ERCIM News No.38 - July 1999

## Statistical Methods for Financial and other Dynamical Stochastic Models

by Kacha Dzhaparidze and Peter Spreij

The high capacity of present day computers has enabled the use of complex stochastic models because data on the system under study can be obtained in huge amounts and analyzed by simulation techniques or other numerical methods. For instance, at the stock exchanges, time and price are recorded for every single trade. Mathematical finance is an example of a field with a vigorous development of new models. The development of statistical methods for stochastic process models, however, lags behind, with the result that far too often statistical methods have been applied that, although they can be relatively sophisticated, suffer from shortcomings because they do not fully take into account and exploit the structure of the new models. Researchers at CWI aim at making a major contribution to the theory of statistical inference for stochastic processes.

The research is carried out in close collaboration with many researchers in The Netherlands and elsewhere in Europe. The theoretical work uses the methods of modern probability theory including stochastic calculus. A more applied project objective is the statistical analysis and modelling of financial data such as stock prices, interest rates, exchange rates and prices of options and other derivative assets, and the development of more realistic models for these than those presently used in the financial industry. There are increasing demands (including new legislation) that banks and other financial institutions improve the management of their risk from holding positions in securities. This will require use of more realistic and sophisticated mathematical models as well as improved statistical procedures to evaluate prices of financial assets.

Mathematical finance is an example of a field where data analysis is, in practice, very often done by means of traditional discrete time models, whereas most of the models used for pricing derivative assets are continuous-time models. Continuous-time models have the additional advantage that they can be analysed by means of the powerful tools of stochastic calculus, so that results can often be obtained even for very complicated models. In many applications, however, one has to take into consideration that data are obtained at discrete time points, so inference methods for discretely observed continuous-time processes are to be applied. In recent years, statistical methods for discrete time observations from diffusion-type processes has started to attract attention and it appears that there are many challenging mathematical problems involved. A survey paper on this subject by Dzhaparidze, Spreij and Van Zanten will soon appear in Statistica Neerlandica.

Very often the complexity of the models in question prevents exact calculation of the statistical properties of the methods developed. An example is calculation of the variances of estimators that are often used to choose the most efficient member of a family of estimators. Computer simulations are then a useful tool, but it is important to have a mathematical theory with which simulation results can be compared. Asymptotic statistical theory can play this role, being therefore an important research objective at CWI. In recent years Dzhaparidze and Spreij have published a number of papers on parameter estimation problems in a general context of semimartingales.

Asymptotic methods can also be used to approximate complex models by simpler ones for inferential purposes. Moreover, the theory of asymptotic equivalence of experiments will be used to simplify decision problems for complex stochastic models to those of Gaussian or Poisson models that approximate them in the deficiency distance. This method can also be used to the approximation of discrete-time models by continuous time-models. Certain rudimentary ideas and facts on the relationship between these models has been reported by Dzhaparidze in a series of three papers in CWI Quarterly. These papers gave rise to a textbook on options valuation which is recently completed and intended for publication at CWI.

The research described above will be further developed in close collaboration with research teams in, eg, Paris, Berlin, Copenhagen, Freiburg, Helsinki and Padova. Most of these teams have been involved in the HCM research programme ‘Statistical Inference for Stochastic Processes’. Contacts between the members of these teams are currently maintained or reinforced at annual workshops, recently in Munzingen (Freiburg). The collaboration with E. Valkeila (Helsinki), in particular, proved to be quite fruitful. A number of joint papers on general parametric families of statistical experiments were published, and others are scheduled for this year.