Upper and lower bounds for sums of random variables

Kaas, Rob; Dhaene, Jan; Goovaerts, Marc

doi:10.1016/s0167-6687(00)00060-3

Cited by 147 publications

(184 citation statements)

References 16 publications

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“…Defining the improved comonotonic upper bound, see relation (8), Kaas et al (2000) introduced implicitly the notion of conditional comonotonicity. This notion is later more formally considered by Jouini and Napp (2004) as a generalization of the classical concept of comonotonicity.…”

Section: Further Developments Of the Theorymentioning

confidence: 99%

“…Therefore, let us assume that apart from the knowledge of the marginals, there exists a random variable Λ with a given distribution function, such that the conditional distributions of the random variables X i , given Λ = λ, are known for all outcomes λ of Λ. Kaas et al (2000) derive the following improved convex order upper bound, denoted S ic , for this particular case:whereBased on an idea that stems from mathematical physics, Kaas et al (2000) propose the following convex order lower bound for S, denoted S , when the information available about the cdf of X is the same as the one that leads to the upper bound in (8):They remark that this lower bound has the nice property that it is a comonotonic sum, provided all terms E [X i | Λ] are increasing (or all are decreasing) functions of Λ. In this case, the quantiles and stop-loss premiums of S = n i=1 E [X i | Λ] follow immediately from the additivity properties of comonotonic sums in (2.1) and (5).…”

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confidence: 99%

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Advanced Mathematical Methods for Finance

Nunno¹,

Øksendal²

2011

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Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this paper, we give an extensive bibliographic overview -without claiming to be complete -of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements over the last five years. These applications range from pricing and hedging of derivatives over risk management to life insurance. ComonotonicityOver the last two decades, researchers in economics, financial mathematics and actuarial science have introduced results related to the concept of comonotonicity in their respective fields of interest. In this paper, we give an overview of the relevant literature in these research fields, with the main emphasis on the development of the theory and its applications in finance and insurance over the last five years. Although it is our intention to give an extensive bibliographic overview, due to the high number of papers on applications of comonotonicity, it is impossible to present here an exhaustive overview of the recent literature. Further, we restrict this paper to a description of how and where comonotonicity comes in and refer to the relevant papers for a detailed mathematical description. In order to make this paper self-contained, we also provide a short overview of the basic definitions and initial main results of comonotonicity theory, hereby referring to part of the older literature on this topic.The concept of comonotonicity is closely related to the following well-known result, which is usually attributed to both Hoeffding (1940) and Fréchet (1951): For any n-dimensional random vector X ≡ (X 1 , X 2 , . . . , X n ) with multivariate cumulative distribution function (cdf) F X and marginal univariate cdf's F X 1 , F X 2 , . . . , F Xn and for any x ≡ (x 1 , x 2 , . . . , x n ) ∈ R n it holds that F X (x) ≤ min (F X 1 (x 1 ) , F X 2 (x 2 ) , . . . , F Xn (x n )) .(1)In the sequel, the notation R n (F X 1 , F X 2 , . . . , F Xn ) will be used to denote the class of all random vectors Y ≡ (Y 1 , Y 2 , . . . , Y n ) with marginals F Y i equal to the respective marginals F X i . The set R n (F X 1 , F X 2 , . . . , F Xn ) is called the Fréchet class related to the random vector X.The upper bound in (1) is reachable in the Fréchet class R n (F X 1 , F X 2 , . . . , F Xn ) in the sense that it is the cdf of an n-dimensional random vector with marginals given by F X i , i = 1, 2, . . . , n. * Department of Mathematics, ECARES, Université Libre de Bruxelles, CP 210, 1050 Brussels, Belgium † Faculty of Business and Economics, Katholieke Universiteit, 3000 Leuven, Belgium. ‡ Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, 9000 Gent, Belgium 1 In order to prove the reachability property, consider a random variable U , uniformly distributed on the unit interval (0, 1). Then one has that F −1 X 1 (U ), F −1 X 2 (U ), . . . , F −1 Xn (U ) ∈ ...

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Section: Further Developments Of the Theorymentioning

confidence: 99%

mentioning

confidence: 99%

Advanced Mathematical Methods for Finance

Nunno¹,

Øksendal²

2011

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“…, Y n−1 ) is not completely specified because one only knows the marginal distribution functions of the random variables Y i . Kaas et al (2000) use comonotonic counterparts to derive bounds for such sums.…”

Section: Comonotonic Upper Boundmentioning

confidence: 99%

Bounds for the price of a European-style Asian option in a binary tree model

Reynaerts

Vanmaele

Dhaene³

et al. 2006

European Journal of Operational Research

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Abstract-Inspired by the ideas of Rogers and Shi (1995), Chalasani, Jha & Varikooty (1998) derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas, Dhaene & Goovaerts (2000) for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of the underlying at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black & Scholes model (1973) found in another paper Vanmaele et al. (2002). We notice that the intervals of Chalasani et al. do not always lie within the Black & Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black & Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.

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“…Ayyub (2002) provides a synoptic overview of the ideas. 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 .…”

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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Upper and lower bounds for sums of random variables

Cited by 147 publications

References 16 publications

Advanced Mathematical Methods for Finance

Advanced Mathematical Methods for Finance

Bounds for the price of a European-style Asian option in a binary tree model

Constructor:synthesizing information about uncertain variables.

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