Geodesic
A guaranteed deterministic approach to superhedging: no arbitrage market condition
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 11 (2019) no. 2, pp. 68-95.

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For a discrete-time superreplication problem, a guaranteed deterministic formulation is considered: the problem is to ensure a complete coverage of the contingent claim on an option under all scenarios which are set using a priori defined compacts, depending on the price history: price increments at each moment of time must lie in the corresponding compacts. The market is considered with trading constraints and without transaction costs. The statement of the problem is game-theoretic in nature and leads directly to the Bellman–Isaacs equations. In this article, we study several notions that formalize the “no arbitrage” property of the market in the context of the deterministic approach and study their properties. A new concept of robustness (structural stability) of the “no arbitrage” properties of the market is introduced.
Keywords: guaranteed estimates, deterministic price dynamics, super-replication, absence of arbitrage opportunities, Bellman-Isaacs equations, multi-valued mapping, robustness, structural stability of the model.
Mots-clés : option, arbitrage 
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Sergey N. Smirnov. A guaranteed deterministic approach to superhedging: no arbitrage market condition. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 11 (2019) no. 2, pp. 68-95. http://geodesic.mathdoc.fr/item/MGTA_2019_11_2_a3/

[1] Andronov A. A., Pontryagin L. S., “Grubye sistemy”, Doklady Akademii Nauk SSSR, XIV:5 (1937), 247–250

[2] Blagodatskikh V. I., Filippov A. F., “Differentsialnye vklyucheniya i optimalnoe upravlenie”, Tr. MIAN SSSR, 169, 1985, 194–252 | Zbl

[3] Borisovich Yu. G., Gelman B. D., Myshkis A. D., Obukhovskii V. V., “Mnogoznachnye otobrazheniya”, Itogi nauki i tekhn. Ser. Mat. anal., 19, 1982, 127–230 | Zbl

[4] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Izdatelstvo «Nauka», M., 1974 | MR

[5] Leikhtveis K., Vypuklye mnozhestva, Perevod s nem., Izdatelstvo «Nauka», M., 1985 | MR

[6] Repovsh D., Semenov P. V., “Teoriya E. Maikla nepreryvnykh selektsii. Razvitie i prilozheniya”, Uspekhi matematicheskikh nauk, 49:6(300) (1994), 151–190 | MR

[7] Polovinkin E. S., Mnogoznachnyi analiz i differentsialnye vklyucheniya, Izdatelstvo «Nauka», M., 2015

[8] Rokhlin D. B., “Kriterii otsutstviya asimptoticheskogo besplatnogo lencha na konechnomernom rynke pri vypuklykh ogranicheniyakh na portfel i vypuklykh operatsionnykh izderzhkakh”, Sibirskii zhurnal industrialnoi matematiki, 1:9 (2002), 133–144 | Zbl

[9] Rokhlin D. B., “Rasshirennaya versiya teoremy Dalanga-Mortona-Villindzhera pri vypuklykh ogranicheniyakh na portfel”, Teoriya veroyatn. i ee primen., 49:3 (2004), 503–521 | DOI

[10] Smirnov S. N., “Garantirovannyi deterministskii podkhod k superkhedzhirovaniyu: model rynka, torgovye ogranicheniya, bezarbitrazhnost i uravneniya Bellmana-Aizeksa”, Matematicheskaya Teoriya Igr i ee Prilozheniya, 10:4 (2018), 59–99 | Zbl

[11] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 1, Fakty. Modeli, Fazis, M., 1998

[12] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, v. 2, Teoriya, Fazis, M., 1998

[13] Cherny A., “General arbitrage pricing model: I-probability approach”, Séminaire de Probabilités, XL, Springer, Berlin–Heidelberg, 2007, 415–445 | MR

[14] Dana R. A., Le Van C., Magnien F., “On the different notions of arbitrage and existence of equilibrium”, Journal of economic theory, 87:1 (1999), 169–193 | DOI | MR | Zbl

[15] Delbaen F., “Representing martingale measures when asset prices are continuous and bounded”, Mathematical Finance, 2:2 (1992), 107–130 | DOI | Zbl

[16] Delbaen F., Schachermayer W., “A general version of the fundamental theorem of asset pricing”, Mathematische annalen, 300:1 (1994), 463–520 | DOI | MR | Zbl

[17] Delbaen F., Schachermayer W., “The fundamental theorem of asset pricing for unbounded stochastic processes”, Mathematische annalen, 312:2 (1998), 215–250 | DOI | MR | Zbl

[18] Harrison J. M., Kreps D. M., “Martingales and arbitrage in multiperiod securities markets”, Journal of Economic theory, 20:3 (1979), 381–408 | DOI | MR | Zbl

[19] Harrison J. M., Pliska S. R., “Martingales and stochastic integrals in the theory of continuous trading”, Stochastic processes and their applications, 11:3 (1981), 215–260 | DOI | MR | Zbl

[20] Jacod J., Shiryaev A. N., “Local martingales and the fundamental asset pricing theorems in the discrete-time case”, Finance And Stochastics, 2:3 (1998), 259–273 | DOI | MR | Zbl

[21] Kreps D. M., “Arbitrage and equilibrium in economies with infinitely many commodities”, Journal of Mathematical Economics, 8:1 (1981), 15–35 | DOI | MR | Zbl

[22] Kneser H., “Sur un theoreme fondamental de la theorie des jeux”, C.R. Acad. Sci. Paris, 234 (1952), 2418–2420 | MR | Zbl

[23] Merton R. C., “Theory of rational option pricing”, The Bell Journal of Economics, 4:1 (1973), 141–183 | DOI | MR | Zbl

[24] Pliska S. R., Introduction to Mathematical Finance: Discrete Time Models, Wiley, 1997

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

[26] Ross S. A., “A simple approach to the valuation of risky streams”, Journal of business, 51:3 (1978), 453–475 | DOI