Abstract
We consider a guaranteed deterministic statement of the problem of discrete-time superreplication: the aim of hedging a contingent claim is to ensure the coverage of possible payout under the option contract for all feasible scenarios. These scenarios are given by a priori given compact sets that depend on the price history: the price increments at each time must lie in the corresponding compact sets. The lack of transaction costs is assumed; the market with trading constraints is considered. The game-theoretic interpretation implies that the corresponding Bellman–Isaacs equations hold. In the present paper, we propose several conditions for the solutions of these equations to be semicontinuous or continuous.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the deterministic approach that we propose, a reference probability measure is not initially specified as it is assumed in the probabilistic approach; see, e.g., [8].
For the purposes of risk management, derivative financial instruments are used, which, as a rule, are nondeliverable contracts.
This includes transactions with underlying assets and a risk-free asset.
This term originated from the fact that it is impossible to replicate contingent liabilities in incomplete markets (which is possible only in complete markets).
In other words, it is the premium charged to the option buyer if the option seller uses pricing that is consistent with superhedging.
The increments are taken “backward,” i.e. \(\Delta X_t=X_t-X_{t-1} \), where \(X_t \) is the vector of discounted prices at time \(t \); the \(i \)th component of this vector is the unit price of the \(i \)th asset.
We assume that the risk-free asset has a constant price equal to one.
The dot designates variables describing price evolution. More precisely, this is the back story, \(\bar {x}_{t-1}= \) \((x_0,\ldots ,x_{t-1})\in (\mathbb {R}^n)^t \) for \(K_t \), while for the functions \(v^*_t \) and \(g_t \) introduced below, this is the history, \(\bar {x}_t=(x_0,\ldots ,x_t)\in (\mathbb {R}^n)^{t+1} \).
The vector \(h \) describes the size of the positions taken in assets, i.e., the \(i \)th component of this vector is the number of units of the \(i \)th asset bought or sold.
The \(\bigvee \) sign denotes the maximum, and \(hy=\langle h,y\rangle \) is the inner product of a vector \(h \) by a vector \(y \).
The neighborhoods of points \(-\infty \) and \(+\infty \) have the form \([-\infty , a) \), \(a \in \mathbb {R}\) and \((b, +\infty ]\), \(b \in \mathbb {R} \), respectively.
Here \(x+A=\{z: \; z-x \in A\} \).
With no trading constraints, this interpretation allows one to give an economically important explanation of the origin of risk-neutral probabilities as one of the properties of the most unfavorable mixed market strategies.
One of our fundamental considerations is that since the description of uncertainty in the market cannot be accurate in practice, fundamental properties such as the “no arbitrage” market (in one sense or another), should not change under small disturbances of the market model. In this regard, in [4] we introduced a new concept of structural stability for the market to be “no arbitrage” and established criteria of a geometric nature for this property.
It is hardly possible to give economic reasons for stochastic price dynamics to be specified by transition kernels (conditional probabilities for a given price history) that do not satisfy the Feller property.
The last condition reflects the realism of deterministic scenarios of price increments.
In this paper, the general case of topological spaces is considered, and the necessary conditions for the existence of a Feller kernel with given supports are weaker than the sufficient conditions for its existence, but in the case of a finite-dimensional Euclidean space, the necessary and sufficient conditions coincide.
In this case, the compactness of \(K_t(\cdot )\), \(t=1,\ldots ,N \) is not required—only closedness is assumed (as in potential topological supports of probability measures). Note that in the case of compact-valued mappings, the lower semicontinuity implies the lower \(h \)-semicontinuity.
As per assumption (C) in [3].
The upper (lower) semicontinuity in the sense of Pompeiu–Hausdorff, in other words, the upper (lower) \(h \)-semicontinuity of the multivalued mapping \(F:X\mapsto \mathcal {N}(Y)\) at the point \(x_0\in X \), is defined for a topological space \(X \) and a metric space \(Y \) with a metric \(\rho \) as the continuity of the numerical function \(x\mapsto e_\rho (F(x),F(x_0))\) (respectively, \(x\mapsto e_\rho (F(x_0),F(x))\)) at the point \(x_0 \), where \(e_\rho (A,B) \) is the Pompeiu deviation of the set \(A \) from the set \(B \), \(e_\rho (A,B)=\sup \{\rho (x,B),x\in A\}\), \(\rho (x,B)=\inf \{\rho (x,x^{\prime}),\;x^{\prime}\in B\}\). Note that the Pompeiu–Hausdorff distance is \(h_\rho (A,B)=e_\rho (A,B)\bigvee e_\rho (B,A) \). A multivalued mapping is upper (lower) \(h \)-semicontinuous if it is upper (lower) semicontinuous at all points in the domain.
Upper (lower) semicontinuity is defined as the openness of the set \(\{x\in X:\;F(x)\subseteq G\} \) for any open \(G\subseteq Y \) (respectively, as the openness of the sets \(\{x\in X:\;F(x)\cap G\neq \emptyset \}\) for any open \(G\subseteq Y \)).
Here image is understood in the sense of a multivalued mapping.
Note that in the book [6], the compactness of \(F \) is included in the definition of upper semicontinuity, in addition to the fact that the set \(\{x\in X:\;F(x)\subseteq G\} \) is open for any open \(G\subseteq Y \).
That is, representable in the form of a countable intersection of open sets.
This result is often called “Berge’s maximum theorem;” in [9], it is Theorem 3.4.
Simultaneously upper and lower semicontinuous multivalued mapping.
That is, \(M(x) \) is the set of maximizers, those \(y\in Y \) for which the maximum is attained in (2.2) for a given \(x\in X \).
The support function of a compact set takes finite values.
The continuity of a multivalued mapping means simultaneously upper and lower semicontinuity. For compact-valued mappings, this is equivalent to continuity in the Pompeiu–Hausdorff metric.
In fact, the normality of the topological space \(Y\) suffices.
Margin trading in the financial market implies the presence of intermediaries (brokers) who allow (upon the conclusion of a general agreement with a market participant) borrowing in securities. At the same time, the regulator usually establishes requirements that the share of own funds in the portfolio of a participant in margin trading should not be lower than the established level; this leads to trading constraints.
\(\alpha _t \) is the maximum allowable debt (for example, bank credit limit) taken with negative sign.
This cone is convex and contains the point \(0\).
The empty set is considered compact.
Lower semicontinuous (the terminology of [11]).
If a function takes finite values on a nonempty convex set and is \(+\infty \) outside it.
Of course, (SSR) implies \(\check {K}_t(x_0)\neq \emptyset \). We can interpret \(\check {K}_t(\cdot ) \) as a new dynamics of the market with uncertainty diminished compared with \(K^*_t(\cdot ) \); in this case, the \( \check {K}_t(\cdot ) \) are convex compact sets.
A multivalued mapping \(F: X \mapsto \mathcal {N}(Y) \) is said to be closed if for Cauchy nets \(x_\alpha \) and \(y_\alpha \) such that \(x_\alpha \to x \), \(y_\alpha \in F(x_\alpha ) \), and \(y_\alpha \to y \) one has \(y \in F(x) \). In other words, the graph of the mapping \(F \) is closed.
This is the result in [2] concerning the Lipschitz property (with constant \(A\)) for support functions.
A closed mapping takes closed values; see, e.g., Remark 2.12 in [9].
This condition appears in Theorem 4.1 in [4] and implies the full size of compact sets \( K_t(\cdot ) \), i.e., \( \mathrm {int}\thinspace (K^*_t(\cdot ))\neq \emptyset \).
Thus, the compact sets \(\check {K}_t(\cdot )\) will also be full-size under the assumptions of Proposition 3.2.
However, to ensure that \(a>0 \) (which is required in the proof of Proposition 3.3), one can simply require \(a>C \).
In the terminology of the book [2], the weakly continuous map \(x\mapsto D_t(\cdot )\) is simultaneously upper and lower weakly semicontinuous; moreover, weak upper semicontinuity is equivalent to closedness (see [2, Theorem 14.7 ]) and weak lower semicontinuity coincides with (ordinary) lower semicontinuity (see [2, Remark 14.1 ]).
REFERENCES
von Leichtweiß, K., Konvexe Mengen, Berlin: VEB Deutscher Verlag der Wissenschaften, 1980. Translated under the title: Vypuklye mnozhestva, Moscow: Nauka, 1985.
Polovinkin, E.S., Mnogoznachnyi analiz i differentsial’nye vklyucheniya (Multivalued Analysis and Differential Inclusions), Moscow: Nauka, 2015.
Smirnov, S.N., A guaranteed deterministic approach to superhedging: financial market model, trading constraints, and the Bellman–Isaacs equations, Autom. Remote Control, 2021, vol. 82, no. 4, pp. 722–743.
Smirnov, S.N., A guaranteed deterministic approach to superhedging: no arbitrage properties of the market, Autom. Remote Control, 2021, vol. 82, no. 1, pp. 345–360.
Smirnov, S.N., A Feller transition kernel with measure supports given by a set-valued mapping, Proc. Steklov Inst. Math., 2020, vol. 308, suppl. 1, pp. 188–S195.
Berge, C., Espaces topologiques: Fonctions multivoques, Paris: Dunod, 1959.
Berge, C., Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces, and Convexity, London: Oliver & Boyd, 1963.
Föllmer, H. and Schied, A., Stochastic Finance. An Introduction in Discrete Time, New York: Walter de Gruyter, 2016. 4th ed.
Hu, S. and Papageorgiou, N., Handbook of Multivalued Analysis: Theory, vol. I. Mathematics and Its Applications. Vol. 419 , Berlin: Springer, 1997.
Kneser, H., Sur un théorème fondamental de la théorie des jeux, CR Acad. Sci. Paris, 1952, vol. 234, pp. 2418–2420.
Rockafellar, R.T., Convex Analysis, Princeton: Princeton Univ. Press, 1970.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1134/S0005117921110163
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117921110163