Mathematical Finance Seminar
May 1, 2003, 5:30 PM to 7:00 PM
Sergei Levendorskii, UT Austin
Optimal stopping in non-Gaussian models, with applications to
pricing of American options and real options
Abstract: A general approach to optimal stopping problems based on the
Wiener-Hopf factorization technique is developed. Jump-diffusion models
and wide classes of exponential L\'evy models in continuous time, and
discrete time models are considered. The failure of the smooth pasting
condition in discrete time models and certain L\'evy models is shown,
and its generalization is suggested. In the infinite time horizon case,
explicit analytical formulas are derived, and in the finite time horizon
case, semi-explicit pricing algorithms are constructed. In continuous
time models, the algorithm is an approximate method: an analog of Carr's
randomization procedure.
For wide classes of random walks and jump-diffusion processes in
continuous time, the pricing algorithm reduces to a series of simple
algebraic problems, which makes the method efficient for numerical
implementation. As applications of the method, pricing of American
options and Bermudan options, and optimal timing of investment under
uncertainty are considered, and the non-standard behavior of the early
exercise boundary in the non-Gaussian case is analyzed.