  Identifying the instances of jumps in a discrete-time-series sample of a jump
diffusion model is a challenging task. We have developed a novel statistical
technique for jump detection and volatility estimation in a return time series
data using a threshold method. The consistency of the volatility estimator has
been obtained. Since we have derived the threshold and the volatility estimator
simultaneously by solving an implicit equation, we have obtained unprecedented
accuracy across a wide range of parameter values. Using this method, the
increments attributed to jumps have been removed from a large collection of
historical data of Indian sectorial indices. Subsequently, we have tested the
presence of regime-switching dynamics in the volatility coefficient using a new
discriminating statistic. The statistic has been shown to be sensitive to the
transition kernel of the regime-switching model. We perform the testing using
the Bootstrap method and find a clear indication of presence of multiple
regimes of volatility in the data. A link to all Python codes is given in the
conclusion. The methodology is suitable for analyzing high-frequency data and
may be applied for algorithmic trading.
