5,002 research outputs found
Optimal periodic dividend strategies for spectrally positive L\'evy risk processes with fixed transaction costs
We consider the general class of spectrally positive L\'evy risk processes,
which are appropriate for businesses with continuous expenses and lump sum
gains whose timing and sizes are stochastic. Motivated by the fact that
dividends cannot be paid at any time in real life, we study
dividend strategies whereby dividend decisions are made according to a separate
arrival process.
In this paper, we investigate the impact of fixed transaction costs on the
optimal periodic dividend strategy, and show that a periodic
strategy is optimal when decision times arrive according to an independent
Poisson process. Such a strategy leads to lump sum dividends that bring the
surplus back to as long as it is no less than at a dividend
decision time. The expected present value of dividends (net of transaction
costs) is provided explicitly with the help of scale functions. Results are
illustrated.Comment: Accepted for publication in Insurance: Mathematics and Economic
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency
postprin
Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes
Fast pricing of American-style options has been a difficult problem since it
was first introduced to financial markets in 1970s, especially when the
underlying stocks' prices follow some jump-diffusion processes. In this paper,
we propose a new algorithm to generate tight upper bounds on the Bermudan
option price without nested simulation, under the jump-diffusion setting. By
exploiting the martingale representation theorem for jump processes on the dual
martingale, we are able to explore the unique structure of the optimal dual
martingale and construct an approximation that preserves the martingale
property. The resulting upper bound estimator avoids the nested Monte Carlo
simulation suffered by the original primal-dual algorithm, therefore
significantly improves the computational efficiency. Theoretical analysis is
provided to guarantee the quality of the martingale approximation. Numerical
experiments are conducted to verify the efficiency of our proposed algorithm
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On finite-time ruin probabilities in a generalized dual risk model with dependence
In this paper, we study the finite-time ruin probability in a reasonably generalized dual risK model, where we assume any non-negative non-decreasing cumulative operational cost function and arbitrary capital gains arrival process. Establishing an enlightening link between this dual risk model and its corresponding insurance risk model, explicit expressions for the finite-time survival probability in the dual risk model are obtained under various general assumptions for the distribution of the capital gains. In order to make the model more realistic and general, different dependence structures among capital gains and inter-arrival times and between both are also introduced and corresponding ruin probability expressions are also given. The concept of alarm time, as introduced in Das and Kratz (2012), is applied to the dual risk model within the context of risk capital allocation. Extensive numerical illustrations are provided
Reduced basis methods for pricing options with the Black-Scholes and Heston model
In this paper, we present a reduced basis method for pricing European and
American options based on the Black-Scholes and Heston model. To tackle each
model numerically, we formulate the problem in terms of a time dependent
variational equality or inequality. We apply a suitable reduced basis approach
for both types of options. The characteristic ingredients used in the method
are a combined POD-Greedy and Angle-Greedy procedure for the construction of
the primal and dual reduced spaces. Analytically, we prove the reproduction
property of the reduced scheme and derive a posteriori error estimators.
Numerical examples are provided, illustrating the approximation quality and
convergence of our approach for the different option pricing models. Also, we
investigate the reliability and effectivity of the error estimators.Comment: 25 pages, 27 figure
Stable Dividends under Linear-Quadratic Optimization
The optimization criterion for dividends from a risky business is most often
formalized in terms of the expected present value of future dividends. That
criterion disregards a potential, explicit demand for stability of dividends.
In particular, within actuarial risk theory, maximization of future dividends
have been intensively studied as the so-called de Finetti problem. However,
there the optimal strategies typically become so-called barrier strategies.
These are far from stable and suboptimal affine dividend strategies have
therefore received attention recently. In contrast, in the class of
linear-quadratic problems a demand for stability if explicitly stressed. These
have most often been studied in diffusion models different from the actuarial
risk models. We bridge the gap between these patterns of thinking by deriving
optimal affine dividend strategies under a linear-quadratic criterion for a
general L\'evy process. We characterize the value function by the
Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the
optimal controls to the classical objective of maximizing expected present
value of future dividends. Thereby we provide a framework within which
stability of dividends from a risky business, as e.g. in classical risk theory,
is explicitly demanded and explicitly obtained
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