統計数学セミナー
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Seminar on Probability and Statistics Monday January 28 2013 Tokyo 006 1:00-2:10 pm
Laplace and Fourier based valuation methods in exponential Levy models
Ernst August Frhr. v. Hammerstein Albert-Ludwigs-Universität Freiburg Abstract A fundamental problem in mathematical finance is the explicit
computation of expectations which arise as prices of derivatives. Closed
formulas that can easily be evaluated are typically only available in
models driven by a Brownian motion. If one considers more sophisticated
jump-type Levy processes as drivers, the problem quickly becomes rather
nontrivial and complicated. Starting with the paper of Carr and Madan
(1999) and the PhD thesis of Raible (2000), Laplace and Fourier based
methods have been used to derive option pricing formulas that can be
evaluated very efficiently numerically. In this talk we review the
initial idea of Raible (2000), show how it can be generalized and
discuss under which precise mathematical assumptions the Laplace and
Fourier approach work. We then give several examples of specific options
and Levy models to which the general framework can be applied to. In the
last part, we present some formulas for pricing options on the supremum
and infimum of the asset price process that use the Wiener-Hopf
factorization.
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