Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty
MathematicS In Action, Tome 11 (2022) no. 1, pp. 193-212.

We solve the problem of super-hedging European or Asian options for discrete-time financial market models where executable prices are uncertain. The risky asset prices are not described by single-valued processes but measurable selections of random sets that allows to consider a large variety of models including bid-ask models with order books, but also models with a delay in the execution of the orders. We provide a numerical procedure to compute the infimum price under a weak no-arbitrage condition, the so-called AIP condition, under which the prices of the non negative European options are non negative. This condition is weaker than the existence of a risk-neutral martingale measure but it is sufficient to numerically solve the super-hedging problem. We illustrate our method by a numerical example.

Publié le :
DOI : 10.5802/msia.24
Classification : 49J53, 60D05, 91G80
Mots clés : Super-hedging prices, Delayed information, Uncertainty, Conditional random sets, AIP Condition
Meriam El Mansour 1 ; Emmanuel Lépinette 1

1 CEREMADE, UMR CNRS 7534, Paris Dauphine University, PSL National Research, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France, and Gosaef, Faculty of Sciences of Tunis, Tunisia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Meriam El Mansour; Emmanuel Lépinette. Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty. MathematicS In Action, Tome 11 (2022) no. 1, pp. 193-212. doi : 10.5802/msia.24. https://msia.centre-mersenne.org/articles/10.5802/msia.24/

[1] Nacira Agram; Bernt Øksendal A financial market with singular drift and no arbitrage, Math. Financ. Econ., Volume 15 (2021) no. 3, pp. 477-500 | DOI | MR | Zbl

[2] Julien Baptiste; Laurence Carassus; Emmanuel Lépinette Pricing without martingale measures (2021) (https://hal.archives-ouvertes.fr/hal-01774150)

[3] Dirk Becherer; Klebert Kentia Good deal hedging and valuation under combined uncertainty about drift and volatility, Probab. Uncertain. Quant. Risk, Volume 2 (2017), 13, 40 pages | MR | Zbl

[4] Dirk Becherer; Klebert Kentia Hedging under generalized good-deal bounds and model uncertainty, Math. Methods Oper. Res., Volume 86 (2017), pp. 171-241 | DOI | MR | Zbl

[5] Dimitris Bertsimas; Vishal Gupta; Nathan Kallus Data-driven robust optimization, Math. Program., Volume 167 (2018) no. 2, pp. 235-292 | DOI | MR | Zbl

[6] Sara Biagini; Bruno Bouchard; Constantinos Kardaras; Marcel Nutz Robust fundamental theorem for continuous processes, Math. Finance, Volume 27 (2017) no. 4, pp. 963-987 | DOI | MR | Zbl

[7] Bruno Bouchard; Marcel Nutz Arbitrage in nondominated discrete-time models, Ann. Appl. Probab., Volume 25 (2015) no. 2, pp. 823-859 | MR | Zbl

[8] Matteo Burzoni; Marco Frittelli; Marco Maggis Universal arbitrage aggregator in discrete-time markets under uncertainty, Finance Stoch., Volume 20 (2016) no. 1, pp. 1-50 | DOI | MR | Zbl

[9] Matteo Burzoni; Frank Riedel; H. Mete Soner Viability and arbitrage under Knightian uncertainty, Swiss Finance Institute Research Paper, Volume 0 (2018), pp. 17-48 | Zbl

[10] Laurence Carassus; Emmanuel Lépinette Pricing without no-arbitrage condition in discrete-time, J. Math. Anal. Appl., Volume 505 (2022) no. 1, 125441, 21 pages | MR | Zbl

[11] Laurence Carassus; Jan Obłój; Johannes Wiesel The robust superreplication problem: a dynamic approach, SIAM J. Financial Math., Volume 10 (2019) no. 4, pp. 907-941 | DOI | MR | Zbl

[12] Patrick Cheridito; Michael Kupper; Ludovic Tangpi Duality formulas for robust pricing and hedging in discrete time, SIAM J. Financial Math., Volume 8 (2017), pp. 738-765 | DOI | MR | Zbl

[13] Christa Cuchiero; Irene Klein; Josef Teichmann A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting, Theory Probab. Appl., Volume 65 (2020) no. 3, pp. 388-404 | DOI

[14] Dimitry De Vallière; Yuri Kabanov; Christophe Stricker No-arbitrage criteria for financial markets with transaction costs and incomplete information, Finance Stoch., Volume 11 (2007) no. 2, pp. 237-251 | DOI | MR | Zbl

[15] Meriam El Mansour; Emmanuel Lépinette Conditional interior and conditional closure of a random set, J. Optim. Theory Appl., Volume 187 (2020) no. 2, pp. 356-369 | DOI | MR | Zbl

[16] Tolulope Fadina; Ariel Neufeld; Thorsten Schmidt Affine processes under parameter uncertainty, Probab. Uncertain. Quant. Risk, Volume 4 (2019), 5, 35 pages | MR | Zbl

[17] David G. Hobson Robust hedging of the lookback option, Finance Stoch., Volume 2 (1998), pp. 329-347 | DOI | Zbl

[18] Tomoyuki Ichiba; Mostafa Mousavi ption pricing with delayed information (2017) (https://www.researchgate.net/publication/318256675_Option_Pricing_with_Delayed_Information)

[19] Yuriy Kazmerchuk; Anatoliy Swishchuk; Jianhong Wu The pricing of options for securities markets with delayed response, Math. Comput. Simul., Volume 75 (2007) no. 3-4, pp. 69-79 | DOI | MR | Zbl

[20] Frank H. Knight Risk, Uncertainty, and Profits, Houghton Mifflin, 1921

[21] Emmanuel Lépinette; Ilya Molchanov Conditional cores and conditional convex hulls of random sets, J. Math. Anal. Appl., Volume 478 (2019) no. 2, pp. 368-392 | DOI | MR | Zbl

[22] Ariel Neufeld; Marcel Nutz Superreplication under volatility uncertainty for measurable claims, Electron. J. Probab., Volume 18 (2013), 48, 14 pages | MR | Zbl

[23] Jan Obłój; Johannes Wiesel Robust estimation of superhedging prices, Ann. Stat., Volume 49 (2021) no. 1, pp. 508-530 | MR | Zbl

[24] Bernt Øksendal; Agnès Sulem; Tusheng Zhang Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. Appl. Probab., Volume 43 (2011) no. 2, pp. 572-596 | DOI | MR | Zbl

[25] Huyen Pham; Xiaoli Wei; Chao Zhou Portfolio diversification and model uncertainty: a robust dynamic mean-variance approach (2018) (https://arxiv.org/abs/1809.01464)

[26] Marie-Claire Quenez Optimal portfolio in a multiple-priors model, Seminar on Stochastic Analysis, Random Fields and Applications IV (Progress in Probability), Volume 58, Birhäuser, 2004 | MR | Zbl

[27] Miklós Rásonyi; Andrea Meireles-Rodrigues On utility maximisation under model uncertainty in discrete-time markets, Math. Finance, Volume 31 (2021), pp. 149-175 | DOI

[28] R. Tyrrell Rockafellar; Roger J.-B. Wets Variational analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer, 1998 | DOI | Zbl

[29] Kristina Rognlien Dahl Pricing of claims in discrete time with partial information, Appl. Math. Optim., Volume 68 (2013) no. 2, pp. 145-155 | DOI | MR | Zbl

[30] Yuri F. Saporito; Jianfeng Zhang Stochastic control with delayed information and related nonlinear master equation, SIAM J. Control Optim., Volume 57 (2019) no. 1, pp. 693-717 | DOI | MR | Zbl

[31] Revaz Tevzadze; Teimuraz Toronjadze; Tamaz Uzunashvili Robust utility maximization for a diffusion market model with misspecified coefficients, Finance Stoch., Volume 17 (2009) no. 3, pp. 535-563 | DOI | MR | Zbl

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