A Nonlinear Filtering Approach to Volatility Estimation With a View Towards High Frequency Data

Abstract

In this paper we consider a nonlinear filtering approach to the estimation of asset price volatility. We are particularly interested in models which are suitable for high frequency data. In order to describe some of the typical features of high frequency data we consider marked point process models for the asset price dynamics. Both jump-intensity and jump-size distribution of this marked point process depend on a hidden state variable which is closely related to asset price volatility. In our setup volatility estimation can therefore be viewed as a nonlinear filtering problem with marked point process observations. We develop efficient recursive methods to compute approximations to the conditional distribution of this state variable using the so-called reference probability approach to nonlinear filtering.

Portfolio Optimization in Discontinuous Markets under Incomplete Information

“…In such a context the price trajectories are typically piecewise constant and jump only at random discrete points in time in reaction to trading or significant new information (for a more detailed discussion on this see e.g. [5]). While in such a context one observes frequent jumps, discontinuous models with less frequent jumps may arise whenever small changes in prices are neglected and only major price movements are registered as a jump.…”

Portfolio Optimization With Markov-Modulated Stock Prices and Interest Rates

“…This is due to the fact that in a diffusion price process model the quadratic variation and thus the volatility can be approximated arbitrarily well by the price process (cf. [9]). Therefore it is in principle sufficient to solve the optimization problem with complete observation.…”

Linear and non-linear filtering in mathematical finance: a review

“…Similar models are used in earlier, seminal work on realized volatility in Barndorf-Nielsen & Shephard (2002). A more complex model where a, β above are functions of a finite state Markov chain is described in Frey & Runggaldier (2001), although we focus on the exposition for the linear state space model in this review. From this point onwards, we can use the calibration and forecasting procedures mentioned in the earlier sections, along with the realised volatility ς n computed from intra-day asset prices, to calibrate the model and to forecast future volatility.…”