Optimal stopping for dynamic convex risk measures

Bayraktar, Erhan; Karatzas, Ioannis; Yao, Song

doi:10.1215/ijm/1336049984

Cited by 54 publications

(59 citation statements)

References 16 publications

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“…But these do not include the European utility indifference valuation, which in general is neither subadditive nor positively homogeneous. The closest article to our work on the American indifference value is Bayraktar et al [2], which solves the problem of optimal stopping under convex risk measures in a Brownian filtration setting. By making use of a representation of convex risky measures from Delbaen et al [8], they consider risk measures that can be written as a worst case expectation of the payoff plus a proper convex penalty function in which the Girsanov kernels of equivalent probability measures are plugged.…”

Section: Indifference Valuementioning

confidence: 99%

“…By this representation, which in turn relies on the predictable representation property of Brownian martingales as stochastic integrals, the problem can be solved by similar methods as for problems of robust (worst-case) combined stochastic control and optimal stopping, see, e.g., Karatzas and Zamfirescu [24]. In one aspect, our assumptions are slightly weaker than those of [2] since we only assume that the filtration is continuous instead of Brownian. But, more importantly, our methods are completely different from theirs.…”

Section: Indifference Valuementioning

confidence: 99%

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Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Kentia¹,

Kühn²

2018

SIAM J. Control Optim.

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Game contingent claims (GCCs) generalize American contingent claims in allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.2010 Mathematics Subject Classification. 91A10, 91A15, 60G40, 91B16, 91G10, 91G20.

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Section: Indifference Valuementioning

confidence: 99%

Section: Indifference Valuementioning

confidence: 99%

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Kentia¹,

Kühn²

2018

SIAM J. Control Optim.

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“…Other contributions to the minimax-relationship (1.1) use the property of recursiveness (see [3], [4], [5], [9])…”

Section: Introductionmentioning

confidence: 99%

Minimax theorems for American options without time-consistency

et al. 2018

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In this paper we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope with respect to a family of absolutely continuous probability measures. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and the path properties of the corresponding density process.

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“…A continuous-time version of Problem (1) has been considered in El , Quenez and Sulem (2014) and Grigorova et al (2015). Related works include, but are not limited to, Bayraktar et al (2010), Bayraktar and Yao (2011). The discrete-time version of Problem (1) has been introduced by Krätschmer and Schoenmakers in Krätschmer and Schoenmakers (2010), Example 2.7, who address the problem under stronger assumptions on the driver g than those made in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

Grigorova

2016

Stochastics

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We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) g-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamadène and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver g without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding g-expectation.

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Optimal stopping for dynamic convex risk measures

Cited by 54 publications

References 16 publications

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Minimax theorems for American options without time-consistency

Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

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