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65
Comonotonic approximations for optimal portfolio selection problems
 Journal of Risk and Insurance
, 2005
"... We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskfree and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of ’constant mix ’ portfolios. First, we consider the portfolio selectio ..."
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Cited by 26 (15 self)
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We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskfree and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of ’constant mix ’ portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari’s dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically. 1
2006b). Bounds for functions of dependent risks
 Finance Stoch
"... Abstract The problem of finding the bestpossible lower bound on the distribution of a nondecreasing function of dependent risks is solved when or a lower bound on the copula of the portfolio is provided. In this paper we correct the statement and the proof of this result, given in Embrechts, Höing ..."
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Cited by 19 (8 self)
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Abstract The problem of finding the bestpossible lower bound on the distribution of a nondecreasing function of dependent risks is solved when or a lower bound on the copula of the portfolio is provided. In this paper we correct the statement and the proof of this result, given in Embrechts, Höing, and Juri (2003). The problem gets much more complicated in arbitrary dimensions when no information on the structure of dependence of the random vector is available. In this case we provide a bound on the distribution function of the sum of risks which we prove to be better than the one generally used in the literature. Key words copulas – dependent risks – dependency bounds – Fréchet bounds JEL Classification G10
The Skorokhod embedding problem and model independent bounds for option prices
 In ParisPrinceton Lecture Notes on Mathematical Finance
, 2010
"... This set of lecture notes is concerned with the following pair of ideas and concepts: 1) The Skorokhod Embedding problem (SEP) is, given a stochastic process X = (Xt)t≥0 and a measure µ on the state space of X, to find a stopping time τ such that the stopped process Xτ has law µ. Most often we take ..."
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Cited by 18 (3 self)
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This set of lecture notes is concerned with the following pair of ideas and concepts: 1) The Skorokhod Embedding problem (SEP) is, given a stochastic process X = (Xt)t≥0 and a measure µ on the state space of X, to find a stopping time τ such that the stopped process Xτ has law µ. Most often we take the process X to be Brownian motion, and µ to be a centred probability measure. 2) The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, riskneutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model? If we know a single call price, then we can calibrate the volatility of the BlackScholes model (but if we know the prices of more than one call then together they will typically be inconsistent with the BlackScholes model). At the other extreme, if we know the prices of call options for all strikes and maturities, then we can find a unique martingale diffusion consistent with those prices. If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but there may be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price process. On the other hand, after a time change it becomes a Brownian motion with a given distribution at a random time. Hence there is a 11 correspondence between candidate price processes which are consistent with observed prices, and solutions of the Skorokhod embedding problem. These notes are about this correspondence, and the idea that extremal solutions of the Skorokhod embedding problem lead to robust, model independent prices and hedges for exotic options. 1
Bounds for Functions of Multivariate Risks
, 2005
"... Li, Scarsini, and Shaked [8] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed ..."
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Cited by 15 (4 self)
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Li, Scarsini, and Shaked [8] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the kdimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications.
Constructing uncertainty sets for robust linear optimization
, 2006
"... doi 10.1287/opre.1080.0646 ..."
Solvency Capital, Risk Measures and Comonotonicity: A Review
 Department of Applied Economics
, 2003
"... In this paper we examine and summarize properties of several wellknown risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship bet ..."
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Cited by 12 (5 self)
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In this paper we examine and summarize properties of several wellknown risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of nonindependent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. (2003).
Risk measurement with equivalent utility principles
 In: Rüschendorf, Ludger (Ed.), Risk Measures: General Aspects and Applications (special issue), Statistics and Decisions
, 2006
"... Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature ..."
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Cited by 10 (5 self)
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Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decisionmaker. In incomplete financial markets, prices are no more unique but depend on the agents ’ attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari’s dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rankdependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities. Key words and phrases: Utility theories, risk measures, coherence, exponential utility, comonotonicity. JELcodes: D81, G10, G20. 1 Introduction and
The hurdlerace problem
 Insurance: Mathematics & Economics
, 2003
"... We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into accou ..."
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Cited by 9 (6 self)
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We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numerical examples illustrating the presented approximations. 1
On the evaluation of savingconsumption plans
 Journal of Pension Economics and Finance
, 2005
"... Knowledge of the distribution function of the stochastically compounded value of a series of future (positive and/or negative) payments is needed for solving several problems in an insurance or finance environment, see e.g. Dhaene et al. (2002 a,b). In Kaas et al. (2000), convex lower bound approxim ..."
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Cited by 8 (2 self)
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Knowledge of the distribution function of the stochastically compounded value of a series of future (positive and/or negative) payments is needed for solving several problems in an insurance or finance environment, see e.g. Dhaene et al. (2002 a,b). In Kaas et al. (2000), convex lower bound approximations for such a sum have been proposed. In case of changing signs of the payments however, the distribution function or the quantiles of the lower bound are not easy to determine, as the approximation for the random compounded value of the payments will in general not be a comonotonic sum. In this paper, we present a method for determining accurate and easy computable approximations for risk measures of such a sum, in case one first has positive payments (savings), followed by negative ones (consumptions). This particular cashflow pattern is observed in ‘saving consumption’ plans. In such a plan, a person saves money on a regular basis for a certain number of years. The amount available at the end of this period is then used to generate a yearly pension for a fixed number of years. Using the results of this paper one can find accurate and easy to compute answers to questions such as: "What is the minimal required yearly savings effort α during a fixed number of years, such that one will be able to meet, with a probability of at least (1 − ε), a given consumption pattern during the withdrawal period?" 1 1