โ Scribed by Damiano Brigo; Bernard Hanzon
No coin nor oath required. For personal study only.
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 Mahieu and Schotman (1997).
(2) A state space approach to estimate multi-factors CIR models of the term structure of interest rates, by Geyer and Pichler (1996).
(3) Risk-minimizing hedging strategies under partial observation in pricing financial derivatives, by Fischer et al. (1996).
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 briefly.
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