Biannual Report - Fachbereich Mathematik - Technische Universität ...

The problem of optimal stopping in discrete time is considered. The algorithm proposed
uses techniques of forecasting of time series and is completely nonparametric in the sense
that it is solely based on observations. It is shown that the expected gain of the corresponding
stopping rule converges to the optimal value whenever the observations are
drawn from a stationary and ergodic sequence. The algorithm is illustrated by applying it
to the problem of optimal exercising an American option.
Contact: D. Jones
References
[1] D. Jones. Data-based optimal stopping via forecasting of time series. Preprint, TU Darmstadt,
2012.
[2] M. Kohler and H. Walk. On data-based optimal stopping under stationarity and ergodicity. To
appear in Bernoulli 2013.
Project: Regression based Monte Carlo methods for pricing Bermudan options
In many applications of regression based Monte Carlo methods for pricing American options
in discrete time parameters of the underlying financial model have to be estimated
from observed data. In this project suitably defined nonparametric regression based Monte
Carlo methods are applied to paths of financial models where the parameters converge towards
true values of the parameters. For various Black-Scholes, Garch and Levy models it
is shown that in this case the price estimated from the approximate model converge to the
true price.
Contact: A. Fromkorth, M. Kohler
References
[1] A. Fromkorth and M. Kohler. On the consistency of regression based monte carlo methods for
pricing bermudan options in case of estimated financial model. Preprint, TU Darmstadt, 2011.
Project: Weakly universally consistent static forecasting of stationary and ergodic
time series via local averaging and least squares estimates
Static forecasting of stationary and ergodic time series is considered in this project, i.e.,
inference of the conditional expectation of the response variable at time zero given the
infinite past. It is shown that the mean squared error of a combination of suitably defined
localized least squares estimates converges to zero for all distributions where the response
variable is square integrable.
Partner: H. Walk, Universität Stuttgart
Support: German National Academic Foundation
Contact: T. Felber, D. Jones, M. Kohler
References
[1] T. Felber, D. Jones, M. Kohler, and H. Walk. Weakly universally consistent static forecasting of
stationary and ergodic time series via local averaging and least squares estimates. Preprint,
TU Darmstadt, 2011.
Project: Fixed design regression estimation based on real and artificial data
86 1 Research