Abstract
Abstract Explicit formulae for maximum likelihood estimates of the parameters of square root processes and Bessel processes and first and second order approximate sufficient statistics are supplied. Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour.
Highlights
Fast and accurate statistical methods in finance and other areas of applied probability are becoming increasingly important due to the enormous amounts of data available that have to be analysed
Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour
We have extremely fast and accurate methods for modelling Gaussian dynamics, where in a continuous-time setting linearly transformed Brownian motions drive model dynamics, for example, as for the logarithm of an asset price under the Black-Scholes model (see Black and Scholes (1973)) or as for the short term interest rate under the Vasicek model. This seems to be not the case for next-generation models that are based on Bessel processes and square root processes like the CIR model in finance
Summary
Fast and accurate statistical methods in finance and other areas of applied probability are becoming increasingly important due to the enormous amounts of data available that have to be analysed. We have extremely fast and accurate methods for modelling Gaussian dynamics, where in a continuous-time setting linearly transformed Brownian motions drive model dynamics, for example, as for the logarithm of an asset price under the Black-Scholes model (see Black and Scholes (1973)) or as for the short term interest rate under the Vasicek model This seems to be not the case for next-generation models that are based on Bessel processes and square root processes like the CIR model in finance. In Overbeck and Ryden (1997), estimators of parameters of (3), employing observations at equispaced times and based on conditional least squares, are studied Another approach involving quasi-maximum likelihood estimation via the Wagner-Platen approximation is introduced in Huang (2013).
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