Comparing Approximations for Risk Measures of Sums of Nonindependent Lognormal Random Variables

Vanduffel, Steven; Hoedemakers, Tom; Dhaene, Jan

doi:10.1080/10920277.2005.10596226

Cited by 66 publications

(48 citation statements)

References 16 publications

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“…Although some error analysis is known for comonotonic bounds of option prices (c.f. Vanduffel et al (2005) and Vanmaele et al (2006)), there appears to be no error estimation in the previous literature on the TVaR of comonotonic lower bound, which we shall use for approximations of risk measures for variable annuity guaranteed benefits. Hence, we first develop a formula for the error estimation.…”

Section: Comonotonicitymentioning

confidence: 99%

“…It is known in the literature (c.f. Vanduffel et al (2005)) that in a multivariate lognormal setup with appropriate choices of Λ, the comonotonic lower bound S l provides a better approximation of S than the comonotonic upper bound. Since (2.1) implies (2.2), we have that for any conditioning random variable Λ,…”

Section: Comonotonic Bounds For Sums Of Random Variablesmentioning

confidence: 99%

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Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior

2015

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractThe computation of various risk metrics is essential to the quantitative risk management of variable annuity guaranteed benefits. The current market practice of Monte Carlo simulation often requires intensive computations, which can be very costly for insurance companies to implement and take so much time that they cannot obtain information and take actions in a timely manner. In an attempt to find low-cost and efficient alternatives, we explore the techniques of comonotonic bounds to produce closed-form approximation of the risk measures for variable annuity guaranteed benefits. The techniques are further developed in this paper to address in a systematic way risk measures for death benefits with the consideration of dynamic policyholder behavior.

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Section: Comonotonicitymentioning

confidence: 99%

Section: Comonotonic Bounds For Sums Of Random Variablesmentioning

confidence: 99%

Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior

2015

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“…This property is particularly of interest in a multivariate lognormal setting. In such a setting, the lower bounds turn out to be very accurate, provided the appropriate choice is made for the conditioning random variable Λ, see, e.g., Vanduffel et al (2005b). The lower bound (9) is applied in Dhaene et al (2002b) to derive accurate approximations for European type Asian options in a Black & Scholes setting, in case of discrete averaging of the stock price.…”

Section: Convex Bounds For Sums Of Random Variablesmentioning

confidence: 99%

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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“…What is the probability of the sum or difference of log-normal random variables? The solution to this question has wide applications in many fields such as finance [1,2], actuarial science [3,4], and physics [5]. Especially, in physics, examples include wave transmittance in random media, dissipation rate of turbulence energy, and temporal fluctuations of some nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Volatility Effects on Correlated Log-Normal Random Variables

2017

Advances in Mathematical Physics

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The transition density function plays an important role in understanding and explaining the dynamics of the stochastic process. In this paper, we incorporate an ergodic process displaying fast moving fluctuation into constant volatility models to express volatility clustering over time. We obtain an analytic approximation of the transition density function under our stochastic process model. Using perturbation theory based on Lie-Trotter operator splitting method, we compute the leading-order term and the first-order correction term and then present the left and right skew scenarios through numerical study.

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Comparing Approximations for Risk Measures of Sums of Nonindependent Lognormal Random Variables

Cited by 66 publications

References 16 publications

Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior

Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior

Advanced Mathematical Methods for Finance

Stochastic Volatility Effects on Correlated Log-Normal Random Variables

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