On three filtering problems arising in mathematical finance

TL;DR

This paper explores three filtering problems in mathematical finance, demonstrating how advanced filtering techniques like projection filters can be applied to estimate volatility, interest rate models, and hedging strategies under partial observation.

Contribution

It introduces the application of projection filtering techniques to three distinct financial problems, highlighting their potential advantages and suggesting ways to develop more general filters.

Findings

Projection filters can be effectively used for volatility estimation.

Different filtering techniques are applicable to interest rate models.

Potential for more general filters in risk management strategies.

Abstract

Three situations in which filtering theory is used in mathematical finance are illustrated at different levels of detail. The three problems originate from the following different works: 1) On estimating the stochastic volatility model from observed bilateral exchange rate news, by R. Mahieu, and P. Schotman; 2) A state space approach to estimate multi-factors CIR models of the term structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3) Risk-minimizing hedging strategies under partial observation in pricing financial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In the first problem we propose to use a recent nonlinear filtering technique based on geometry to estimate the volatility time series from observed bilateral exchange rates. The model used here is the stochastic volatility model. The filters that we propose are known as projection filters, and a brief derivation of such filters is given. The second problem is introduced in detail, and a possible use of different filtering techniques is hinted at. In fact the filters used for this problem in 2) and part of the literature can be interpreted as projection filters and we will make some remarks on how more general and possibly more suitable projection filters can be constructed. The third problem is only presented shortly.