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On three filtering problems arising in mathematical finance

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  • Damiano Brigo
  • Bernard Hanzon

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.

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  • Damiano Brigo & Bernard Hanzon, 2008. "On three filtering problems arising in mathematical finance," Papers 0812.4050, arXiv.org.
    • Handle: RePEc:arx:papers:0812.4050
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    References listed on IDEAS

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    1. Hanzon, B. & Hut, R., 1991. "New results on the projection filter," Serie Research Memoranda 0023, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    2. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    3. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    4. Ronald J. Mahieu & Peter C. Schotman, 1998. "An empirical application of stochastic volatility models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 13(4), pages 333-360.
    5. Martin Schweizer, 1994. "Risk‐Minimizing Hedging Strategies Under Restricted Information," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 327-342, October.
    6. Geyer, Alois L J & Pichler, Stefan, 1999. "A State-Space Approach to Estimate and Test Multifactor Cox-Ingersoll-Ross Models of the Term Structure," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 22(1), pages 107-130, Spring.
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