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The numéraire portfolio in semimartingale financial models
 Finance Stoch
"... Abstract. We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingal ..."
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Cited by 81 (12 self)
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Abstract. We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a nofreelunchtype notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk ” (NFLVR) is not needed, only the weaker “No Unbounded Profit with Bounded Risk ” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal apriori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks. 0.1. Background and Discussion of Results. A broad class of models, that have been
Utility maximization in incomplete markets with random endowment
 Finance & Stochastics
, 2001
"... This paper solves a longstanding open problem in mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dua ..."
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Cited by 80 (3 self)
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This paper solves a longstanding open problem in mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is indeed possible if the dual problem and its domain are carefully defined. More
Robust utility maximization in a stochastic factor model
, 2006
"... We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and longterm trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a ..."
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Cited by 77 (6 self)
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We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and longterm trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with negative risk aversion and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.
Necessary and sufficient conditions in the problem of oprimal investment in incomplete markets
, 2002
"... Following [10] we continue the study of the problem of expected utility maximization in incomplete markets. Our goal is to find minimal conditions on a model and a utility function for the validity of several key assertions of the theory to hold true. In [10] we proved that a minimal condition on t ..."
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Cited by 54 (3 self)
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Following [10] we continue the study of the problem of expected utility maximization in incomplete markets. Our goal is to find minimal conditions on a model and a utility function for the validity of several key assertions of the theory to hold true. In [10] we proved that a minimal condition on the utility function alone, i.e. a minimal market independent condition, is that the asymptotic elasticity of the utility function is strictly less than 1. In this paper we show that a necessary and sufficient condition on both, the utility function and the model, is that the value function of the dual problem is finite. Key words: utility maximization, incomplete markets, Legendre transformation, duality theory.
Varianceoptimal hedging for processes with stationary and independent increments. The Annals of Applied Probability
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Duality theory for optimal investments under model uncertainty
, 2005
"... Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utili ..."
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Cited by 45 (7 self)
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Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaRλ, concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.
Optimal investment with random endowments in incomplete markets
 Ann. Appl. Prob
, 2004
"... In this paper, we study the problem of expected utility maximization of an agent who, in addition to an initial capital, receives random endowments at maturity. Contrary to previous studies, we treat as the variables of the optimization problem not only the initial capital but also the number of uni ..."
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Cited by 40 (3 self)
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In this paper, we study the problem of expected utility maximization of an agent who, in addition to an initial capital, receives random endowments at maturity. Contrary to previous studies, we treat as the variables of the optimization problem not only the initial capital but also the number of units of the random endowments. We show that this approach leads to a dual problem, whose solution is always attained in the space of random variables. In particular, this technique does not require the use of finitely additive measures and the related assumption that the endowments are bounded. 1. Introduction. A
Optimal Investments for Risk and AmbiguityAverse Preferences: A Duality Approach
, 2006
"... Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in te ..."
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Cited by 39 (8 self)
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Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in terms of convex risk measures. In this paper we study the corresponding problem of optimal investment over a given time horizon, using a duality approach and building upon the results by Kramkov and Schachermayer (1999, 2001).
Utility indifference pricing  an overview
 Indifference Pricing
, 2005
"... The idea of gamblers ranking risky lotteries by their expected utilities dates back to Bernoulli [4]. An individual’s certainty equivalent amount is the certain amount of money that makes them indifferent between the return from the gamble and this amount, as described in Chapter 6 of MasColell et ..."
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Cited by 32 (4 self)
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The idea of gamblers ranking risky lotteries by their expected utilities dates back to Bernoulli [4]. An individual’s certainty equivalent amount is the certain amount of money that makes them indifferent between the return from the gamble and this amount, as described in Chapter 6 of MasColell et al [59]. Certainty equivalent amounts and the principle of equimarginal
Minimizing Expected Loss Of Hedging In Incomplete And Constrained Markets
, 1998
"... We study the problem of minimizing the expected discounted loss E e \Gamma R T 0 r(u)du (C \Gamma X x;ß (T )) + when hedging a liability C at time t = T , using an admissible portfolio strategy ß(\Delta) and starting with initial wealth x. The existence of an optimal solution is establ ..."
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Cited by 32 (0 self)
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We study the problem of minimizing the expected discounted loss E e \Gamma R T 0 r(u)du (C \Gamma X x;ß (T )) + when hedging a liability C at time t = T , using an admissible portfolio strategy ß(\Delta) and starting with initial wealth x. The existence of an optimal solution is established in the context of continuoustime, Ito processes incomplete market models, by studying an appropriate dual problem. It is shown that the optimal strategy is of the form of a knockout option with payoff C, where the "domain of the knockout" depends on the value of the optimal dual variable. We also discuss a dynamic measure for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C. Key words: expected loss, hedging, incomplete markets, portfolio constraints, dynamic measures of risk. AMS 1991 subject classifications: Primary 90A09, 90A46; secondary 93E20, 60H30. Research supported in part by the Nationa...