What is quadratic variations?

Quadratic variations are a measure of the total variation or fluctuations of a stochastic process over time. They are especially useful in measuring the volatility or risk of financial assets or other stochastic processes.

In discrete time, the quadratic variation of a stochastic process is defined as the sum of the squared differences between consecutive values of the process. In continuous time, the quadratic variation is defined as the integral of the square of the process's rate of change over time.

Quadratic variations are often used in the context of stochastic calculus and mathematical finance to quantify the volatility of financial assets and other stochastic processes. They play a crucial role in the construction of stochastic models and the pricing of financial derivatives.

One prominent application of quadratic variations is in the calculation of realized volatility, which is an important measure of risk and volatility in financial markets. By calculating the quadratic variation of asset prices or returns over a certain period, investors can estimate the level of volatility and make informed decisions about their investments.

Overall, quadratic variations are a useful tool for quantifying the fluctuations and risk associated with stochastic processes, especially in the field of finance.