Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lorden's inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

In this paper we extend some recent results on the comparison of multivariate risk vectors with respect to supermodular and related orderings. We introduce a dependence notion called the ‘weakly conditional increasing in sequence order’ that allows us to conclude that ‘more dependent’ vectors in this ordering are also comparable with respect to the supermodular ordering. At the same time, this ordering allows us to compare two risks with respect to the directionally convex order if the marginals increase convexly. We further state comparison criteria with respect to the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally, we discuss Fréchet bounds with respect to Δ-monotone functions when multivariate marginals are given. It turns out that, in the case of multivariate marginals, comonotone vectors no longer yield necessarily the largest risks but, in some cases, may even be vectors which minimize risk.

The purpose of this paper is to analyse the real-time trading of electricity. We address a model for an auction-like trading which captures key features of real-world electricity markets. Our main result establishes that, under certain conditions, the expected total payment for electricity is independent of the particular auction type. This result is analogous to the revenue-equivalence theorem known for classical auctions and could contribute to an improved understanding of different electricity market designs and their comparison.

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

In this paper, we discuss max-sum equivalence and convolution closure of heavy-tailed distributions. We generalize the well-known max-sum equivalence and convolution closure in the class of regular variation to two larger classes of heavy-tailed distributions. As applications of these results, we study asymptotic behaviour of the tails of compound geometric convolutions, the ruin probability in the compound Poisson risk process perturbed by an α-stable Lévy motion, and the equilibrium waiting-time distribution of the M/G/k queue.

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.

This paper proposes a class of complete financial market models, the benchmark models, with security price processes that exhibit intensity-based jumps. The benchmark or reference unit is chosen to be the growth-optimal portfolio. Primary security account prices, when expressed in units of the benchmark, turn out to be local martingales. In the proposed framework an equivalent risk-neutral measure need not exist. Benchmarked fair derivative prices are obtained as conditional expectations of future benchmarked prices under the real-world probability measure. This concept of fair pricing generalizes the classical risk-neutral approach and the actuarial present-value pricing methodology.

In this paper, we obtain sharp estimates for the expected payoffs and prices of European call options on an asset with an absolutely continuous price in terms of the price density characteristics. These techniques and results complement other approaches to the derivative pricing problem. Exact analytical solutions to option-pricing problems and to Monte-Carlo techniques make strong assumptions on the underlying asset's distribution. In contrast, our results are semi-parametric. This allows the derivation of results without knowing the entire distribution of the underlying asset's returns. Our results can be used to test different modelling assumptions. Finally, we derive bounds in the multiperiod binomial option-pricing model with time-varying moments. Our bounds reduce the multiperiod setup to a two-period setting, which is advantageous from a computational perspective.

In this paper, insurance claims X on [0, ∞) with tail distributions which are O(x −δ) for some δ > 1 are considered. Markets are assumed arbitrageable, the insurer setting a premium P > E[X]. Setting a premium as a fixed quantile of the loss distribution presents difficulties; for Pareto distributions with F(x) = 1 – (x + l)–δ ‘ultimately' (as δ ↓ 1) E[X] is larger than any quantile. When δ is near 1, premiums determined by weighting outcomes and a rule analogous to the expected utility principle are highly sensitive to change in δ, which is generally unknown or known only approximately. Under these circumstances, to protect insurers' interests, strategies are needed which provide some ‘premium stability' across a range of δ-values. We introduce a class of pricing functions which are functionally dependent on the governing loss distribution, and which are themselves distribution functions. We demonstrate that they provide a coherent framework for pricing insurance premiums when the loss distribution is fat tailed, and enable some degree of premium stability to be established.

Sapir (1998) calculated the probabilities of the expert rule and of the simple majority rule being optimal under the assumption of exponentially distributed logarithmic expertise levels. Here we find the analogous probabilities for the family of restricted majority rules, including the above two extreme rules as special cases, and the family of balanced expert rules. We compare the two families, the rules within each family, and all rules of the two families with the extreme rules.

The inability to predict the future growth rates and earnings of growth stocks (such as biotechnology and internet stocks) leads to the high volatility of share prices and difficulty in applying the traditional valuation methods. This paper attempts to demonstrate that the high volatility of share prices can nevertheless be used in building a model that leads to a particular cross-sectional size distribution. The model focuses on both transient and steady-state behavior of the market capitalization of the stock, which in turn is modeled as a birth-death process. Numerical illustrations of the cross-sectional size distribution are also presented.

This paper aims at enhancing the understanding of long-range dependence (LRD) by focusing on mechanisms for generating this dependence, namely persistence of signs and/or persistence of magnitudes beyond what can be expected under weak dependence. These concepts are illustrated through a discussion of fractional Brownian noise of index H ∈ (0,1) and it is shown that LRD in signs holds if and only if ½ < H < 1 and LRD in magnitudes if and only if ¾ ≤ H < 1. An application to discrimination between two risky asset finance models, the FATGBM model of Heyde and the multifractal model of Mandelbrot, is given to illustrate the use of the ideas.

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesney et al. (1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

In this paper, we consider the compound Poisson process that is perturbed by diffusion (CPD). We derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD and also present some examples of the CPD with an exponential individual claim amount distribution and a mixture exponential individual claim amount distribution.

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∊ [0,1], claims increase at rate β ∊ [0,1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping time and to specify the expected capital at that time.

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

A time-series consisting of white noise plus Brownian motion sampled at equal intervals of time is exactly orthogonalized by a discrete cosine transform (DCT-II). This paper explores the properties of a version of spectral analysis based on the discrete cosine transform and its use in distinguishing between a stationary time-series and an integrated (unit root) time-series.

A new representation for the characteristic function of the multivariate strictly geo-stable distribution is presented. The representation is appealing from a parametric viewpoint: its parameters have an intuitive probabilistic interpretation; and it is particularly useful for estimating the parameters of the geo-stable distribution.