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Workshop on
Optimal Stopping, Sequential Methods and Related Topics
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Abstracts
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| Program and Abstracts (pdf) |
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| April 28–29, 2011, Freiburg, Germany, Room 404, Eckerstr. 1 |
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Elena Boguslavskaya, London School of Economics and Political Science (United Kingdom)
Solving optimal stopping problems with Appell functions
Recently, there were series of papers by Novikov and Shiryaev, Kyprianou, Levendorski, Mordecki, Salminen,
Surya, where the solution of optimal stopping problems for Lévy processes and random walks were found
in terms of the maximum/minimum of the process. We continue in the same spirit. In the present paper we propose
the method similar to the one presented in the papers by Surya, and Novikov and Shiryaev. To find the optimal
stopping boundary explicitly, we construct the Appell function, and find the desired boundary as a root of this function.
The so-called Appell function happens to be an Esscher–Laplace transform, and we also show how the Appell
function can be represented as series in Appell polynomials. As Appell polynomials are complete Bell
polynomials in cumulants, it offers an easy way to calculate coefficients of Appell polynomials as
"inverse moments". Moreover, we show how the above mentioned representation leads naturally to the martingale
property of Appell polynomials generated by Lévy processes.
To conclude, we provide several examples, including the cases when the underlying processes are Brownian motion,
Poisson process, spectrally negative or spectrally positive processes.
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Sören Christensen, University of Kiel
A method for pricing American options using semi-infinite linear programming
A new approach for the numerical pricing of American options is introduced. The main idea
is to choose a finite number of suitable excessive functions (randomly) and to find the
smallest majorant of the gain function in the span of these functions. The resulting problem is a
linear semi-infinite programming problem, that can be solved using standard algorithms. This leads
to good upper bounds for the original problem. For our algorithms no discretization of space and
time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems.
The algorithm provides an approximation of the value not only for one starting point, but for the
complete value function on the continuation set, so that the optimal exercise region and e.g. the
Greeks can be calculated. We apply the algorithm to (one- and) multidimensional diffusions and to
Lévy processes, and show it to be fast and accurate.
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Yan Dolinsky, ETH Zurich (Switzerland)
Application of strong approximation theorems to optimal stopping
We derive error estimates for discrete approximations of optimal stopping values which are defined on a
d-dimensional Brownian probabiltiy space. We consider a general path-dependent payoffs with some
regularity properties. Our main tool is based on strong approximation theorems for i.i.d. random vectors,
which were obtained by Sakhanenko (2002). We also show how to use this method to approximate American options
values in Jump-diffusion models.
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| Christian-Oliver Ewald, University of Sydney (Australia)
Asymptotic solutions for real options under stochastic volatility
We derive asymptotic solutions for real options under stochastic volatility. We consider modelling the project's
value with processes that resemble both a geometric Brownian motion and a geometric mean reverting process,
but their variances being modelled using a CIR process. We further derive the relationship between the threshold
(the point at which to undertake a project or not) and the parameters used in the stochastic volatility model.
We compare these results to the classical case of non-stochastic volatility.
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| Pavel V. Gapeev, London School of Economics and Political Science (United Kingdom)
About two-dimensional Bayesian disorder problems
We study the Bayesian disorder detection problems in a model in which two observable constantly correlated Wiener processes
change their drift rates at some independent exponential times which are inaccessible for observation.
The initial problems are reduced to optimal stopping problems for multi-dimensional continuous Markov
processes called sufficient statistics. We derive stochastic differential equations for the sufficient
statistics in the disorder problem for the case of linear and exponential delay penalty costs. The optimal stopping
times of alarm are sought as the first times at which sufficient statistics processes exit certain regions restricted
by non-constant boundaries. By means of the change-of-variable formula with local time on surfaces,
it is shown that the optimal stopping boundaries can be uniquely characterized as solutions of the associated
free-boundary problems. We also derive several explicit estimates for the initial Bayesian risk functions and the
optimal stopping boundaries.
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| A. V. Gnedin, Utrecht University (The Netherlands)
Sequential selection of random chains in self-similar posets
We consider sampling from a space S endowed with a partial order and a probability measure.
All upper cones in $S$ are assumed similar up to a scaling factor. The standard example is a d-dimensional cube
with the uniform distribution. The problem is to select, in nonanticipating fashion, an increasing
subsequence of independent n-sample from S.
We compare the lengths of selected sequences under a greedy policy (selecting the "chain records") and a policy which
may reject some of the options that satisfy the order constraint.
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| Daniel Jones, Technical University Darmstadt
Optimal exercising of American options in discrete time via
forecasting of stationary and ergodic time series
The problem of exercising an American option in discrete time in an optimal way is considered, i.e. maximization of the expected discounted payoff. The algorithm proposed uses techniques of forecasting of time series and is completely nonparametric in the sense that it is solely based on observations of the underlying asset. It is shown that the expected payoff of the corresponding stopping rule converges to the optimal value whenever the returns of the underlying asset are stationary and ergodic.
Joint work with Michael Kohler, Technical University of Darmstadt.
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| Claudia Kirch, Karlsruhe Institute of Technology (KIT)
On the bootstrap for sequential change-point tests
Change-point analysis deals with the detection of structural breaks in time series. In a sequential setting the data arrive one by one and after each new observation one checks whether a change has occurred. This approach is natural for many applications such as monitoring intensive care patients, financial time series or climate data. Critical values are based on distributional asymptotics but do not work well if there are only few historical observations (control data).
Bootstrapping methods have widely been used in a non-sequential setting, however, there is hardly any literature on variations or the validity of the bootstrap in a sequential setup.
In this talk we will investigate these questions using a simple mean change model as well as a linear regression model. In particular we consider how to make use of the new incoming observations which can improve the bootstrap estimate. From a practical point of view a repeated bootstrap after each observation has high computational costs so variations are of interest. From a theoretical point of view the critical values change with each incoming observation, so the question is whether this procedure remains consistent.
This is joint work with Marie Husková (Prague).
References
- Husková, M., Kirch, C.: Bootstrapping sequential change-point tests for linear regression. Metrika, 2011. To appear.
- Kirch, C.: Bootstrapping sequential change-point tests. Seq. Anal., 27:330--349, 2008.
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| A T W O R K |
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