Arbitrage and duality in nondominated discrete-time models

By Bruno Bouchard, Marcel Nutz. 2013

257 citations recorded by the source corpus.

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/cd1fb531f1bdb3863bef406b103f93826c3324d5

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In Management Science and Operations Research

Citations in the corpus

Stochastic control for a class of nonlinear kernels and applications (2015, unknown): “The extension to possibly incomplete markets has been carried out notably by [10] in discrete-time and more recently by [69] in continuous time for models possibly incorporating jumps.”
Martingale optimal transport and robust hedging in continuous time (2012, unknown): “However, several recent papers [3,8] study this question in discrete time.”
Model-free superhedging duality (2015, unknown): “In [5], a superhedging theorem is proven in the case of a nondominated class of priors P ′ ⊆ P .”
Martingale Benamou–Brenier: A probabilistic perspective (2017, unknown): “[17, 30, 49]) and the Skorokhod embedding problem [7, 35].”
Martingale optimal transport in the Skorokhod space (2014, background): “MSC: 91G10; 60G44 Keywords: Model-free Hedging; Martingale Optimal Transport; Skorokhod Space”
Exponential utility maximization under model uncertainty for unbounded endowments (2016, methodology): “In fact, Bouchard and Nutz deduce (28) from (27), and (27) from (26); see Theorem 4.9 as well as equation (5.1) and Theorem 5.1 in [8] respectively.”
Optimal transport and Skorokhod embedding (2013, background): “bson’s time change argument ([25, 26]), Theorem 5.2 is comparable to the work of Dolinsky and Soner [18, 17]. Similar duality results in a discrete time framework are established by Bouchard and Nutz [9] among others. Using the same argument as above, we see that it suces to establish Theorem 5.2 in”
An Explicit Martingale Version of the One-Dimensional Brenier's Theorem with Full Marginals Constraint (2016, unknown): “For models in discrete-time, we refer to Acciaio, Beiglböck, Penkner & Schachermayer[1], Beiglböck & Nutz [8], Beiglböck, Henry-Labordère & Touzi [6], Bouchard & Nutz [11], Campi, Laachir & Martini [14], Fahim & Huang [28], De Marco & Henry-Labordère [24].”
All adapted topologies are equal (2019, unknown): “[34, 13, 27, 23, 17, 33, 18, 12, 14] among many others).”
ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES (2014, background): “We emphasize that this result necessitates the continuity of S; it is to be compared to the discrete-time case of [6].”
Adapted Wasserstein distances and stability in mathematical finance (2019, methodology): “e eld, see the surveys of Hobson [34] and Ob loj [45]. Recently this has been complemented by techniques coming from (martingale) optimal transport, early papers which advance this viewpoint include [35, 14, 26, 15, 19, 24, 22, 13]. The literature on ‘local’ misspecication of volatility in a sense m”
Complete Duality for Martingale Optimal Transport on the Line (2015, background): “, [1, 8, 12, 13, 14, 19, 33] for models in discrete time and [6, 10, 15, 17, 18, 21, 22, 29, 31, 32, 35, 37] for continuous-time models.”
UTILITY MAXIMIZATION UNDER MODEL UNCERTAINTY IN DISCRETE TIME (2013, methodology): “We shall work under the no-arbitrage condition NA(P) of [6]; that is, given h ∈ R, h∆S ≥ 0 P-q.”