Suppose I have a model of stock prices developed using Brownian motion. I have a second time series derived from the first. At each price point in the first time series, I take the arithmetic mean up to that point and that is the data point for my second time series. The volatility of the second time series is lower because the arithmetic mean has a dampening effect. Are there any statistical tools to model the volatility of the second time series?
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Let's say your stock price series is $S_t$ so that your arithmetic average series is $X_t = \frac{1}{t}\sum_{j=1}^t S_j$. The series of returns that you use to calculate the volatility can be defined in several ways, the most popular being the log-returns series $R_t = \frac{1}{\delta t} \log \frac{S_t}{S_{t-1}}$ and the simple returns series $R_t = \frac{1}{\delta t} \left( \frac{S_t}{S_{t-1}} - 1 \right)$ where $\delta t$ is the time between observations (measured as a fraction of a year). Both of these involve the ratio $S_i / S_{i-1}$, so to look at the volatility of the series $X$ we should calculate this ratio: $\frac{X_t}{X_{t-1}} = \frac{t-1}{t} \left( 1 + \frac{S_t}{S_1+\cdots+S_{t-1}} \right)$ Either way you cut it, the returns series for $X$ is not going to be pretty. The most straightforward approach is to look at the simple returns series for $X$, let's call it $Q$: $Q_t = \frac{1}{\delta t} \frac{t-1}{t} \left( \frac{S_t}{S_1+\cdots+S_{t-1}} - \frac{1}{t-1} \right)$ The volatility of the series $X$ is the standard deviation of this series. Note that even if the volatility of $S$ is constant, the volatility of $X$ won't be constant, because of the dependence of the series on $t$. In fact, under some mild assumptions on the nature of the series $S$ (eg bounded returns) you should be able to show that the volatility of $X$ decreases to zero as $t\to\infty$. This doesn't fully answer your question, but it points out some of the difficulties involved in what you're trying to do. Perhaps a better idea (depending on what you're trying to achieve) is to take a moving average of your stock price series, taking the arithmetic average of the previous $T$ observations. At least this way your averaged series won't have an explicit dependence on the time index $t$. You might also consider taking the geometric mean rather than the arithmetic mean. Although conceptually less intuitive, it has the advantage that it is equivalent to taking the arithmetic average of the log-returns series, which greatly simplifies the algebra later. |
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