Nonstationarity of real-life time series requires model adaptation. In classical approaches like ARMA-ARCH there is assumed some arbitrarily chosen dependence type. To avoid their bias, we will focus on novel more agnostic approach: moving estimator, which estimates parameters separately for every time $t$: optimizing $F_t=\sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ local log-likelihood with exponentially weakening weights of the old values. In practice such moving estimates can be found by EMA (exponential moving average) of some parameters, like $m_p=E[|x-μ|^p]$ absolute central moments, updated by $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p-m_{p,t})$. We will focus here on its applications for alpha-Stable distribution, which also influences Hurst exponent, hence can be used for its adaptive estimation. Its application will be shown on financial data as DJIA time series - beside standard estimation of evolution of center $μ$ and scale parameter $σ$, there is also estimated evolution of $α$ parameter allowing to continuously evaluate market stability - tails having $ρ(x) \sim 1/|x|^{α+1}$ behavior, controlling probability of potentially dangerous extreme events.