Structural Stability of the Financial Market Model: Continuity of Superhedging Price and Model Approximation

doi:10.1007/s40305-023-00524-x

Abstract

Abstract: The present paper continues the topic of our recent paper in the same journal, aiming to show the role of structural stability in financial modeling. In the context of financial market modeling, structural stability means that a specific “no-arbitrage” property is unaffected by small (with respect to the Pompeiu–Hausdorff metric) perturbations of the model’s dynamics. We formulate, based on our economic interpretation, a new requirement concerning “no arbitrage” properties, which we call the “uncertainty principle”. This principle in the case of no-trading constraints is equivalent to structural stability. We demonstrate that structural stability is essential for a correct model approximation (which is used in our numerical method for superhedging price computation). We also show that structural stability is important for the continuity of superhedging prices and discuss the sufficient conditions for this continuity.

Key words: Uncertainty, Structural stability, No arbitrage, Continuity of superhedging price, Compact-valued multifunction, Financial market model approximation, Trading constraints

CLC Number:

91A80
91G80

Sergey N. Smirnov. Structural Stability of the Financial Market Model: Continuity of Superhedging Price and Model Approximation[J]. Journal of the Operations Research Society of China, 2024, 12(1): 215-241.

TrendMD

References

[1] Smirnov, S.N.: Realistic models of financial market and structural stability. J. Math. (2021). https://doi.org/10.1155/2021/6651324
[2] Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19-48 (2004)
[3] Bayraktar, E., Zhang, Y., Zhou, Z.: A note on the fundamental theorem of asset pricing under model uncertainty. Risks 2(4), 425-433 (2014)
[4] Ostrovski, V.: Stability of no-arbitrage property under model uncertainty. Stat. Probab. Lett. 83, 89-92 (2013)
[5] Hou, Z., Obƚój, J.: Robust pricing-hedging dualities in continuous time. Finance Stoch. 22, 511-567 (2018)
[6] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: no arbitrage properties of the market. Autom. Remote. Control. 82, 172-187 (2021)
[7] Davis, M., Hobson, D.: The range of traded option prices. Math. Finance 17, 1-14 (2007)
[8] Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Financ. 26, 233-251 (2013)
[9] Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823-859 (2015)
[10] Burzoni, M., Frittelli, M., Maggis, M.: Universal arbitrage aggregator in discrete-time markets under uncertainty. Finance Stoch. 20, 1-50 (2016)
[11] Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27, 1452-1477 (2017)
[12] Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., Obƚój, J.: Pointwise arbitrage pricing theory in discrete time. Math. Oper. Res. 44, 1034-1057 (2018)
[13] Kolokoltsov, V.N.: Nonexpansive maps and option pricing theory. Kybernetika 34, 713-724 (1998)
[14] Bernhard, P., Engwerda, J.C., Roorda, B., et al.: The Interval Market Model in Mathematical Finance: Game-Theoretic Methods. Springer, New York (2013)
[15] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: sensitivity of solutions of Bellman-Isaacs equations and numerical methods. Comput. Math. Model. 31, 384-401 (2020)
[16] Carassus, L., Obƚój, J., Wiesel, J.: The robust superreplication problem: a dynamic approach. SIAM J. Financ. Math. 10, 907-941 (2019)
[17] Carassus, L., Obƚój, J., Wiesel, J.: Erratum to “The Robust Superreplication Problem: A Dynamic Approach”. SIAM Journal on Financial Mathematics 13, 653-655 (2022)
[18] Smirnov, S.N.: Guaranteed deterministic approach to superhedging: the semicontinuity and continuity properties of solutions of the Bellman-Isaacs equations. Autom. Remote. Control. 82, 2024-2040 (2021)
[19] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: financial market model, trading constraints, and the Bellman-Isaacs equations. Autom. Remote. Control. 82, 722-743 (2021)
[20] Carassus, L., Vargiolu, T.: Super-replication price: It can be OK. ESAIM Proc. Surv. 65, 241-281 (2018)
[21] Smirnov, S.N.: Geometric criterion for a robust condition of no sure arbitrage with unlimited profit. Mosc. Univ. Comput. Math. Cybern. 44, 146-150 (2020)
[22] Jacod, J., Shiryaev, A.N.: Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2, 259-273 (1998)
[23] Carassus, L., Lépinette, E.: Pricing without no-arbitrage condition in discrete time. J. Math. Anal. Appl. 505, 125441 (2022)
[24] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
[25] Cherif, D., Lépinette, E.: No-arbitrage conditions and pricing from discrete-time to continuous-time strategies. Ann. Finance 19, 141-168 (2023)
[26] Gerhold, S., Krühner, P.: Dynamic trading under integer constraints. Finance Stoch. 2018(22), 919-957 (2018)
[27] Smirnov, S.N.: The guaranteed deterministic approach to superhedging: Lipschitz properties of solutions of the Bellman-Isaacs equations. In: Frontiers of Dynamics Games: Game Theory and Management, Birkhäuser, St. Petersburg. pp. 267-288 (2019)
[28] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: relation between “deterministic” and “stochastic” settings in the case of no trading constraints. Theory Probab. Appl. 67, 548-569 (2023)
[29] Smirnov, S.N.: Structural stability threshold for the condition of robust no deterministic sure arbitrage with unbounded profit. Mosc. Univ. Comput. Math. Cybern. 45, 34-44 (2021)
[30] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: structural stability and approximation. Comput. Math. Model. 32, 129-146 (2021)
[31] Goberna, M.A., González, E., Martínez-Legaz, J.E., Todorov, M.I.: Motzkin decomposition of closed convex sets. J. Math. Anal. Appl. 364, 209-221 (2010)
[32] Smirnov, S.N.: Guaranteed deterministic approach to superhedging: mixed strategies and game equilibrium. Autom. Remote. Control. 83, 2019-2036 (2022)
[33] Smirnov, S.N.: A note on transition kernels for the most unfavourable mixed strategies of the market. J. Oper. Res. Soc, China (2023). https://doi.org/10.1007/s40305-023-00490-4
[34] Leichtweiß, K.: Konvexe Mengen. Springer-Verlag, Berlin Heidelberg (1979)
[35] Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis: Theory, Mathematics and Its Applications, vol. I. Springer, Berlin (1997)
[36] Smirnov, S.N.: A guaranteed deterministic approach to superhedging: a game equilibrium in the case of no trading constraints. J. Math. Sci. 248, 105-115 (2020)
[37] Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis A Hitchhiker’s Guide, 3rd edn. Springer-Verlag, Berlin (2006)
[38] Karlin, S.: Extreme points of vector functions. Proc. Am. Math. Soc. 4, 603-610 (1953)
[39] Tulcea, C., Ionescu, T.: Mesures dans les espaces produits. Atti Accad. Naz. Lincei Rend., 7, pp. 208-211 (1949)
[40] Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 4nd edition, Walter de Gruyter, New York (2016)
[41] Engelking, R.: General Topology. Heldermann, Berlin (1989)