A guaranteed deterministic approach to superhedging: financial market model,trading constraints and Bellman--Isaacs equations

Sergey N. Smirnov

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 10 (2018) no. 4, pp. 59-99.

Résumé

A guaranteed deterministic problem setting of super-replication with discrete time is proposed, as an alternative to the traditional probabilistic approach based on the use of the reference measure. Within the proposed framework, the reference measure is not needed, and aim of hedging of contingent claim is to guarantee coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The presentation focuses on achieving clarity, without aiming the greatest possible generality; this is the reason for the nature of a number of assumptions. The absence of transaction costs is assumed, the market is considered both with trade restrictions, and without trade restrictions. The game-theoretical approach immediately allows us to write down the corresponding Bellman–Isaac equations using economic interpretation of the problem.

Détail

Keywords: guaranteed estimates, deterministic price dynamics, super-replication, absence of arbitrage opportunities, Bellman–Isaacs equations, multi-valued mapping.
Mots-clés : option, arbitrage

Bibliographie

[1] Andreev N. A., Smirnov S. N., “Garantirovannyi podkhod k zadacham investirovaniya i khedzhirovaniya”, «Tikhonovskie chteniya»: nauchnaya konferentsiya, Tezisy dokladov: posvyaschaetsya pamyati akademika Andreya Nikolaevicha Tikhonova (29 oktyabrya–2 noyabrya 2018 g.), MAKS Press, M., 2018, 11

[2] Arkhipov V. M., Zakharov I. Yu., Naumenko V. V., Smirnov S. N., Predposylki vvedeniya kolichestvennykh mer effektivnosti dlya GER, Vysshaya shkola ekonomiki. Seriya WP16 «Finansovaya inzheneriya, risk-menedzhment i aktuarnaya nauka». Preprint No WP16/2007/05, 2007

[3] Bertsekas D., Shriv S., Stokhasticheskoe optimalnoe upravlenie: sluchai diskretnogo vremeni, Perevod s angl., ed. A.A. Yushkevich, Izdatelstvo «Nauka». Glavnaya redaktsiya fiziko-matematicheskoi literatury, M., 1985, 287 pp.

[4] Zakharov A. V., Mussa D. A., Garantirovannyi podkhod k zadache tsenoobrazovaniya i khedzhirovaniya dlya sluchaya obuslovlennogo obyazatelstva s neskolkimi riskovymi aktivami, Dep. VINITI, No 1092–B01, 2001, 19 pp.

[5] Zverev O. V., Khametov V. M., “Minimaksnoe khedzhirovanie optsionov evropeiskogo tipa na nepolnykh rynkakh (diskretnoe vremya)”, Obozrenie prikladnoi i promyshlennoi matematiki, 18:1 (2011), 26–54 | MR

[6] Zverev O. V., Khametov V. M., “Minimaksnoe khedzhirovanie optsionov evropeiskogo tipa na nepolnykh rynkakh (diskretnoe vremya). II”, Obozrenie prikladnoi i promyshlennoi matematiki, 18:2 (2011), 193–204 | MR

[7] Zverev O. V., Khametov V. M., “Minimaksnoe khedzhirovanie optsionov evropeiskogo tipa na kompaktnom (1,S)-rynke”, Obozrenie prikladnoi i promyshlennoi matematiki, 18:11 (2011), 121–122

[8] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Izdatelstvo «Nauka». Glavnaya redaktsiya fiziko-matematicheskoi literatury, M., 1977, 392 pp.

[9] Molchanov S. A., “Silno fellerovskoe svoistvo diffuzionnykh protsessov na gladkikh mnogoobraziyakh”, Teoriya veroyatnostei i ee primeneniya, 13:3 (1968), 493–498 | MR | Zbl

[10] Mussa D. A., Modelirovanie finansovykh rynkov metodami stokhasticheskikh differentsialnykh uravnenii, Dissertatsiya kand. fiz.-mat. nauk: 01.01.02, M., 2001, 72 pp. | Zbl

[11] Smirnov S. N., “Obschaya teorema teorii antagonisticheskikh igr o konechnom nositele smeshannoi strategii”, Doklady Akademii Nauk, 480:1 (2018), 25–28 | DOI | MR | Zbl

[12] Smirnov S. N., “Fellerovskoe perekhodnoe yadro s nositelyami mer, zadannymi mnogoznachnym otobrazheniem”, Trudy instituta matematiki i mekhaniki UrO RAN, 25, no. 1, 2019

[13] Smirnov S. N., Zakharov A. V., Polimatidi I. V., Balabushkin A. N., Sposob elektronnoi birzhevoi torgovli proizvodnymi finansovymi instrumentami, sposoby opredeleniya urovnya depozitnoi marzhi, sposoby uregulirovaniya situatsii s defitsitom marzhi, Patent #2226714, 2004

[14] Khametov V. M., Chalov D. M., “Evropeiskii optsion — eto beskonechnaya antagonisticheskaya igra”, Obozrenie prikladnoi i promyshlennoi matematiki, 11:2 (2004), 264–265

[15] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 1, Fakty. Modeli, FAZIS, M., 1998, 512 pp.

[16] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 2, Teoriya, FAZIS, M., 1998, 544 pp.

[17] Shiryaev A. N., Veroyatnost–1, MTsNMO, M., 2004, 520 pp.

[18] Aksamit A., Deng S., Obloj J., Tan X., “Robust pricing–hedging duality for American options in discrete time financial markets”, Financial Mathematics, 2018 | DOI | MR | Zbl

[19] Bayraktar E., Zhang Y., “Fundamental theorem of asset pricing under transaction costs and model uncertainty”, Mathematics of Operations Research, 41:3 (2016), 1039–1054 | DOI | MR | Zbl

[20] Bayraktar E., Zhou Z., “On arbitrage and duality under model uncertainty and portfolio constraints”, Mathematical Finance, 27:4 (2017), 988–1012 | DOI | MR | Zbl

[21] Bellman R., Dynamic Programming, Princeton Univ. Press, Princeton, 1957, 363 pp. | MR | Zbl

[22] Bernhard P., “The robust control approach to option pricing and interval models: An overview”, Numerical Methods in Finance, eds. Breton M., Ben-Ameur H., Springer, New York, 2005, 91–108 | DOI | MR | Zbl

[23] Bernhard P., Engwerda J. C., Roorda B., Schumacher J., Kolokoltsov V., Saint-Pierre P., Aubin J.-P., The Interval Market Model in Mathematical Finance: Game-Theoretic Methods, Springer, New York, 2013, 348 pp. | MR | Zbl

[24] Bertsekas D. P., Shreve S. E., Stochastic Optimal Control: The Discrete-Time Case, Academic Press, 1978, 340 pp. | MR | Zbl

[25] Bouchard B., Nutz M., “Arbitrage and duality in nondominated discrete-time models”, Annals of Applied Probability, 25:2 (2015), 823–859 | DOI | MR | Zbl

[26] Burzoni M., Frittelli M., Hou Z., Maggis M., Obloj J., Pointwise arbitrage pricing theory in discrete time, 2016, arXiv: 1612.07618

[27] Burzoni M., Frittelli M., Hou Z., Maggis M., “Universal arbitrage aggregator in discrete-time markets under uncertainty”, Finance and Stochastics, 20:1 (2016), 1–50 | DOI | MR | Zbl

[28] Burzoni M., Frittelli M., Maggis M., “Model-free superhedging duality”, The Annals of Applied Probability, 27:3 (2017), 1452–1477 | DOI | MR | Zbl

[29] Carassus L., Gobet E., Temam E., “A class of financial products and models where super-replication prices are explicit”, The 6th the Ritsumeikan International Conference on Stochastic processes and applications to mathematical finance (2006)

[30] Carassus L., Vargiolu T., Super-replication price for asset prices having bounded increments in discrete time, , 2010 hal.archives-ouvertes.fr/hal-0051166

[31] Cvitanić J., Shreve S., Soner H., “There is no nontrivial hedging portfolio for option pricing with transaction costs”, Annals of Applied Probability, 5 (1995), 327–355 | MR | Zbl

[32] Dana R.-A., Jeanblanc-Picqué M., Marchés financiers en temps continu, 1st edition, Economica, Paris, 1994, 330 pp.

[33] Denis L., Martini C., “A theoretical framework for the pricing of contingent claims in the presence of model uncertainty”, Annals of Applied Probability, 16:2 (2006), 827–852 | DOI | MR | Zbl

[34] F{ö}llmer H., Kabanov Y., “Optional decomposition and Lagrange multipliers”, Finance and Stochastics, 2:1 (1997), 69–81 | DOI | MR

[35] F{ö}llmer H., Schied A., Stochastic Finance. An Introduction in Discrete Time, 4nd edition, Walter de Gruyter, New York, 2016, 608 pp. | MR

[36] Hobson D., “Robust hedging of the lookback option”, Finance and Stochastics, 2:4 (1998), 329–347 | DOI | MR | Zbl

[37] Karoui N. E., Quenez M., “Dynamic programming and pricing of contingent claims in an incomplete market”, SIAM journal on Control and Optimization, 33:1 (1995), 29–66 | DOI | MR | Zbl

[38] Knight F. H., Risk, uncertainty and profit, Houghton Mifflin Co., New York, 1921, 381 pp.

[39] Kolokoltsov V. N., “Nonexpansive maps and option pricing theory”, Kybernetika, 34:6 (1998), 713–724 | MR | Zbl

[40] Kramkov D., “Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets”, Probability Theory and Related Fields, 105:4 (1996), 459–479 | DOI | MR | Zbl

[41] Neufeld A., Nutz M., “Superreplication under volatility uncertainty for measurable claims”, Electronic Journal of Probability, 18:48 (2013), 1–14 | MR

[42] Mycielski J., Sẃierczkowski S., “On the Lebesgue measurability and the axiom of determinateness”, Fundamenta Mathematicae, 54:1 (1964), 67–71 | DOI | MR | Zbl

[43] Obloj J., Wiesel J., A unified Framework for Robust Modelling of Financial Markets in discrete time, 2018, arXiv: 1808.06430

[44] Smirnov S. N., “Thoughts on Financial Risk Modeling: the Role of Interpretation”, Intelligent Risk, 2:2 (2012), 12–15

[45] Rockafellar R. T., Convex Analysis, Princeton University Press, Princeton, 1970, 451 pp. | MR | Zbl

[46] Solovay R., “A model of set-theory in which every set of reals is Lebesgue measurable”, Ann. Math., 92:1 (1970), 1–56 | DOI | MR | Zbl

[47] Soner H. M., Touzi N., Zhang J., “Dual formulation of second order target problems”, Annals of Applied Probability, 23:1 (2013), 308–347 | DOI | MR | Zbl