Volatility of the stock price is the key to the pricing problem of stock related derivatives in finance. Volatility appears in the diffusion term of the usual modeling of stock prices. One popular approach is to take volatility to be stochastic, and assumes that it satisfies a stochastic differential equation. Taking the stock price to be the observation, we may then pose the filtering problem of estimating the volatility on line based on the stock price data. This is an unconventional filtering problem which we solve in this paper. But even more interesting is the fact that this filtering algorithm is inherently not robust. In the rest of the paper we derive the robust form of this filter.