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Some fractional extensions of the Poisson process
Enzo Orsingher (Sapienza University of Rome, Italy)
11:00-12:00, salle Extranef 110
In this talk we present three different fractional generalizations of the classical homogeneous Poisson process. The first one is the time-fractional Poisson process where the governing difference-differential equations involve Dzerbayshan-Caputo derivatives. In this case we obtain explicit probability distributions and we show also that the time-fractional Poisson process is a time-changed classical Poisson process. We analyze also the renewal structure of the time fractional Poisson process with Mittag-Leffler distributed intertimes. Another generalization is the so called space-fractional Poisson process which is a process with independent , homogeneous increments (this is the main difference with respect to the time-fractional Poisson process). We give its explicit distribution and analyze some of its properties. A third version of the fractional Poisson process is constructed as a weighted sum of homogeneous processes and discuss some of its applications to random flights. At the end we present a large class of time-rescaled Poisson processes with independent increments and present in detail some particular cases.
| 3 juillet 2015
Valuation of Guaranteed Minimum Maturity Benefits in variable annuities with surrender options
Jonathan Ziveyi (University of New South Wales, Sydney, Australia)
14:00-15:00, salle Extranef 126
We consider the pricing of guaranteed minimum maturity benefits (GMMB) embedded in variable annuity contracts in the case where the guarantees can be surrendered any time prior to maturity. Surrender charges are imposed as a way of discouraging early termination of the variable annuity contract. We formulate the problem as an American put option and derive the corresponding pricing partial differential equation (PDE) using hedging arguments and Ito’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density PDE whose solution is well known in literature. An explicit integral expression for the pricing PDE is then presented with the aide of Duhamel’s principle. An expression for the delta of the surrender option which can be used for risk management purposes is also derived. We then outline the algorithm for implementing the integral expression for the price and the corresponding early exercise boundary for the surrender option. We wrap up the paper by presenting numerical results for the prices, early exercise boundaries, deltas and the corresponding sensitivities.
| 9 octobre 2015
Boualem Djehiche (KTH Stockholm)
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