36 research outputs found
The scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends
First passage problems for spectrally negative L\'evy processes with possible
absorbtion or/and reflection at boundaries have been widely applied in
mathematical finance, risk, queueing, and inventory/storage theory.
Historically, such problems were tackled by taking Laplace transform of the
associated Kolmogorov integro-differential equations involving the generator
operator. In the last years there appeared an alternative approach based on the
solution of two fundamental "two-sided exit" problems from an interval (TSE). A
spectrally one-sided process will exit smoothly on one side on an interval, and
the solution is simply expressed in terms of a "scale function" (Bertoin
1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a
second scale function (Avram, Kyprianou and Pistorius 2004).
Since many other problems can be reduced to TSE, researchers produced in the
last years a kit of formulas expressed in terms of the " alphabet" for a
great variety of first passage problems. We collect here our favorite recipes
from this kit, including a recent one (94) which generalizes the classic De
Finetti dividend problem. One interesting use of the kit is for recognizing
relationships between apparently unrelated problems -- see Lemma 3. Last but
not least, it turned out recently that once the classic are replaced with
appropriate generalizations, the classic formulas for (absorbed/ reflected)
L\'evy processes continue to hold for:
a) spectrally negative Markov additive processes (Ivanovs and Palmowski
2012),
b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion
or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with
Omega killing (Li and Palmowski 2017)
Alternative approach to the optimality of the threshold strategy for spectrally negative Levy processes
Consider the optimal dividend problem for an insurance company whose
uncontrolled surplus precess evolves as a spectrally negative Levy process. We
assume that dividends are paid to the shareholders according to admissible
strategies whose dividend rate is bounded by a constant. The objective is to
find a dividend policy so as to maximize the expected discounted value of
dividends which are paid to the shareholders until the company is ruined.
Kyprianou, Loeffen and Perez [28] have shown that a refraction strategy (also
called threshold strategy) forms an optimal strategy under the condition that
the Levy measure has a completely monotone density. In this paper, we propose
an alternative approach to this optimal problem.Comment: 16 page
Brownian excursions and Parisian barrier options
In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long
Optimization of consumption with labor income
We present the solution of a portfolio optimization problem for an economic agent endowed with a stochastic insurable stream, under a liquidity constraint over the time interval [0,T]. Generally, the existence of labor income complicates the agent's decisions. Moreover, in the real world the economic agents are restricted in their ability to borrow against their future labor income. We deal with this kind of liquidity constraint following the lines of American option valuation which allows us to give a precise characterization of the optimal consumption as well as the terminal wealth. In a Markovian case, with infinite horizon and HARA utility, we obtain a closed form solution.Portfolio optimization, labor income, American option, optimal stopping problem
Some combinations of Asian, Parisian, and barrier options
This article addresses some of the valuation problems, in the Black and Scholes setting of a geometric Brownian motion for the underlying asset dynamics, for options whose pay-off is related to the terminal price of the stock and an arithmetic average of fixing and/or involves stopping times related to excursions. In all cases, we are able to provide at least the Laplace transform in time of the option price under a form whose complexity varies with the number of exotic features. We emphasize that we do not give closed form formulas for the general case, but we aim to develop a methodology which may be used in many cases