sigmaT2

sigmaT2 is a notation used in probability theory and financial mathematics to denote the variance of a stochastic process over a fixed time horizon T. It represents the time-aggregated or integrated variance over the interval and is central to models of asset price dynamics. In many contexts sigmaT2 is written as Var[log(S_{t+T}/S_t)] or simply as the variance of the log-return over the horizon.

In the simplest setting, where log returns over a horizon T are normally distributed with variance sigma^2 T, sigmaT2 equals sigma^2 T. More generally, sigmaT2 may be defined as the integral of the instantaneous variance process: sigmaT2 = ∫_0^T sigma_s^2 ds, or as the quadratic variation [log S]_T for continuous-price models. This quantity captures the total uncertainty or risk accumulated over the period.

Applications include option pricing under stochastic volatility models, risk management, and the construction of variance-based financial products such as variance swaps and VIX-like measures. It is also used to describe the term structure of volatility: all else equal, longer horizons tend to yield larger sigmaT2.

Estimation methods include historical estimation from the sample variance of log returns, realized variance computed from high-frequency data, and implied variance derived from option prices. sigmaT2 should be distinguished from instantaneous volatility sigma_t, which is the instantaneous standard deviation at a specific time t.