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Mathematics and Financial Economics

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07-11-2023

On intermediate marginals in martingale optimal transportation

Author: Julian Sester

Published in: Mathematics and Financial Economics | Issue 4/2023

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Abstract

We study the influence of additional intermediate marginal distributions on the value of the martingale optimal transport problem. From a financial point of view, this corresponds to taking into account call option prices not only, as usual, for those call options where the respective future maturities coincide with the maturities of some exotic derivative but also additional maturities and then to study the effect on model-independent price bounds for the exotic derivative. We characterize market settings, i.e., combinations of the payoff of exotic derivatives, call option prices and marginal distributions that guarantee improved price bounds as well as those market settings that exclude any improvement. Eventually, we showcase in numerous examples that the consideration of additional price information on vanilla options may have a considerable impact on the resultant model-independent price bounds.

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Note that in practice the expiration date of an exchange-traded call option is often the third Friday of a month. Very often, for shorter maturities more expiration dates are available, often on a weekly basis.

A probability measure \(\mu _2\in \mathcal {P}(\mathbb {R})\) is larger in convex order than \(\mu _1\in \mathcal {P}(\mathbb {R})\), abbreviated by \(\mu _1 \preceq \mu _2\), if \(\int _\mathbb {R}f(x) {\mathop {}\!\textrm{d}}\mu _1(x) \le \int _\mathbb {R}f(x) {\mathop {}\!\textrm{d}}\mu _2(x)\) for all convex functions \(f:\mathbb {R}\rightarrow \mathbb {R}\) such that the integrals are finite.

Moreover, note that if the marginals increase in convex order, then the prices of European call options computed as expectations with respect to these marginals also increase, compare [35, Lemma 7.24]. That prices of European call options increase with an increasing maturity is well-known in arbitrage free markets, hence the assumption appears to be natural.

This assumption is mainly imposed to be able to apply a duality result (see [12] or [8]), and could be relaxed to a certain degree, for example by considering semi-continuous payoff functions instead of continuous functions. To avoid the need of distinguishing between lower-semi continuous payoff functions for lower bounds and upper semi-continuous payoff functions for upper bounds, we decided to circumvent this issue by only considering continuous payoff functions. The linear growth condition could be slightly relaxed to functions that do not grow stronger than a sum of \(\mu _i\)-integrable functions, however it is standard to use the linear growth condition as in the definition of \(\mathcal {C}_{{\text {lin}}}(\mathbb {R}^d)\), see e.g. [12, Theorem 1.1].

Compare e.g. [54, Proposition 2.2.] for a proof of the compactness with respect to the Wasserstein distance and the non-emptiness, which both are consequences of Berge’s maximum theorem.

We refer for more details on semi-static trading strategies and its properties to [2, 21, 28] and [56].

A probability kernel is a map where for fixed first component the map is a probability measure, and for fixed second component the map is Borel-measurable.

For any \(\mu ,\nu \in \mathcal {P}(\mathbb {R}^d)\) the Wasserstein distance of order 1 (or Wasserstein 1-distance) is defined as \( W_1(\mu ,\nu ):=\inf _{\mathbb {P}\in \Pi (\mu ,\nu )}\int _{\mathbb {R}^d \times \mathbb {R}^d} \Vert x-y\Vert ~\mathbb {P}({\mathop {}\!\textrm{d}}x, {\mathop {}\!\textrm{d}}y), \) where \(\Vert \cdot \Vert \) denotes the Euclidean norm on \(\mathbb {R}^d\), and where \(\Pi (\mu ,\nu )\) denotes the set of joint distributions of \(\mu \) and \(\nu \), compare also for example [62, Definition 6.1.].

For an extension of the presented setting that enables to formulate model-free super-hedging in a market with frictions, we refer the reader to [20, Section 2] [24, 31, Section 3], [33, Section 6.3] or [53, Appendix A.1].

We denote by \(\delta _x\) the Dirac measure centered on \(x \in \mathbb {R}\), i.e., for any measurable set \(A \subset \mathbb {R}\) we have \(\delta _x(A) = 1\) if \(x\in A\) and 0 else.

As shown in [43], under the unique optimal measure for the maximization problem, the conditional law of \(S_{t_3}\) is supported only on two values, and therefore Proposition 3.7 is applicable to the analogue maximization problem.

We use the mid-prices, i.e., the mean of observed bid and ask prices.

Note that the largest maturities correspond to marginals that are the largest with respect to the convex order.

Here we apply a variant of Lemma 4.1, where we only substitute two of the three \(L^1\)-integrand by functions from \(\mathcal {C}_b(\mathbb {R})\). This is possible due to the argumentation from [12, Appendix] showing that for all \(u\in L^1(\mu _i)\) and for all \(\varepsilon >0\) there exists some \(\widetilde{u} \in \mathcal {C}_b(\mathbb {R})\) such that \(\widetilde{u} \le u\) and \(\mathbb {E}_{\mu _i}[u]-\mathbb {E}_{\mu _i}[\widetilde{u}]<\varepsilon \).

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(2), 233–251 (2016)MathSciNetMATHCrossRef

Acciaio, B., Larsson, M., Schachermayer, W.: The space of outcomes of semi-static trading strategies need not be closed. Finance Stochast. 21(3), 741–751 (2017)MathSciNetMATHCrossRef

Aksamit, A., Hou, Z., Obłój, J.: Robust framework for quantifying the value of information in pricing and hedging. SIAM J. Finan. Math. 11(1), 27–59 (2020)MathSciNetMATHCrossRef

Alfonsi, A., Corbetta, J., Jourdain, B.: Sampling of one-dimensional probability measures in the convex order and computation of robust option price bounds. Int. J. Theoret. Appl. Finance 22(03), 1950002 (2019)MathSciNetMATHCrossRef

Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media (2005)

Ansari, J., Lütkebohmert, E., Neufeld, A., Sester, J.: Improved robust price bounds for multi-asset derivatives under market-implied dependence information. arXiv:2204.01071 (2022)

Backhoff-Veraguas, J., Pammer, G.: Stability of martingale optimal transport and weak optimal transport. Ann. Appl. Probab. 32(1), 721–752 (2022)MathSciNetMATHCrossRef

Bartl, D., Cheridito, P., Kupper, M.: Robust expected utility maximization with medial limits. J. Math. Anal. Appl. 471(1–2), 752–775 (2019)MathSciNetMATHCrossRef

Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stochast. 24(1), 215–248 (2020)MathSciNetMATHCrossRef

10.

Bäuerle, N., Schmithals, D.: Consistent upper price bounds for exotic options. Int. J. Theoret. Appl. Finance 24(02), 2150011 (2021)MathSciNetMATHCrossRef

11.

Bayraktar, E., Deng, S., Norgilas, D.: Supermartingale shadow couplings: the decreasing case. arXiv:2207.11732 (2022)

12.

Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices-a mass transport approach. Finance Stochast. 17(3), 477–501 (2013)MathSciNetMATHCrossRef

13.

Beiglböck, M., Hobson, D., Norgilas, D.: The potential of the shadow measure. Electron. Commun. Probab. 27, 1–12 (2022)MathSciNetMATHCrossRef

14.

Beiglböck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44(1), 42–106 (2016)MathSciNetMATHCrossRef

15.

Beiglböck, M., Juillet, N.: On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44(1), 42–106 (2016)MathSciNetMATHCrossRef

16.

Beiglböck, M., Juillet, N.: Shadow couplings. Trans. Am. Math. Soc. 374(7), 4973–5002 (2021)MathSciNetMATHCrossRef

17.

Beiglböck, M., Lim, T., Obłój, J.: Dual attainment for the martingale transport problem. Bernoulli 25(3), 1640–1658 (2019)MathSciNetMATHCrossRef

18.

Beiglböck, M., Nutz, M., Touzi, N.: Complete duality for martingale optimal transport on the line. Ann. Probab. 45(5), 3038–3074 (2017)MathSciNetMATHCrossRef

19.

Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51(4), 621–651 (1978)CrossRef

20.

Burzoni, M.: Arbitrage and hedging in model-independent markets with frictions. SIAM J. Financ. Math. 7(1), 812–844 (2016)MathSciNetMATHCrossRef

21.

Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452–1477 (2017)MathSciNetMATHCrossRef

22.

Chang, J.T., Pollard, D.: Conditioning as disintegration. Stat. Neerl. 51(3), 287–317 (1997)MathSciNetMATHCrossRef

23.

Cheridito, P., Kiiski, M., Prömel, D.J., Mete Sonerm, H.: Martingale optimal transport duality. Math. Ann. 379(3), 1685–1712 (2021)MathSciNetMATHCrossRef

24.

Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time. SIAM J. Financ. Math. 8(1), 738–765 (2017)MathSciNetMATHCrossRef

25.

Cohen, S.N., Reisinger, C., Wang, S.: Detecting and repairing arbitrage in traded option prices. Appl. Math. Finance 27(5), 345–373 (2020)MathSciNetMATHCrossRef

26.

Cousot, L.: Conditions on option prices for absence of arbitrage and exact calibration. J. Bank. Finance 31(11), 3377–3397 (2007)CrossRef

27.

Cox, A. M. G., Vidmar, M.: The structure of non-linear martingale optimal transport problems. arXiv:1903.06606 (2019)

28.

Davis, M.H.A., Hobson, D.G.: The range of traded option prices. Math. Financ. 17(1), 1–14 (2007)MathSciNetMATHCrossRef

29.

Aquino, L.G., Bernard, C.: Bounds on multi-asset derivatives via neural networks. Int. J. Theoret. Appl. Finance 23(08), 2050050 (2020)MathSciNetMATHCrossRef

30.

Doldi, A., Frittelli, M.: Entropy martingale optimal transport and nonlinear pricing-hedging duality. arXiv:2005.12572 (2020)

31.

Dolinsky, Y., Soner, H.M.: Robust hedging with proportional transaction costs. Finance Stochast. 18(2), 327–347 (2014)MathSciNetMATHCrossRef

32.

Eckstein, S., Guo, G., Lim, T., Obłój, J.: Robust pricing and hedging of options on multiple assets and its numerics. SIAM J. Financ. Math. 12(1), 158–188 (2021)MathSciNetMATHCrossRef

33.

Eckstein, S., Kupper, M.: Martingale transport with homogeneous stock movements. Quant. Finance 21(2), 271–280 (2021)MathSciNetMATHCrossRef

34.

Ekren, I., Soner, H.M.: Constrained optimal transport. Arch. Ration. Mech. Anal. 227(3), 929–965 (2018)MathSciNetMATHCrossRef

35.

Föllmer, H., Schied, A.: Stochastic finance. In Stochastic Finance. de Gruyter, (2016)

36.

Ghoussoub, N., Kim, Y.-H., Lim, T.: Structure of optimal martingale transport plans in general dimensions. Ann. Probab. 47(1), 109–164 (2019)MathSciNetMATHCrossRef

37.

Guo, G., Obłój, J.: Computational methods for martingale optimal transport problems. Ann. Appl. Probab. 29(6), 3311–3347 (2019)MathSciNetMATHCrossRef

38.

Henry-Labordere, P.: Automated option pricing: numerical methods. Int. J. Theoret. Appl. Finance 16(08), 1350042 (2013)MathSciNetMATHCrossRef

39.

Henry-Labordère, P.: Model-free hedging. A martingale optimal transport viewpoint. Financial Mathematics Series. Chapman and Hall/CRC (2017)

40.

Henry-Labordère, P., Tan, X., Touzi, N.: An explicit martingale version of the one-dimensional Brenier’s theorem with full marginals constraint. Stoch. Process. Appl. 126(9), 2800–2834 (2016)MathSciNetMATHCrossRef

41.

Henry-Labordère, P., Touzi, N.: An explicit martingale version of the one-dimensional Brenier theorem. Finance Stochast. 20(3), 635–668 (2016)MathSciNetMATHCrossRef

42.

Hobson, D., Klimmek, M.: Robust price bounds for the forward starting straddle. Finance Stochast. 19(1), 189–214 (2015)MathSciNetMATHCrossRef

43.

Hobson, D., Neuberger, A.: Robust bounds for forward start options. Math. Financ. 22(1), 31–56 (2012)MathSciNetMATHCrossRef

44.

Hobson, D., Norgilas, D.: Robust bounds for the American put. Finance Stochast. 23(2), 359–395 (2019)MathSciNetMATHCrossRef

45.

Hobson, D.G., Norgilas, D.: The left-curtain martingale coupling in the presence of atoms. Ann. Appl. Probab. 29(3), 1904–1928 (2019)MathSciNetMATHCrossRef

46.

Hou, Z., Obłój, J.: Robust pricing-hedging dualities in continuous time. Finance Stochast. 22(3), 511–567 (2018)MathSciNetMATHCrossRef

47.

Hull, J.C.: Options futures and other derivatives. Pearson Education India (2003)

48.

Juillet, N.: Stability of the shadow projection and the left-curtain coupling. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 52, pp 1823–1843. Institut Henri Poincaré, (2016)

49.

Kellerer, H.G.: Markov-komposition und eine anwendung auf martingale. Math. Ann. 198(3), 99–122 (1972)MathSciNetMATHCrossRef

50.

Lütkebohmert, E., Sester, J.: Tightening robust price bounds for exotic derivatives. Quant. Finance 19(11), 1797–1815 (2019)MathSciNetMATHCrossRef

51.

Lux, T., Papapantoleon, A.: Improved fréchet-hoeffding bounds on \( d \)-copulas and applications in model-free finance. Ann. Appl. Probab. 27(6), 3633–3671 (2017)MathSciNetMATHCrossRef

52.

Neufeld, A., Papapantoleon, A., Xiang, Q.: Model-free bounds for multi-asset options using option-implied information and their exact computation. Manage. Sci. (2022)

53.

Neufeld, A., Sester, J.: Model-free price bounds under dynamic option trading. SIAM J. Financ. Math. 12(4), 1307–1339 (2021)MathSciNetMATHCrossRef

54.

Neufeld, A., Sester, J.: On the stability of the martingale optimal transport problem: a set-valued map approach. Stat. Probabi. Lett. 176, 109131 (2021)MathSciNetMATHCrossRef

55.

Nutz, M., Stebegg, F., Tan, X.: Multiperiod martingale transport. Stoch. Process. Appl. (2019)

56.

Nutz, M., Wiesel, J., Zhao, L.: Limits of semistatic trading strategies. arXiv:2204.12251 (2022)

57.

Rothschild, M., Stiglitz, J. E. (1978) Increasing risk: I. A definition. In Uncertainty in Economics, pp. 99–121

58.

Sester, J.: Robust bounds for derivative prices in Markovian Models. Int. J. Theoret. Appl. Finance 23(03), 2050015 (2020)MathSciNetMATHCrossRef

59.

Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer (2007)

60.

Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36(2), 423–439 (1965)MathSciNetMATHCrossRef

61.

Tankov, P.: Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 48(2), 389–403 (2011)MathSciNetMATHCrossRef

62.

Villani, C., et al. Optimal transport: old and new, Vol. 338. Springer (2009)

63.

Wiesel, J.: Continuity of the martingale optimal transport problem on the real line. arXiv preprint arXiv:1905.04574 (2019)

64.

Zaev, D.A.: On the Monge-Kantorovich problem with additional linear constraints. Math. Notes 98(5), 725–741 (2015)MathSciNetMATHCrossRef

Title: On intermediate marginals in martingale optimal transportation
Author: Julian Sester
Publication date: 07-11-2023
Publisher: Springer Berlin Heidelberg
Published in: Mathematics and Financial Economics / Issue 4/2023
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-023-00345-9

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