Stochastic Upper Bounds for Present Value Functions

Goovaerts, Marc; Dhaene, Jan; Schepper, Ann De

doi:10.2307/253674

Cited by 40 publications

(30 citation statements)

References 20 publications

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“…From a mathematical point of view, the results for the infinite market case described in Theorem 1 and Theorem 3 are very similar to finding a best upper bound for a stop-loss premium of a sum of non-independent random variables in terms of stop-loss premia of the marginals involved, as described in Goovaerts et al (2000) and Kaas et al (2000). Early references to solutions for this problem are Meilijson & Nadas (1979) and Rüschendorf (1983).…”

Section: Definitionmentioning

confidence: 74%

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Static super-replicating strategies for a class of exotic options

Chen

Deelstra

Dhaene

et al. 2008

Insurance: Mathematics and Economics

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In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity.We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. (2000) who considered this problem for Asian options in the infinite market case. Laurence and Wang (2004) and Hobson et al. (2005) considered this problem for basket options, in the infinite as well as in the finite market case.As opposed to Hobson et al. (2005) who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.

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Section: Definitionmentioning

confidence: 74%

“…Taking the left hand border of this interval to be the value of the inverse cdf at p, leads to the inverse as defined in (12). Similarly, we define F −1+ X (p) as the right hand border of the interval:…”

Section: The Inverse Of a Cumulative Distribution Function Is Usuallymentioning

confidence: 99%

Static super-replicating strategies for a class of exotic options

Chen

Deelstra

Dhaene

et al. 2008

Insurance: Mathematics and Economics

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show abstract

“…for U ∼ U[0, 1], see derivations in Goovaerts et al (2000). This result means that the total loss Y T in the convex order sense, comprised of the most risky joint vector of losses with given marginals, has the comonotonous joint distribution.…”

Section: Scale: Development and Accident Year Variance Model Structuresmentioning

confidence: 99%

“…Figure 16 demonstrates how the estimated risk marginp i changes across accident years, superimposed with its creditable interval. Figure 17 displays the corresponding changes in estimated variance and skewness using the variance and skewness equations in (12) and (13) respectively. The risk marginp starts at 0.895 at accident year 1 when the variance is quite high.…”

Section: Skewnessmentioning

confidence: 99%

Risk Margin Quantile Function via Parametric and Non-Parametric Bayesian Quantile Regression

2014

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We develop quantile regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modelling frameworks are considered based around parametric and nonparametric quantile regression models which we develop specifically in this insurance setting.In the parametric quantile regression framework, several models including the flexible generalized beta distribution family, asymmetric Laplace (AL) distribution and power Pareto distribution are considered under a Bayesian regression framework. The Bayesian posterior quantile regression models in each case are studied via Markov chain Monte Carlo (MCMC) sampling strategies.In the nonparametric quantile regression framework, that we contrast to the parametric Bayesian models, we adopted an AL distribution as a proxy and together with the parametric AL model, we expressed the solution as a scale mixture of uniform distributions to facilitate implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.Asymmetric Laplace distribution, Bayesian inference, Markov chain Monte Carlo methods, Quantile regression, loss reserve, risk margin, central estimate.

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“…Several applications of the theory have been made in risk analysis (e .g., Kriegler and Held 2004 ;Bernat et al 2004 ;Ferson and Hajagos 2004 ;EPA 2003a,b ;Ferson and Tucker 2003 ;EPA 2002 ;Regan et al 2002a,b ;Kaas et al . 2000;Goovaerts et al . 2000) .…”

Section: 2 Probability Boxes (P-boxes)mentioning

confidence: 99%

Constructor:synthesizing information about uncertain variables.

Tucker

Ferson

Hajagos

et al. 2005

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Constructor is software for the Microsoft Windows microcomputer environment that facilitates the collation of empirical information and expert judgment for the specification of probability distributions, probability boxes, random sets or Dempster-Shafer structures from data, qualitative shape information, constraints on moments, order statistics, densities, and coverage probabilities about uncertain unidimensional quantities . These quantities may be real-valued, integer-valued or logical values .

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Stochastic Upper Bounds for Present Value Functions

Cited by 40 publications

References 20 publications

Static super-replicating strategies for a class of exotic options

Static super-replicating strategies for a class of exotic options

Risk Margin Quantile Function via Parametric and Non-Parametric Bayesian Quantile Regression

Constructor:synthesizing information about uncertain variables.

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