Numerical estimation of volatility values from discretely observed diffusion data

We consider a Black-Scholes type model, but with volatility being a Markov Chain process. Assuming that the stock price is observed at discrete, possibly random times, the goal is to estimate the current volatility value. The model parameters, that is, the possible volatility values and transition probabilities, are estimated using the Multiscale Trend Analysis method of Zaliapin, Gabrielov and Keilis-Borok [17], adapted to our framework. Once these are given, the volatility is estimated using the filtering formula of Cvitanić, Liptser and Rozovskii [3]. Our numerical implementation shows that the estimation is of very high quality under a range of conditions.