{t - 1}}$

Then if a day consists of $n$ time-steps of size $\Delta t$, $(n \Delta t = 1)$ we can write:

\displaystyle \begin{aligned} R_{1} &= ln \frac{P_{t}}{P_{t - n \Delta t}}\\ \\&= ln \frac{P_{t}}{P_{t - \Delta t}}\times \frac{P_{t - \Delta t}}{P_{t - 2 \Delta t}}\times\frac{P_{t - 2\Delta t}}{P_{t - 3 \Delta t}}\times\cdots \times \frac{P_{t - (n - 1) \Delta t}}{P_{t - n \Delta t}} \\ \\&= ln \frac{P_{t}}{P_{t - \Delta t}}+ ln \frac{P_{t - \Delta t}}{P_{t - 2\Delta t}}+ ln \frac{P_{t - 2 \Delta t}}{P_{t - 3\Delta t}}+ \cdots +ln \frac{P_{t - (n - 1) \Delta t}}{P_{t - n\Delta t}}\\ \\&=R_{\Delta t} + R_{\Delta t} + R_{\Delta t} + \cdots +R_{\Delta t} \\ \\&= n R_{\Delta t} \\ \\ \Rightarrow R_{\Delta t} &= \frac{1}{n} R_1 \\ \\ \mbox{or } R_{\Delta t} &= \Delta t R_1\end{aligned}

where we have assumed that returns over any time-step $\Delta t$ are ‘identical’ over time, i.e. return over time $t - \Delta t$  to $t$ behaves ‘similarly’ to the return over time $t - (n - 1) \Delta t$ and $t - n \Delta t$ and so on (technically speaking, statistically stationary).

Taking expectations on both sides then gives:

\begin{aligned} E[R_{\Delta t}] &= E[\Delta t R_1] \\ \\&= \Delta t E[R_1] \\ \\&= \mu \Delta t \end{aligned}

i.e. mean of returns over time-step $\Delta t$ behaves as $\mu \Delta t$. Now to the variance/standard deviation.

Given the approximation $R_1 = R_{\Delta t}+ R_{\Delta t}+R_{\Delta t}+ \cdots +R_{\Delta t}$, and assuming that returns are uncorrelated over different time-steps (again, a property of a stationary process), variance over time-step $\Delta t$ is easily derived via:

\begin{aligned} Var[R_1] &= Var[R_{\Delta t}]+ Var[R_{\Delta t}]+ Var[R_{\Delta t}]+ \cdots +Var[R_{\Delta t}] \\ \\&= n \times Var[R_{\Delta t}]\\ \\ \Rightarrow Var[R_{\Delta t}] &= \frac{1}{n} Var[R_1] \\ \\&= \Delta t \sigma^2 \end{aligned}

i.e. variance of returns over time-step $\Delta t$ behaves as $\Delta t$, or that standard deviation of returns over time-step $\Delta t$ behaves as $\sqrt{\Delta t}$.

Now we are ready to write our first stock returns model for any time-step $\Delta t$ as:

$R_{\Delta t} = \mu \Delta + \sigma \epsilon \sqrt{\Delta t}$

or in terms of stock prices

\displaystyle \begin{aligned} \frac{S_{t + \Delta t} - S_t}{S_t} &= \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} \\ \\ \mbox{or } S_{t + \Delta t} &= \mu S_t \Delta t + \sigma S_t \epsilon \sqrt{\Delta t} \end{aligned}

or alternatively, in a more familiar form as:

$\boxed{\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}}$