Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

[PDS] Modeling Stock Returns: II

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(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}}

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Written by Vineet

January 17, 2013 at 12:40 am

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