364 research outputs found
A method for pricing American options using semi-infinite linear programming
We introduce a new approach for the numerical pricing of American options.
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 L\'evy processes, and show it to be fast and accurate
Optimal decision under ambiguity for diffusion processes
In this paper we consider stochastic optimization problems for an ambiguity
averse decision maker who is uncertain about the parameters of the underlying
process. In a first part we consider problems of optimal stopping under drift
ambiguity for one-dimensional diffusion processes. Analogously to the case of
ordinary optimal stopping problems for one-dimensional Brownian motions we
reduce the problem to the geometric problem of finding the smallest majorant of
the reward function in a two-parameter function space. In a second part we
solve optimal stopping problems when the underlying process may crash down.
These problems are reduced to one optimal stopping problem and one Dynkin game.
Examples are discussed
On Poisson constrained control of linear diffusions
The classical setting in optimal stopping and optimal control theory assumes that the agent controlling the system can operate continuously in time. In optimal stopping this setting is highly stylized for many applications, for example, in mathematical finance due to illiquid markets. In optimal stochastic control this setting often leads to optimal strategies being singular with respect to the Lebesgue measure, and thus the strategies are not feasible in practice. Hence, it is of importance to study these problems from such a perspective that their solutions are practically more implementable.
In this thesis we alter the classical setting by introducing an exogenous constraint, in the form of a signal process, for the control opportunities of the agent. In order to keep the problems more tractable, especially time-homogeneous and Markovian, the signal process is assumed to be a Poisson process with constant intensity. Consequently, the agent can only have influence on the system at discrete times. We call these control problems Poisson constrained control problems and study them when the dynamics are governed by linear diffusion processes.
Linear diffusions are particular enough to have a rich theory but still general enough to offer a class of interesting dynamics that are applicable in various situations. A key factor is also that many control problems with diffusions will lead to closed-form solutions. This thesis investigates to which extent the classical theory of diffusion can be applied in this class of control problems to form closed-form solutions
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