A generalised Gerber–Shiu measure for Markov-additive risk processes with phase-type claims and capital injections

In this paper we consider a risk reserve process where the arrivals (either claims or capital injections) occur according to a Markovian point process. Both claim and capital injection sizes are phase-type distributed and the model allows for possible correlations between these and the inter-claim times. The premium income is modelled by a Markov-modulated Brownian motion which may depend on the underlying phases of the point arrival process. For this risk reserve model we derive a generalised Gerber–Shiu measure that is the joint distribution of the time to ruin, the surplus immediately before ruin, the deficit at ruin, the minimal risk reserve before ruin, and the time until this minimum is attained. Numeral examples illustrate the influence of the parameters on selected marginal distributions.     

On the severity of ruin in a Markov-modulated risk model

We consider a Markov-modulated risk model in which the claim inter-arrivals, amounts and premiums are influenced by an external Markovian environment process. A system of Laplace transforms of the probabilities of the severity of ruin, given the initial environment state, is established from a system of integro-differential equations derived by Snoussi [The severity of ruin in Markov-modulated risk models Schweiz Aktuarver. Mitt., 2002, 1, 31–43]. In the two-state model, explicit formulas for probabilities of the severity of ruin are derived, when the initial reserve is zero or when both claim amount distributions are from the rational family. Numerical illustrations are also given.     

Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy

In this paper, we use a Markov-modulated regime switching approach to model various states of the economy, and study the pricing of vulnerable European options when the dynamics of the underlying asset value and the asset value of the counterparty follow two correlated jump-diffusion processes under regime switching. The correlation is modelled by both the diffusion parts and the pure jump parts which describe the uncertainty of the value of the risky assets. We develop a method to determine an equivalent martingale measure and a parsimonious representation of the risk-neutral density is provided. Based on this, we derive an analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, and implement the formula through numerical Laplace inversion.     

An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.     

Ruin Probabilities for Insurance Models Involving Investments

In this paper we study the ruin problem for insurance models that involve investments. Our risk reserve process is an extension of the classical Cramér-Lundberg model, which will contain stochastic interest rates, reserve-dependent expense loading, diffusion perturbed models, and many others as special cases. By introducing a new type of exponential martingale parametrized by a general rate function, we put various Cramér-Lundberg type estimations into a unified framework. We show by examples that many existing Lundberg-type bounds for ruin probabilities can be recovered by appropriately choosing the rate functions.     

A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications

We provide closed-form solutions for a continuous time, Markov-modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications. We further demonstrate that the two-state model provides the leptokurtic return features, volatility smile, and volatility clustering observed empirically for the Dow Jones Industrial Average (DJIA) and its component stocks. Using 10 years of stock return data, we confirm the existence of jump intensity switching and clustering, illustrate transition probabilities, and verify superior empirical fit over competing Poisson-style models.     

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.      

Traditional versus non-traditional reinsurance in a dynamic setting

We consider a stochastic risk reserve process whose risk exposure can be controlled dynamically by applying proportional reinsurance and by issuing CAT Bonds. The CAT Bond payments are only partly correlated with the insurers losses. The aim is to minimize the probability of ruin. Using a two-dimensional diffusion approximation we obtain a controlled diffusion problem which can be solved explicitly with the help of the HJB equation. We present some numerical results and discuss to which extend the proportional reinsurance can be replaced by issuing CAT Bonds.     

Lundberg parameters for non standard risk processes

We consider risk processes with delayed claims in a Markovian environment, and we study the asymptotic behaviour of finite and infinite horizon ruin probabilities under the small claim assumption. We also consider multivariate risk processes of the same kind, and we give upper and lower bounds for the Lundberg parameters of the corresponding total reserve. Our results have strong analogies with those one in the paper by Juri (Super modular order and Lundberg exponents, 2002).     

The Moments of the Time of Ruin in Markovian Risk Models

Abstract

We present an approach based on matrix-analytic methods to find moments of the time of ruin in Markovian risk models. The approach is applicable when claims occur according to a Markovian arrival process (MAP) and claim sizes are phase distributed with parameters that depend on the state of the MAP. The method involves the construction of a sample-path-equivalent Markov-modulated fluid flow for the risk model. We develop an algorithm for moments of the time of ruin and prove the algorithm is convergent. Examples show that the proposed approach is computationally stable.     

Risk theory in a Markovian environment

Abstract

We consider risk processes _{t t?0} with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Z_t } _t?0 such that β=β _i and B=B_i when Z_t=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.     

Computation of Multivariate Barrier Crossing Probability and its Applications in Credit Risk Models

Abstract

In this paper we consider computational methods of finding exit probabilities for a class of multivariate diffusion processes. Although there is an abundance of results for one-dimensional diffusion processes, for multivariate processes one has to rely on approximations or simulation methods. We adopt a Large Deviations approach to approximate barrier crossing probabilities of a multivariate Brownian Bridge. We use this approach in conjunction with simulation methods to develop an efficient method of obtaining barrier crossing probabilities of a multivariate Brownian motion. Using numerical examples, we demonstrate that our method works better than other existing methods. We mainly focus on a three-dimensional process, but our framework can be extended to higher dimensions. We present two applications of the proposed method in credit risk modeling. First, we show that we can efficiently estimate the default probabilities of several correlated credit risky entities. Second, we use this method to efficiently price a credit default swap (CDS) with several correlated reference entities. In a conventional approach one normally adopts an arbitrary copula to capture dependency among counterparties. The method we propose allows us to incorporate the instantaneous variance-covariance structure of the underlying process into the CDS prices.     

Das stochastische Re-Reserving

Next to premium risk, reserve risk is an essential part of underwriting risk for property and casualty insurers. Up to now, the literature on non-life reserve risk for the most part only considered an ultimative point of view, which means that the reserve risk uncertainty would be quantified up to final settlement. In the context of the new Solvency II project, as well as against the background of internal models in a several year context, for now an intensified discussion about a one-year horizon has occurred. Thereby, the reserve risk uncertainty would only be quantified up to a one-year claims development result. In this paper we would like to present a simulation method, called stochastic Re-Reserving for modelling the reserve risk on a yearly basis. In addition to that, we would like to make a paradigmatic comparison between stochastic Re-Reserving and an analytical approach, called claims development result method (see Merz and Wuethrich, Modelling the claims development result for solvency purposes. In: CAS E-Forum, pp. 542–568, 2008).     

The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion

Abstract

A Markov-modulated risk process perturbed by diffusion is considered in this paper. In the model the frequencies and distributions of the claims and the variances of the Wiener process are influenced by an external Markovian environment process with a finite number of states. This model is motivated by the flexibility in modeling the claim arrival process, allowing that periods with very frequent arrivals and ones with very few arrivals may alternate. Given the initial surplus and the initial environment state, systems of integro-differential equations for the expected discounted penalty functions at ruin caused by a claim and oscillation are established, respectively; a generalized Lundberg’s equation is also obtained. In the two-state model, the expected discounted penalty functions at ruin due to a claim and oscillation are derived when both claim amount distributions are from the rational family. As an illustration, the explicit results are obtained for the ruin probability when claim sizes are exponentially distributed. A numerical example also is given for the case that two classes of claims are Erlang(2) distributed and of a mixture of two exponentials.     

A ruin model with a resampled environment

ABSTRACT

This paper considers a Cramér–Lundberg risk setting, where the components of the underlying model change over time. We allow the more general setting of the cumulative claim process being modeled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs, we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes. We extend the classical Cramér–Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.     

Optimal reinsurance with expectile

In this paper, we study optimal reinsurance treaties that minimize the liability of an insurer. The liability is defined as the actuarial reserve on an insurer’s risk exposure plus the risk margin required for the risk exposure. The risk margin is determined by the risk measure of expectile. Among a general class of reinsurance premium principles, we prove that a two-layer reinsurance treaty is optimal. Furthermore, if a reinsurance premium principle in the class is translation invariant or is the expected value principle, we show that a one-layer reinsurance treaty is optimal. Moreover, we use the expected value premium principle and Wang’s premium principle to demonstrate how the parameters in an optimal reinsurance treaty can be determined explicitly under a given premium principle.     

On pricing barrier control in a regime-switching regulated market

We study a pricing barrier control problem in a regime-switching regulated market. In doing so, we analyze a class of one-dimensional reflected regime-switching diffusion processes. Such diffusion models arise as the key approximating processes in a regulated financial market system with the presence of regime changes. Our main goal is to determine optimal pricing barriers as solutions of long-run average mean–variance optimization problems. More precisely, the optimal barrier, if exists, will be to maximize the long-run average expected return (i.e. steady-state mean) subject to a selected level of long-run average risk (i.e. steady-state variance).     

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.