On a generalization of the Gerber–Shiu function to path-dependent penalties☆
Introduction
In a now classical series of papers, Gerber and Shiu, 1997, Gerber and Shiu, 1998a, Gerber and Shiu, 1998b introduced the concept of Expected Discounted Penalty Function (EDPF), a functional of the ruin time (i.e., the first time the surplus of a firm becomes negative), the surplus prior to ruin, and the deficit at ruin. The joint analysis of these random variables, which had been traditionally studied separately, allowed them to offer an elegant characterization of the ruin event in terms of a renewal equation. Applications of the EDPF are natural in the context of solvency requirements and option pricing. In the first case, the EDPF can be used to determine the initial capital required by a company to avoid insolvency with a minimum level of confidence. More generally, the EDPF is useful whenever one wishes to place a value on cashflows triggered by the first passage of a process across a given barrier. This is the case for credit risky securities, whose cashflows depend on a firm’s assets falling below its liabilities, or for American options, whose exercise is triggered by the underlying security’s market value crossing an exercise boundary. Gerber and Landry (1998) and Gerber and Shiu (1999), for example, used the EDPF to price perpetual American options and reset guarantees.
The penalty delivered by the classical EDPF has local nature, in the sense that the surplus prior to ruin and the deficit at ruin only characterize the surplus in a neighborhood of the ruin time. In this paper we explore the possibility of introducing path-dependent variables in the EDPF.
A first motivation for this comes from the computation of capital requirements that may allow for different penalization of the ruin event based on the path characteristics of the surplus. An example is provided by the last minimum of the surplus before ruin. The lower this minimum, the worse the financing conditions that can be negotiated by the company with capital providers. Similarly, the closer the last minimum was to the bankruptcy level, and the shorter the time elapsed since that minimum, the more urgent was the need to correct the course and steer away from dangerous waters. Understanding whether ex post penalization of different ruin-related quantities provides the right ex ante incentives is clearly very relevant. A proper answer to this question would require endogenizing the surplus dynamics, for example by allowing for controllable premiums or dynamic selection of risk exposures. Here we limit ourselves to the classical risk model, where the risk process dynamics is taken as given, and focus on providing explicit expressions for capital requirements implied by ruin-related penalties that may be path-dependent.
A second motivation for the extension of the EDPF to path-dependent penalties comes from the pricing of some exotic American options. By allowing the penalty function to act on the last minimum (or maximum) before ruin, we obtain a pricing functional for American options with lookback features, as studied for example by Dai and Kwok (2005) in a geometric Brownian motion setting. Similarly, we could price reset guarantees with path-dependent payoffs that generalize the cases analyzed in Gerber and Shiu (1999).
In this work we extend the definition of the classical EDPF to include a new random variable, the last minimum of the surplus before ruin. We work with a surplus process with nonpositive jumps, stationary and independent increments, to provide an analytical characterization of the generalized EDPF in terms of convolutions, extending results found in Gerber and Landry (1998), Tsai and Willmot (2002), and Morales (2007), among others. The characterization is obtained by using a fluctuation identity for Lévy processes recently developed by Doney and Kyprianou (2006). We further show that the fluctuation identity provides an effective tool to approach the classical EDPF as well. As an example, we show how to recover and extend the results obtained by Morales (2007) in a perturbed subordinator model.
The paper is organized as follows. In Section 2, we introduce a perturbed subordinator model for the surplus and provide the definitions of the classical and generalized EDPF. In Section 3, we review two useful results from the fluctuation theory for Lévy processes. In Section 4, we provide a characterization of the generalized EDPF in terms of convolutions and the corresponding defective renewal equation. We then provide an example of how the fluctuation identity of Doney and Kyprianou (2006) can be used to study the classical EDPF in a perturbed subordinator model. Section 5 offers some concluding remarks.
Section snippets
Risk model and discounted penalty functions
We consider a setup that generalizes the perturbed risk model of Dufresne and Gerber (1991). We replace their compound Poisson risk process by a subordinator , i.e., a Lévy process with nondecreasing paths representing cumulated claims, and their Brownian perturbation by a spectrally negative Lévy process , i.e., a Lévy process with only negative jumps accounting for any fluctuations in the risk process, such as claims arrivals, premium income and investment returns. We set , and
First-passage times for Lévy processes
Exit problems for Lévy processes are well understood and extensively studied in the literature (see Bertoin, 1996, Kyprianou, 2006). Recent results developed in Doney and Kyprianou (2006) provide a characterization of the joint law of five random variables related to the first-passage time of two-sided Lévy processes. In particular, more explicit expressions are available for the case of spectrally one-sided Lévy processes.
Let be a spectrally positive Lévy process with Lévy triplet
Characterization of the generalized EDPF
Let us consider the risk model defined in (1). Setting , we have that is a spectrally positive Lévy process with Lévy triplet and Laplace exponent Since , the original ruin problem is equivalent to studying the first-passage time of across , and the quintuple law (11) is all we need to derive the analytical characterization of the generalized EDPF (10). In particular, since the time of ruin can be identified with the
Conclusion
In this paper we have generalized the Expected Discounted Penalty Function introduced by Gerber and Shiu, 1997, Gerber and Shiu, 1998a, Gerber and Shiu, 1998b to include the last minimum of the surplus before ruin, in addition to the surplus before ruin and the deficit at ruin. We have studied the generalized EDPF in the context of a perturbed subordinator risk model, where the perturbation is channeled by a spectrally negative Lévy process. By using some recent developments in the theory of
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We are grateful to the reviewers for the careful reading of the manuscript and for helping us improve the paper substantially. We are also indebted to Professors Alejandro Balbas, Jose Garrido, and Andreas Kyprianou for their very helpful comments and suggestions. The usual disclaimer applies. This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) operating grant RGPIN-311660, by Le Fonds québécois de la recherche sur la nature et les technologies (FQRNT) operating grant NC-113809 and by the Society of Actuaries through the CKER Research Grant. An earlier version of this paper was previously circulated under the title ‘On the expected penalty function of three ruin-related random variables in a general Lévy risk model’.