Skip to main content
SpringerLink
Log in

A Guaranteed Deterministic Approach to Superhedging: Financial Market Model, Trading Constraints, and the Bellman–Isaacs Equations

  • MATHEMATICAL GAME THEORY AND APPLICATIONS
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

A guaranteed deterministic problem setting of superreplication in 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 payoff under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compact sets, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compact sets. 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, and the market is considered both with and without trade constraints. The game-theoretic approach immediately allows us to write down the corresponding Bellman–Isaacs equations using economic interpretation of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In this case, in essence, only the class of “almost impossible” (zero probability) events is important, but the measure itself is not of fundamental importance.

  2. In our proposed “deterministic” approach, in contrast to the “probabilistic” approach, no reference measure is initially specified.

  3. A guaranteed approach for the general case of a control problem under uncertainty is presented in the book by A.B. Kurzhansky [8].

  4. A number of authors, in particular Fölmer and Shied [35], qualify statements of the problem that are not related to probabilistic ones as formalization of Knightian uncertainty. Such a name, however, seems to be incorrect, because Knight [38, Ch. 8] speaks of uncertainty that cannot be quantified (as opposed to risk).

  5. In 2013, Pierre Bernhard and six other authors published the book [23], which contains results close to ours.

  6. The invited article [44], published in the issue of Intelligent Risk magazine timed to coincide with PRMIA’s Global Risk Conference dedicated to the 10th anniversary of the Professional Risk Managers’ International Association (PRMIA), presents our reflections on the importance of interpreting the concepts, assumptions, and mathematical results of financial models, in particular, in relation to risk-neutral assessment of financial instruments. It is shown that this is of fundamental importance for the correct understanding of the phenomenon of “bubbles” in financial markets.

  7. These are Rainbow options.

  8. Perhaps this is due to the fact that Kolokoltsov’s work considers only the case of no trading constraints.

  9. In the terminology in [23, Ch. 12], \(K \) is “strongly positively complete” for NDAO and “positively complete” for NDSA. The acronyms NDAO and NDSA are derived from No Deterministic Arbitrage Opportunities and No Deterministic Sure Arbitrage, respectively.

  10. The model in [23, Chaps. 13 and 14] corresponds to a fairly regular behavior (Lipschitz property) of multivalued mappings, so that the deterministic Kolokoltsov model, as a rule, leads to coincidence of the results with the stochastic approach (the question also depends on the “smoothness” of the payout functions, which can lead to a discrepancy between the results of two approaches, for example, for binary options).

  11. However, a prudent economist can nevertheless theoretically admit (but not justify) such a possibility, in a very short term, given the increased role of algorithmic (high-frequency) trading in recent years, which, as practice shows, is capable of provoking a sharp collapse in prices (flash crash) under some (not 100% clear) situation in the market.

  12. In particular, within the framework of the robust approach for the case of continuous time, a number of authors have studied models with volatility uncertainty; see, for example, [33, 41, 47].

  13. The acronym for “Robust No Deterministic Sure Arbitrage with Unbounded Profit.”

  14. In reality, we are talking about a boundary value, defined as some infimum; under certain conditions of the “smoothness” of solutions, it can be achieved (and then it is correct to speak of a minimum).

  15. In contrast to the models of classical mechanics, where continuous time is natural and discrete time appears during a numerical solution, in financial models, in our opinion, discrete time is natural and continuous time appears during approximation.

  16. The very concept of a market price is ambiguous—it can be (for different purposes) the closing price, the current price, or the price determined by the market regulator as “market price”; see, e.g., information on price indicators on the Moscow Exchange website.

  17. It is assumed that some (incomplete) information on the price behavior is available; this will be formalized in Assumption (UD) below. Uncertain variables can be considered random after mixed strategies are introduced.

  18. ‘Observable information’ below means not only the availability of this information but also that it is this information that will be used when describing price increment restrictions, trading constraints, and hedging strategies.

  19. See, e.g., the survey [2].

  20. The assumption of the presence of a risk-free asset is standard in financial theory, although in practice, especially after the global financial crisis, its absence is recognized. The theory can also be constructed without Assumption (RA), but at the cost of technical complication, in particular, due to the need to consider arbitrage of the first and second kind.

  21. In the absence of trading constraints, having at least two risk-free assets behaving differently would lead to arbitrage opportunities—this is standard economic reasoning to justify the existence of a single risk-free asset in a market model.

  22. Zero prices can lead to arbitrage opportunities if they can subsequently take on positive values.

  23. The United States Securities and Exchange Commission is the US government agency that oversees and regulates the American securities market.

  24. The latest information can be obtained from the Final Report document—Amendment to Commission Delegated Regulation (EU) 2017/588 (RTS 11), 12 December 2018, ESMA 70-156-834.

  25. Using probabilistic terminology, we can say that the process \(\hat {H} \) (as well as the price \(X_0 \) of the risk-free asset) is predictable with respect to the filtration generated by the asset price process. The probabilistic way of describing dependences through measurability with respect to \(\sigma \)-algebras generated by random elements is mathematically more elegant, but we consistently adhere to the deterministic way of description in terms of functional dependence. In any case, there is no logical contradiction here: a random variable \(\eta \) is measurable with respect to the \(\sigma \)-algebra \(\mathcal {F}_\xi \) generated by a random element \(\xi \) ranging in a measurable space \((E,\mathcal {E}) \) if and only if it is representable as a measurable function \(\varphi \) of \(\xi \), i.e., \(\eta =\varphi \circ \xi \); see, e.g., [17, Ch. II, Sec. 4, Theorem 3].

  26. Thus, (2.2), just as (L), relies on Assumption (TC) about the lack of any direct or indirect transaction costs.

  27. If all prices \(X^i_t \) are expressed (as will be assumed) in a single currency, then discounted prices are dimensionless.

  28. The notation \(\hat {X}_t\), \(\hat {H}_t \), and \(\hat {D}_t \) was used for \(n+1 \) assets, including the risk-free one.

  29. The selected notation for uncertain variables admits some analogy with the conditional expectation of random variables; for a random variable \(Y \), there exists a measurable function \(\varphi \) such that \( \mathbb {E}^{\mathcal {F}_X} Y =\mathbb {E}(Y|X)=\varphi (X) \) a.c. and \( \mathbb {E} (Y|X=x)=\varphi (x)\) (the last relation is also defined almost everywhere with respect to the measure \( P_X \) of the distribution \( X \)).

  30. The statement of the problem without such a simplifying assumption is possible and is discussed in Sec. 3 of the present paper.

  31. Here and below, the term “support” means the topological support—smallest closed set of full measure; it coincides with the set of distribution growth points, that is, points whose any neighborhood has a positive probability.

  32. We are talking about regular conditional distributions, which are determined almost surely with respect to the reference probability measure.

  33. To a certain extent, the choice of \(\check {C} \) as a parallelepiped could be characterized as coordinate independence of multiplicative factors, at least when it comes to satisfying the “no-arbitrage” condition coordinatewise.

  34. Thus, the process describing price dynamics is a multidimensional Brownian motion.

  35. In formulas (2.11) and (2.12), we interpret \(\Delta X_t \) and \(M_t \) in matrix form as column vectors; the symbol “ \(T \)” designates the transposition of a matrix.

  36. The statement of the problem for this example is discussed in Sec. 3 of the present paper.

  37. In practice, it is possible that margin trading is possible only for part of positions.

  38. In fact, as we will see later, this setting, under certain “no-arbitrage” conditions covers cases of payments at specific points in time, including options of the European and Bermuda types.

  39. Usually, such quantities are the strike prices; for example for “plain vanilla” American “call” options \( G_t=(X_t-S_t)_{+}\), and for the “put” options \( G_t=(S_t-X_t)_{+}\) the quantities \(S_t \) are constant and expressed in monetary units.

  40. It suffices to have upper semicontinuity of the functions \(g_t(\cdot ) \) and the upper semicontinuity of \(K_t(\cdot ) \) in the sense of multivalued mappings; this will be shown in a separate publication.

  41. See [21, p. 83].

  42. As a consequence, one has formula (L).

  43. The symbol \(\lor \) stands for maximum, and \( hy \) stands for the inner product of a vector \( h \) by a vector \( y \).

  44. In Eqs. (BA), the functions \(v^*_t \), as well as the corresponding suprema and infima, assume values in the extended set of real numbers \(\mathbb {R}\cup \{-\infty , +\infty \}=[-\infty , +\infty ]\), that is, the two-point compactification of \(\mathbb {R}\) (i.e., the neighborhoods of points \(-\infty \) and \(+\infty \) have the form \([\infty , a) \), \(a \in \mathbb {R} \) and \((b, +\infty ] \), \(b \in \mathbb {R} \), respectively). We also adopt the convention that \( a+(+\infty ) = (+\infty )+a=+\infty \) for \( a \in (-\infty , +\infty ] \) and \( b+(-\infty ) = (-\infty )+b=-\infty \) for \( b \in [-\infty , +\infty ) \).

  45. It relies on Assumption (TC) of no transaction costs.

  46. Here we mean intervals on the set of nonnegative integers.

  47. It was by solving an integer optimization problem that the strategy for the central counterparty to conduct balancing transactions for the defaulter with a shortage of collateral in the already mentioned invention [13] was determined.

  48. However, from the point of view of an analytical study of Eqs. (BA), this is unlikely to be helpful.

  49. One of the suggested solutions to the problem is to use universal measurability.

  50. Such a change of the axiom does not lead to a contradiction in the framework of the Zermelo–Fraenkel axioms.

  51. We adopt the convention \(\inf \emptyset = + \infty \), \(\sup \emptyset = - \infty \).

REFERENCES

  1. Andreev, N.A. and Smirnov, S.N., Guaranteed approach to investment and hedging problems, in Tikhonovskie chteniya: nauchnaya konferentsiya: tezisy dokladov: posvyashchaetsya pamyati akad. A.N. Tikhonova: 29 oktyabrya–2 noyabrya 2018 g. (Tikhonov Readings: Abstr. Rep. Sci. Conf. Dedicated to Acad. A.N. Tikhonov, October 29–November 2, 2018), Moscow: MAKS Press, 2018, p. 11.

    Google Scholar 

  2. Arkhipov, V.M., Zakharov, I.Yu., Naumenko, V.V., and Smirnov, S.N., Preconditions for introducing quantitative performance measures for EMH, Preprint of Higher School of Economics, Moscow, 2007, Ser. WP16 Financial Engineering, Risk Management, and Actuarial Science, no. WP16/2007/05.

  3. Bertsekas, D.P. and Shreve, S.E., Stochastic Optimal Control. The Discrete Time Case, New York–San Francisco–London: Academic Press, 1978. Translated under the title: Stokhasticheskoe optimal’noe upravlenie: sluchai diskretnogo vremeni, Moscow: Nauka, 1985.

    MATH  Google Scholar 

  4. Zakharov, A.V. and Mussa, D.A., Garantirovannyi podkhod k zadache tsenoobrazovaniya i khedzhirovaniyadlya sluchaya obuslovlennogo obyazatel’stva s neskol’kimi riskovymi aktivami (Guaranteed approach to the problem of pricing and hedging for a contingent liability with multiple risky assets), Available from VINITI, 2001, Moscow, no. 1092–B01.

  5. Zverev, O.V. and Khametov, V.M., Minimax hedging of European-style options in incomplete markets (discrete time), Obozr. Prikl. Prom. Mat., 2011, vol. 18, no. 1, pp. 26–54.

    Google Scholar 

  6. Zverev, O.V. and Khametov, V.M., Minimax hedging of European-style options in incomplete markets (discrete time). II, Obozr. Prikl. Prom. Mat., 2011, vol. 18, no. 2, pp. 193–204.

    Google Scholar 

  7. Zverev, O.V. and Khametov, V.M., Minimax hedging of European-style options in a compact (1, S)-market, Obozr. Prikl. Prom. Mat., 2011, vol. 18, no. 11, pp. 121–122.

    Google Scholar 

  8. Kurzhansky, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertainty), Moscow: Nauka, 1977.

    Google Scholar 

  9. Molchanov, S.A., Strong Feller property of diffusion processes on smooth manifolds, Teoriya Veroyatn. Ee Primen., 1968, vol. 13, no. 3, pp. 493–498.

    MathSciNet  MATH  Google Scholar 

  10. Mussa, D.A., Modeling financial markets by methods of stochastic differential equations, Cand. Sci. (Phys.-Math.) Dissertation, Moscow, 2002.

  11. Smirnov, S.N., General theorem of the theory of antagonistic games on the ultimate support of a mixed strategy, Dokl. Ross. Akad. Nauk, 2018, vol. 480, no. 1, pp. 25–28.

    Google Scholar 

  12. Smirnov, S.N., A Feller transition kernel with measure supports given by a set-valued mapping, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2019, vol. 25, no. 1, pp. 219–228.

    MathSciNet  Google Scholar 

  13. Smirnov, S.N., Zakharov, A.V., Polimatidi, I.V., and Balabushkin, A.N., Method of electronic exchange trading in derivative financial instruments, methods of determining the level of the deposit margin, methods of resolving the situation with a margin deficit, RF Patent no. 2226714, 2004.

  14. Khametov, V.M. and Chalov, D.M., European-style option is an endless antagonistic game, Obozr. Prikl. Prom. Mat., 2004, vol. 11, no. 2, pp. 264–265.

    Google Scholar 

  15. Shiryaev, A.N., Osnovy stokhasticheskoi finansovoi matematiki. Tom 1. Fakty. Modeli (Basics of Stochastic Financial Mathematics. Vol. 1. Facts. Models), Moscow: FAZIS, 1998.

    Google Scholar 

  16. Shiryaev, A.N., Osnovy stokhasticheskoi finansovoi matematiki. Tom 2. Teoriya (Basics of Stochastic Financial Mathematics. Vol. 2. Theory), Moscow: FAZIS, 1998.

    Google Scholar 

  17. Shiryaev, A.N., Veroyatnost’—1 (Probability—1), Moscow: MTsNMO, 2004.

    Google Scholar 

  18. Aksamit, A., Deng, S., Obloj, J., and Tan, X., Robust pricing–hedging duality for American options in discrete time financial markets, Financ. Math., 2018. https://doi.org/10.1111/mafi.12199

  19. Bayraktar, E. and Zhang, Y., Fundamental theorem of asset pricing under transaction costs and model uncertainty, Math. Oper. Res., 2016, vol. 41, no. 3, pp. 1039–1054.

    Article  MathSciNet  Google Scholar 

  20. Bayraktar, E. and Zhou, Z., On arbitrage and duality under model uncertainty and portfolio constraints, Math. Finance, 2017, vol. 27, no. 4, pp. 988–1012.

    Article  MathSciNet  Google Scholar 

  21. Bellman, R., Dynamic Programming, Princeton: Princeton Univ. Press, 1957.

    MATH  Google Scholar 

  22. Bernhard, P., The robust control approach to option pricing and interval models: an overview, in Numerical Methods in Finance, Breton, M. and Ben-Ameur, H., Eds., New York: Springer, 2005, pp. 91–108.

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

    Book  Google Scholar 

  24. Bertsekas, D.P. and Shreve, S.E., Stochastic Optimal Control: The Discrete-Time Case, New York: Academic Press, 1978.

    MATH  Google Scholar 

  25. Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models, Ann. Appl. Probab., 2015, vol. 25, no. 2, pp. 823–859.

    Article  MathSciNet  Google Scholar 

  26. Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., and Obloj, J., Pointwise arbitrage pricing theory in discrete time, Preprint, 2016. .

  27. Burzoni, M., Frittelli, M., Hou, Z., and Maggis, M., Universal arbitrage aggregator in discrete-time markets under uncertainty, Finance Stochastics, 2016, vol. 20, no. 1, pp. 1–50.

    Article  MathSciNet  Google Scholar 

  28. Burzoni M, Frittelli, M., and Maggis, M., Model-free superhedging duality, Ann. Appl. Probab., 2017, vol. 27, no. 3, pp. 1452–1477.

    Article  MathSciNet  Google Scholar 

  29. Carassus, L., Gobet, E., and Temam, E., A class of financial products and models where super-replication prices are explicit, The 6th the Ritsumeikan Int. Conf. Stochastic Process. Appl. Math. Finance (2006).

  30. Carassus, L. and 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., and Soner, H., There is no nontrivial hedging portfolio for option pricing with transaction costs, Ann. Appl. Probab., 1995, vol. 5, pp. 327–355.

    MathSciNet  MATH  Google Scholar 

  32. Dana, R.-A. and Jeanblanc-Picqué, M., Marchés financiers en temps continu, Paris: Economica, 1994.

    Google Scholar 

  33. Denis, L. and Martini, C., A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab., 2006, vol. 16, no. 2, pp. 827–852.

    Article  MathSciNet  Google Scholar 

  34. Föllmer, H. and Kabanov, Y., Optional decomposition and Lagrange multipliers, Finance Stochastics, 1997, vol. 2, no. 1, pp. 69–81.

    Article  MathSciNet  Google Scholar 

  35. Föllmer, H. and Schied, A., Stochastic Finance. An Introduction in Discrete Time, New York: Walter de Gruyter, 2016, 4th ed.

    Book  Google Scholar 

  36. Hobson, D., Robust hedging of the lookback option, Finance Stochastics, 1998, vol. 2, no. 4, pp. 329–347.

    Article  Google Scholar 

  37. Karoui, N.E. and Quenez, M., Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optim., 1995, vol. 33, no. 1, pp. 29–66.

    Article  MathSciNet  Google Scholar 

  38. Knight, F.H., Risk, Uncertainty, and Profit, New York: Houghton Mifflin, 1921.

    Google Scholar 

  39. Kolokoltsov, V.N., Nonexpansive maps and option pricing theory, Kybernetika, 1998, vol. 34, no. 6, pp. 713–724.

    MathSciNet  MATH  Google Scholar 

  40. Kramkov, D., Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probab. Theory Relat. Fields, 1996, vol. 105, no. 4, pp. 459–479.

    Article  MathSciNet  Google Scholar 

  41. Neufeld, A. and Nutz, M., Superreplication under volatility uncertainty for measurable claims, Electron. J. Probab., 2012, vol. 18, no. 48, pp. 1–14.

    MathSciNet  MATH  Google Scholar 

  42. Mycielski, J. and Swierczkowski, S., On the Lebesgue measurability and the axiom of determinateness, Fundam. Math., 1964, vol. 54, no. 1, pp. 67–71.

    Article  MathSciNet  Google Scholar 

  43. Obloj, J. and Wiesel, J., A unified framework for robust modelling of financial markets in discrete time, Preprint, 2018. .

  44. Smirnov, S.N., Thoughts on financial risk modeling: the role of interpretation, Intell. Risk, 2012, vol. 2, no. 2, pp. 12–15.

    Google Scholar 

  45. Rockafellar, R.T., Convex Analysis, Princeton: Princeton Univ. Press, 1970.

    Book  Google Scholar 

  46. Solovay, R., A model of set-theory in which every set of reals is Lebesgue measurable, Ann. Math., 1970, vol. 92, no. 1, pp. 1–56.

    Article  MathSciNet  Google Scholar 

  47. Soner, H.M., Touzi, N., and Zhang, J., Dual formulation of second order target problems, Ann. Appl. Probab., 2013, vol. 23, no. 1, pp. 308–347.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    S. N. Smirnov

Authors
  1. S. N. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Smirnov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1134/S0005117921040081

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117921040081

Keywords

Search

Navigation