Estimating the implicit interest rate of a risky asset
Hidden Markov models characterize stochastic phenomena, where the Markov behavior is observed only noisily or indirectly. In a recent paper, the first author had studied filtering problems related to a Markov chain, and obtained finite-dimensional filters that provide revised estimates for the parameters in the model. In the present paper, these results are applied to a scalar model that arises in finance. The model is that of the price, \(S_ t\), of a risky asset, described by the stochastic differential equation \(dS_ t= S_ t(\rho_ t dt+ \sigma dw)\), where \(\rho_ t\) the implicit interest rate at time \(t\), behaving according to a finite state Markov chain, which is the quantity of interest for estimation. In this estimation problem, the logarithm of \(S_ t\) is used as the observation process. The results obtained allow a re-estimation of all the parameters of the model, except the volatility \(\sigma\) which is assumed to be a known constant. Among these parameters are the states of the Markov chain, and the elements of the rate matrix.
scientific article; zbMATH DE number 519766