Exponential Hedging and Entropic Penalties

Delbaen, Freddy; Grandits, Peter; Rheinländer, Thorsten; Samperi, Dominick; Schweizer, Martin; Stricker, Christophe

doi:10.1111/1467-9965.02001

Cited by 304 publications

(365 citation statements)

References 26 publications

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“…The problem of expected utility maximization in incomplete markets has been widely studied in the mathematical finance literature in recent years, and it has been shown that there is a duality relationship between maximization of expected utility and minimization of an appropriate divergence (e.g., Frittelli 2000, Rouge and El Karoui 2000, Goll and Rüschendorf 2001, Delbaen et al 2002, Slomczyński and Zastawniak 2004, Ilhan et al 2008, Samperi 2005. Most of this literature has focused on the case of exponential utility, for which the dual problem is the minimization of the reverse KL divergence D KL q p , as well as on issues that arise in multiperiod or continuous-time markets.…”

Section: Utility/entropy Duality In Incomplete Marketsmentioning

confidence: 99%

Scoring Rules, Generalized Entropy, and Utility Maximization

Jose

Nau

Winkler

2008

Operations Research

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Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative entropy), decision analysis (where new information can lead to improved decisions), and finance (where investors optimize portfolios based on their private information and risk preferences). In this paper, we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well-known generalized entropies and utility functions, shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization. In particular, we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy (divergence) with convenient properties, and they also correspond exactly to the solutions of expected utility maximization problems in which a risk-averse decision maker whose utility function belongs to the linear-risk-tolerance family interacts with a risk-neutral betting opponent or a complete market for contingent claims in either a one-period or a two-period setting. When the market is incomplete, the corresponding problems of maximizing linear-risk-tolerance utility with the risk-tolerance coefficient are the duals of the problems of minimizing the pseudospherical or power divergence of order between the decision maker's subjective probability distribution and the set of risk-neutral distributions that support asset prices.

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Section: Utility/entropy Duality In Incomplete Marketsmentioning

confidence: 99%

Scoring Rules, Generalized Entropy, and Utility Maximization

Jose

Nau

Winkler

2008

Operations Research

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“…It is related to the portfolio choice problem: under indifference valuation, the price of a claim is such that the agent is indifferent between selling and not selling the claim, provided that each of the two alternatives is combined with an optimal portfolio choice that maximizes utility. Particularly popular is exponential indifference valuation due to its analytical tractability on the one hand -the exponential form induces a convenient translation invariance property -and its theoretically appealing properties, especially in a dynamic context, on the other (El Karoui and Rouge [24], Delbaen et al [18], Kabanov and Stricker [42], Mania and Schweizer [53]). See also Hu, Imkeller and Müller [38], Becherer [4], Morlais [57,58], and Cheridito and Hu [12] for recent work on the problems of portfolio choice and indifference valuation.…”

Section: Introductionmentioning

confidence: 99%

Robust Portfolio Choice and Indifference Valuation

Laeven

Stadje²

2014

Mathematics of OR

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We solve, theoretically and numerically, the problems of optimal portfolio choice and indifference valuation in a general continuous-time setting. The setting features (i) ambiguity and ambiguity averse preferences, (ii) discontinuities in the asset price processes, with a general and possibly infinite activity jump part next to a continuous diffusion part, and (iii) general and possibly non-convex trading constraints. We characterize our solutions as solutions to Backward Stochastic Differential Equations (BSDEs). We prove existence and uniqueness of the solution to the general class of BSDEs with jumps having a drift (or driver) that grows at most quadratically, encompassing the solutions to our portfolio choice and valuation problems as special cases. We provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium, and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the jump part. We further compute our solutions in a few examples by numerically solving the corresponding BSDEs using regression techniques.

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“…We follow Tehranchi (2004), and define A as the set of strategies p satisfying E sup t2½0;T expðÀg 0 X p t Þo1 for some g 0 4g. There are other possible characterisations of admissibility in exponential utility maximisation problems, and these are discussed in depth in Delbaen et al (2002) and Schachermayer (2001), to which the interested reader is referred.…”

Section: Article In Pressmentioning

confidence: 99%

“…It is well known (for example Delbaen et al, 2002;Schachermayer, 2001) that the value functions uðxÞ and vðyÞ are conjugate:…”

Section: Article In Pressmentioning

confidence: 99%

The minimal entropy measure and an Esscher transform in an incomplete market model

Monoyios

2007

Statistics & Probability Letters

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We consider an incomplete market model with one traded stock and two correlated Brownian motions W ; e W . The Brownian motion W drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration e F:¼ð f F t Þ 0ptpT generated by e W . We show that the projections of the minimal entropy and minimal martingale measures onto f F T are related by an Esscher transform involving the correlation between W ; e W , and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an f F T -measurable European claim. r

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Exponential Hedging and Entropic Penalties

Cited by 304 publications

References 26 publications

Scoring Rules, Generalized Entropy, and Utility Maximization

Scoring Rules, Generalized Entropy, and Utility Maximization

Robust Portfolio Choice and Indifference Valuation

The minimal entropy measure and an Esscher transform in an incomplete market model

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