The necessity of more trustworthy methods for measuring the risk (volatility) of financial assets has come to the surface with the global market downturn This dissertation aims to propose sample arc length of a time series, which provides a measure of the overall magnitude of the one-step-ahead changes over the observation time period, as a new approach for quantifying the risk. The Gaussian functional central limit theorem is proven under finite second moment conditions. Without loss of generality we consider equally spaced time series when first differences of the series follow a variety of popular stationary models including autoregressive moving average, generalized auto regressive conditional heteroscedastic, and stochastic volatility. As applications we use CUSUM statistic to identify changepoints in terms of volatility of Dow Jones Index returns from January, 2005 through December, 2009. We also compare asset series to determine if they have different volatility structures when arc length is used as the tool of quantification. The idea is that processes with larger sample arc lengths exhibit larger fluctuations, and hence suggest greater variability.
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