Fourier Inversion Formulas in Option Pricing and Insurance

Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max (S – K, 0) also arises in excess-of-loss or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for payoffs which occur in insurance. Numerical examples are provided.      

Estimation of heavy-tailed probability density function with application to Web data

Summary  Common non-parametric estimators of a probability density function (PDF) show bad performance for heavy-tailed PDFs. Using a parametric approximation of the true cumulative distribution function (CDF), the transformation-retransformation of the data is explored here as a useful tool for the reliable PDF prediction. The PDF estimators are compared by their capacity to solve a classification problem. Simulation results and an application to Web data analysis are presented, too.     

Multivariate Gaussian criteria in SMAA

We consider stochastic multicriteria decision-making problems with multiple decision makers. In such problems, the uncertainty or inaccuracy of the criteria measurements and the partial or missing preference information can be represented through probability distributions. In many real-life problems the uncertainties of criteria measurements may be dependent. However, it is often difficult to quantify these dependencies. Also, most of the existing methods are unable to handle such dependency information.In this paper, we develop a method for handling dependent uncertainties in stochastic multicriteria group decision-making problems. We measure the criteria, their uncertainties and dependencies using a stochastic simulation model. The model is based on decision variables and stochastic parameters with given distributions. Based on the simulation results, we determine for the criteria measurements a joint probability distribution that quantifies the uncertainties and their dependencies. We then use the SMAA-2 stochastic multicriteria acceptability analysis method for comparing the alternatives based on the criteria distributions. We demonstrate the use of the method in the context of a strategic decision support model for a retailer operating in the liberated European electricity market.     

Designing a contaminated soil sampling strategy for human health risk assessment

Human health risk assessment is a site-based approach used to identify the potential health hazards which are induced by an old site contamination. For a proper evaluation of the daily doses of contaminants to which people will be exposed given the future occupation of the site, both a characterization and a quantification of soil pollution are needed. Such information can be provided by soil sampling. Thus the choice of the location, the number, depth and type of soil samples is very important and ought to follow a well-defined strategy. A review of contaminated site sampling practices in Europe and North America could not identify any completely formalized sampling strategy for human health risk assessment. On the contrary there are several approaches which can be roughly classified into two categories: a systematic sampling scheme over the whole site, on the one hand, and a sampling design driven by an initial knowledge of the contamination sources and fitted to the suspected pollution pattern, on the other. The first approach provides a complete coverage of the site but it may be rather expensive and entail useless sampling. The performance of the second depends on the quality of prior information. Actually both methods can be combined as explained hereafter. In view of the specificity of each site, the requirements of health risk assessment and the time and cost constraints, it seems difficult to work out a typical soil sampling strategy suitable for all sites. However, some recommendations can be made according to the site dimensions, the nature, degree and heterogeneity of contamination, and the (future) use of the site. The scientist should thus rely on a thorough examination of all available information (site history, geology and hydrogeology, soil properties, contaminants behaviour , etc.) to delimit contaminated areas as homogeneous as possible and then distribute the sampling points (e.g.using a sampling grid). They should also take the potential exposure paths into account in order to define the areas and soil strata to be sampled as a priority. Statistical and geostatistical tools can be helpful for formulating a sampling strategy as well as for interpreting the collected data. Received: 7 December 2001 Accepted: 24 February 2002     

Conditional VaR estimation using Pearson’s type IV distribution

This paper presents a new value at risk (VaR) estimation model for equity returns time series and tests it extensively on Stock Indices of 14 countries. Two most important stylized facts of such series are volatility clustering, and non-normality as a result of fat tails of the return distribution. While volatility clustering has been extensively studied using the GARCH model and its various extensions, the phenomenon of non-normality has not been comprehensively explored, at least in the context of VaR estimation. A combination of extreme value theory (EVT) and GARCH has been explored to analyze financial data showing non-normal behavior. This paper proposes a combination of the Pearson’s Type IV distribution and the GARCH (1, 1) approach to furnish a new method with superior predictive abilities. The approach is back tested for the entire sample as well as for a holdout sample using rolling windows.     

Asset and liability modelling for participating policies with guarantees

We study the problem of asset and liability management of participating insurance policies with guarantees. We develop a scenario optimization model for integrative asset and liability management, analyze the tradeoffs in structuring such policies, and study alternative choices in funding them. The nonlinearly constrained optimization model can be linearized through closed form solutions of the dynamic equations. Thus large-scale problems are solved with standard methods. We report on an empirical analysis of policies offered by Italian insurers. The optimized model results are in general agreement with current industry practices. However, some inefficiencies are identified and potential improvements are highlighted.     

Asymmetric impacts of global risk appetite on the risk premium for an emerging market

This paper analyzes the impact of global risk appetite on the risk premium utilizing high-frequency data. Taking the Turkish economy as our laboratory, we find that the risk premium volatility responds only to a worsening in the risk appetite for the Turkish economy, which is a result that we do not observe for the other emerging markets. Then, we investigate the role of current account dynamics on this asymmetric effect, by focusing also on an economy with similar current account performance. The empirical results find supporting evidence for the role of current account dynamics on the estimated asymmetry.     

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.     

Dual characterization of properties of risk measures on Orlicz hearts

We extend earlier representation results for monetary risk measures on Orlicz hearts. Then we give general conditions for such risk measures to be Gateaux-differentiable, strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical results are used to analyze various specific examples of risk measures. We thank Andreas Hamel and Michael Kupper for fruitful discussions and helpful comments. P. Cheridito has been supported by NSF Grant DMS-0642361, a Rheinstein Award and a Peek Fellowship. T. Li has been supported by a Marshall Scholarship and a Merage Fellowship.