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

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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.

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Notes

  1. We believe, however, that this mistake is of “local” character, so it cannot have any implications for the other results in [16].

  2. In probabilistic terminology, this corresponds to the predictability of \(H_t\) with respect to the filtration, generated by asset prices process.

  3. If \(h=(h^1,\cdots ,h^n)\) and \(y=(y^1,\cdots ,y^n)\) then \(hy=h^1y^1 + \cdots + h^ny^n\).

  4. See the definition of this metric, e.g., in [35, Subsection 1.1, Chapter 1].

  5. In the probabilistic setting, this geometric criterion, understood almost surely, was found in [22].

  6. This recurrence relation is the starting point of our research and, in principle, can also be regarded as a kind of axiom.

  7. The sufficient conditions for the validity of (BI), that are formulated in [19, Theorem 3.1], are discussed in [19, Remark 3.1].

  8. Specifically, a particular case of this statement is needed, namely, the support function of a set is non-negative iff the point 0 belongs to closure of convex hull of this set.

  9. Recall that \(\mathop {\textrm{cl}}\limits \bigl (\mathop {\textrm{conv}}\limits (K_t(\cdot ))\bigr )= \mathop {\textrm{conv}}\limits (K_t(\cdot ))\) for compact \(K_t(\cdot )\).

  10. Continuity with respect to the Pompeiu–Hausdorff metric.

  11. Note that in this case RNDSAUP is equivalent to RNDAO.

  12. In this case \(D_t(\cdot )\equiv \mathbb {R}^n\), so that the condition 3) of Theorem 1 is satisfied.

  13. In fact, in [30] we prove a more general result (with constructive estimates of uniform approximation).

  14. Recall that for a non-empty set B in a linear space X, the recession cone of B (denoted \(\mathop {\textrm{rec}}\limits (B)\)) is the set of all vectors \(y\in X\) such that the half-lines \(\{x+\lambda y, \lambda \in [0, + \infty ) \}\) lie in B for all \(x \in B\). If B is convex, its recession cone is convex as well. Refer e. g. to [24, Section 8].

  15. The domain \(\mathcal {A}\) of Q, i. e., the \(\sigma \)-algebra of all subsets of \(\mathbb {R}^n\), can be chosen depending on a particular class \(\mathcal {P}_t(\cdot )\). For example, we can take as \(\mathcal {A}\) the class of all subsets if \(\mathcal {P}_t(\cdot )\) consists of probability measures concentrated in a finite set of points. In other cases, we can take, e.g., a Borel \(\sigma \)-algebra, which is natural when the functions \(v^*_t\) are upper semicontinuous.

  16. Hedger’s mixed strategies make no sense because the bracketed expression in (8) is a linear function of h (due to no transaction costs), and the set \(D_t(\cdot )\) is convex.

  17. Note that the following inequality always holds: \(\rho _t(\cdot ) \geqslant \rho '_t(\cdot )\).

  18. For example, as \(\mathcal {P}_t(\cdot )\) we can choose the class \(\mathcal {P}^*(K_t(\cdot ))\) of measures concentrated in a finite set of points from \(K_t(\cdot )\) (to avoid additional requirements on the functions \(v^*_t\)). An alternative approach involves, e.g., the universal measurability of \(v^*_t\) (or any other smoothness property, e.g., semicontinuity).

  19. Recall that \(\mathop {\textrm{ri}}\limits (A)\) is the relative interior of convex set A.

  20. That is, for every closed \(C \subseteq \mathbb {R}^n\), the set \(L^-(C)= \{ x \in X:\, L(x) \cap C \ne \varnothing \}\) is closed.

  21. In the case of one risky asset, i.e., \(n=1 \), the condition NDAO is tantamount to RNDAO.

  22. Note that Proposition 3.8 of [16] is nevertheless valid, since in one-dimensional case the condition RNDAO is equivalent to NDAO, so that the condition of structural stability is fulfilled.

  23. We are referring here to a theorem from [39]; although it is about countable products of measurable spaces, it holds for finite products as well.

  24. Under a martingale measure the (discounted) price process is a martingale.

  25. Recall that \(\delta _x \) stand for a Dirac measure (see Definition 6).

  26. Recall that according to our assumptions about trading constraints the vector 0 must lie in \(D_t(\cdot )\). Moreover, the sets \(D_t(\cdot )\) are convex.

  27. More intuitively clear arguments follows from the alternative geometric criterion of RNDSAUP, as proven in [21, Theorem 2]: in our case it is evident that \(\mathop {\textrm{bar}}\limits (D_t(\cdot ))\) and \(\mathop {\textrm{conv}}\limits (K_t (\cdot ) )\) intersect transversally; therefore, RNDSAUP holds.

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  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 119991, Russian Federation

    Sergey N. Smirnov

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The work was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No75-15-2022-284.

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Smirnov, S.N. Structural Stability of the Financial Market Model: Continuity of Superhedging Price and Model Approximation. J. Oper. Res. Soc. China 12, 215–241 (2024). https://doi.org/10.1007/s40305-023-00524-x

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