We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace-or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.
We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations. Spectrally negative Lévy processes and scale functionsLet X = {X t : t ≥ 0} be a Lévy process defined on a filtered probability space (Ω, F, F, P), where {F t : t ≥ 0} is the filtration generated by X satisfying the usual conditions. For x ∈ R denote by P x the law of X when it is started at x and write simply P 0 = P. Accordingly we shall write E x and E for the associated expectation operators. In this paper we shall assume throughout that X is spectrally negative meaning here that it has no positive jumps and that it is not the negative of a subordinator. It is well known that the latter allows us to talk about the Laplace exponent ψ(θ) := log E[e θX1 ] for (θ) ≥ 0 where in particular we have the Lévy-Khintchine representationwhere a ∈ R, σ ≥ 0 is the Gaussian coefficient and Π is a measure concentrated on (−∞, 0) satisfying (−∞,0) (1∧x 2 )Π(dx) < ∞. The, so-called, Lévy triple (a, σ, Π) completely characterises the process X. For later reference we also introduce the function Φ : [0, ∞) → [0, ∞) as the right inverse of ψ on (0, ∞) so that for all q ≥ 0 Φ(q) = sup{θ ≥ 0 : ψ(θ) = q}.(2)
In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale trasnform for exponential processes, and the minimal entropy martingale measure for exponential Lévy models and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail
We study the density of the supremum of a strictly stable Lévy process. We prove that for almost all values of the index α -except for a dense set of Lebesgue measure zero -the asymptotic series which were obtained in [13] are in fact absolutely convergent series representations for the density of the supremum.
We consider the following optimization problem for an insurance companyHere U (x) = (1 − exp(−γx))/γ denotes an exponential utility function with risk aversion parameter γ, C denotes the accumulated dividend process, and β a discount factor. We show that -assuming that a certain integral equation has a solution -the optimal strategy is a barrier strategy. The barrier function is a solution of the integral equation and turns out to be time-dependent. In addition we study the problem from a different point of view, namely by using a certain ansatz for the value function and the barrier.Keywords: optimal dividend payment, optimal control, free boundary value problem IntroductionInsurance companies face a certain dilemma between choosing a strategy maximizing their dividends (in one way or the other) and a more conservative approach, which can be expressed e.g., by demanding a low probability of ruin. The literature on both topics is vast. We mention just a few of the articles. In Gaier, Grandits, and Schachermayer (2003), Hipp and Plum (2000), Schmidli (2005), Paulsen (2002), Kalashnikov and Norberg (2002) or Grandits (2004), insurance companies are considered, which invest in the stock market (or face a stochastic interest rate on their reserves). The problem is to find estimates for the ruin probability, or to find an investment strategy, which minimizes this probability.As regards the maximization of dividends several approaches are possible. On the one hand side one can distinguish between the ways one models the reserve of the company, e.g. diffusion approximation, compound Poisson process or a general renewal model are used. On the other hand one has the possibility to fix the strategy (e.g. a linear dividend barrier as in Gerber (1981) or a nonlinear one as in Albrecher and Kainhofer (2002)) and to try to estimate the expected discounted dividends by analytical or numerical methods). Another approach is to try to find the optimal strategy in a model, which is simple enough. E.g. the article by Gerber (1969) for the compound Poisson case and Asmussen and Taksar (1997) for the diffusion approximation
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