Consistent price systems under model uncertainty

Abstract

We develop a version of the fundamental theorem of asset pricing for discrete-time markets with proportional transaction costs and model uncertainty. A robust notion of no-arbitrage of the second kind is defined and shown to be equivalent to the existence of a collection of strictly consistent price systems.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Before the final revision of the present paper was completed, we received the revised version of [3] in which a multivariate extension is provided using the strict no-arbitrage condition of [18].

  2. 2.

    As can be seen here, we can dispense with a condition that is slightly weaker than (2.2), but aesthetically less pleasant: the existence of a constant \(c\) and selectors \(S_{t}\) such that \(y^{i}/S^{i}_{t+1}(\omega,\cdot) \leq cy^{1}/S^{1}_{t+1}(\omega ,\cdot)\) for all \(y\in K^{*}_{t+1}(\omega,\cdot)\).

References

  1. 1.

    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. Finance (2013). doi:10.1111/mafi.12060

    Google Scholar 

  2. 2.

    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  3. 3.

    Bayraktar, E., Zhang, Y.: Fundamental theorem of asset pricing under transaction costs and model uncertainty. Math. Oper. Res., to appear. Available online at arXiv:1309.1420v2

  4. 4.

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

    Google Scholar 

  5. 5.

    Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823–859 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bouchard, B., Taflin, E.: No-arbitrage of second kind in countable markets with proportional transaction costs. Ann. Appl. Probab. 23, 427–454 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Burzoni, M., Frittelli, M., Maggis, M.: Universal arbitrage aggregator in discrete time markets under uncertainty. Finance Stoch. (2015). doi:10.1007/s00780-015-0283-x

    Google Scholar 

  8. 8.

    Dalang, R.C., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29, 185–201 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

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

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin (2006)

    Google Scholar 

  11. 11.

    Deparis, S., Martini, C.: Superhedging strategies and balayage in discrete time. In: Dalang, R.C., et al. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability, vol. 58, pp. 205–219. Birkhäuser, Basel (2004)

    Google Scholar 

  12. 12.

    Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields 160, 391–427 (2014)

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

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

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Fernholz, D., Karatzas, I.: Optimal arbitrage under model uncertainty. Ann. Appl. Probab. 21, 2191–2225 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter, Berlin (2004)

    Google Scholar 

  16. 16.

    Kabanov, Yu.M., Rásonyi, M., Stricker, Ch.: No-arbitrage criteria for financial markets with efficient friction. Finance Stoch. 6, 371–382 (2002)

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Kabanov, Yu.M., Safarian, M.: Markets with Transaction Costs. Mathematical Theory. Springer Finance. Springer, Berlin (2009)

    Google Scholar 

  18. 18.

    Kabanov, Yu.M., Stricker, Ch.: The Harrison–Pliska arbitrage pricing theorem under transaction costs. J. Math. Econ. 35, 185–196 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Nutz, M.: Superreplication under model uncertainty in discrete time. Finance Stoch. 18, 791–803 (2014)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Nutz, M., van Handel, R.: Constructing sublinear expectations on path space. Stoch. Process. Appl. 123, 3100–3121 (2013)

    MATH  Article  Google Scholar 

  21. 21.

    Rásonyi, M.: Arbitrage under transaction costs revisited. In: Delbaen, F., et al. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance. The Kabanov Festschrift, pp. 211–225. Springer, Berlin (2009)

    Google Scholar 

  22. 22.

    Riedel, F.: Financial economics without probabilistic prior assumptions. Decis. Econ. Finance 38, 75–91 (2015)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Gossez, J.P., et al. (eds.) Nonlinear Operators and the Calculus of Variations. Lecture Notes in Math., vol. 543, pp. 157–207. Springer, Berlin (1976)

    Google Scholar 

  24. 24.

    Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    Yan, J.A.: Caractérisation d’une classe d’ensembles convexes de \(L^{1}\) ou \(H^{1}\). In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités, XIV, Paris, 1978/1979. Lecture Notes in Math., vol. 784, pp. 220–222. Springer, Berlin (1980)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bruno Bouchard.

Additional information

Research of first author supported by ANR Liquirisk and Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).

Research of second author supported by NSF Grant DMS-1208985.

Appendix

Appendix

A.1 Measure theory

Given a measurable space \((\varOmega,\mathcal{A})\), let \(\mathfrak {P}(\varOmega)\) be the set of all probability measures on \(\mathcal{A}\). The universal completion of \(\mathcal{A}\) is the \(\sigma\)-field \(\bigcap_{P\in \mathfrak{P} (\varOmega)} \mathcal{A}^{P}\), where \(\mathcal{A}^{P}\) is the \(P\)-completion of \(\mathcal{A}\). When \(\varOmega\) is a topological space with Borel \(\sigma \)-field \(\mathcal{B}(\varOmega)\), we always endow \(\mathfrak{P}(\varOmega )\) with the topology of weak convergence. If \(\varOmega\) is Polish, then \(\mathfrak {P}(\varOmega )\) is also Polish. A subset \(A\subset\varOmega\) is analytic if it is the image of a Borel subset of another Polish space under a Borel-measurable mapping. Analytic sets are stable under countable union and intersection, under forward and inverse images of Borel functions, but not under complementation; the complement of an analytic set is called co-analytic and is not analytic unless it is Borel. Any Borel set is analytic, and any analytic set is universally measurable, i.e., measurable for the universal completion of \(\mathcal{B} (\varOmega)\). We refer to [4, Chap. 7] for these results and further background.

A.2 Random sets

Let \((\varOmega,\mathcal{A})\) be a measurable space. A mapping \(\varPsi\) from \(\varOmega\) into the power set \(2^{\mathbb{R}^{d}}\) is called a random set in \(\mathbb{R}^{d}\) and its graph is defined as

$$\operatorname{graph}(\varPsi)=\{(\omega,x):\, \omega\in\varOmega,\, x\in\varPsi(\omega )\}\subset\varOmega\times\mathbb{R}^{d}. $$

We say that \(\varPsi\) is \(\mathcal{A}\) -measurable (weakly \(\mathcal{A} \) -measurable) if

$$\{\omega\in\varOmega:\, \varPsi(\omega)\cap O\neq\emptyset\}\in \mathcal{A} \quad\mbox{for all closed (open)}\ O\subset\mathbb{R}^{d}. $$

Moreover, \(\varPsi\) is called closed (convex, etc.) if \(\varPsi(\omega)\) is closed (convex, etc.) for all \(\omega\in\varOmega\). We emphasize that measurability is not defined via the measurability of the graph, as it is sometimes done in the literature.

Lemma A.1

Let \((\varOmega,\mathcal{A})\) be a measurable space and \(\varPsi\) a closed, nonempty random set in \(\mathbb{R}^{d}\). The following are equivalent:

  1. (i)

    \(\varPsi\) is \(\mathcal{A}\)-measurable.

  2. (ii)

    \(\varPsi\) is weakly \(\mathcal{A}\)-measurable.

  3. (iii)

    The distance function \(d(\varPsi,y)=\inf\{x\in\varPsi:\, |x-y|\}\) is \(\mathcal{A}\)-measurable for all \(y\in\mathbb{R}^{d}\).

  4. (iv)

    There exist \(\mathcal{A}\)-measurable functions \((\psi _{n})_{n\geq1}\) with \(\varPsi=\overline{\{\psi_{n}, n \geq1\}}\) (“Castaing representation”).

Moreover, (i)–(iv) imply that

  1. (v)

    \(\mathrm{graph}(\varPsi)\) is \(\mathcal{A}\times\mathcal{B}(\mathbb {R}^{d})\)-measurable;

  2. (vi)

    The dual cone \(\varPsi^{*}\) is \(\mathcal{A}\)-measurable;

  3. (vii)

    \(\operatorname{graph}(\operatorname{int}\varPsi^{*})\) is \(\mathcal{A}\times\mathcal{B}(\mathbb{R} ^{d})\)-measurable;

  4. (viii)

    There exists an \(\mathcal{A}\)-measurable selector \(\psi \) of \(\varPsi^{*}\) which satisfies \(\psi\in\operatorname{int}\varPsi^{*}\) on \(\{\operatorname{int} \varPsi^{*}\neq\emptyset\}\).

If \(\mathcal{A}\) is universally complete, then (v) is equivalent to (i)–(iv).

Proof

We refer to [23, Sect. 1] for the results concerning (i)–(vi). If (iv) holds, then we have the representation

$$\begin{aligned} \operatorname{graph}(\operatorname{int}\varPsi^{*}) &=\big\{ (\omega,y)\in\varOmega\times\mathbb{R}^{d}:\, \langle x,y \rangle>0\mbox{ for all } x\in\varPsi(\omega)\setminus\{0\}\big\} \\ &=\bigcap_{n\geq1}\{(\omega,y)\in\varOmega\times\mathbb{R}^{d}:\, \langle\psi _{n}(\omega),y \rangle>0 \mbox{ or }\psi_{n}(\omega )=0\} \end{aligned}$$

which readily implies (vii).

Finally, let (vi) hold; then \(\varPsi^{*}\) has a Castaing representation \((\phi_{n})\). If we define \(\phi:=\sum_{n} 2^{-n} \phi_{n}\), then \(\phi\) is \(\mathcal{A}\)-measurable and \(\varPsi^{*}\)-valued since \(\varPsi^{*}\) is closed and convex. Let \(\omega\in\varOmega\) be such that \(\operatorname{int}\varPsi ^{*}(\omega)\neq\emptyset\). By their denseness, at least one of the points \(\phi_{n}(\omega)\in\varPsi^{*}(\omega)\) must lie in the interior of \(\varPsi^{*}(\omega)\). Moreover, since \(\varPsi^{*}(\omega)\) is convex, we observe that a nondegenerate convex combination of a point in \(\varPsi^{*}(\omega)\) with an interior point of \(\varPsi^{*}(\omega)\) is again an interior point. These two facts yield that \(\phi(\omega)\in \operatorname{int}\varPsi^{*}(\omega)\) as desired. (This applies to any closed and nonempty convex random set, not necessarily of the form \(\varPsi^{*}\).) □

In some cases, we need to select from random sets in infinite-dimensional spaces, or random sets that are not closed. The following is sufficient for our purposes.

Lemma A.2

(Jankov–von Neumann)

Let \(\varOmega, \varOmega'\) be Polish spaces and \(\varGamma\! \subset \varOmega\times\varOmega'\) an analytic set. Then the projection \(\pi_{\varOmega}(\varGamma)\subset \varOmega\) is universally measurable, and there exists a universally measurable function \(\psi:\pi_{\varOmega}(\varGamma)\to\varOmega'\) whose graph is contained in \(\varOmega'\).

We refer to [4, Proposition 7.49] for a proof. In many applications, we start with a random set \(\varPsi:\varOmega\to 2^{\varOmega'}\) such that \(\varGamma:=\operatorname{graph}(\varPsi)\) is analytic. Noting that \(\pi_{\varOmega}(\varGamma)=\{\varPsi\neq\emptyset\}\), Lemma A.2 then yields a universally measurable selector for \(\varPsi\) on the set \(\{\varPsi\neq\emptyset\}\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bouchard, B., Nutz, M. Consistent price systems under model uncertainty. Finance Stoch 20, 83–98 (2016). https://doi.org/10.1007/s00780-015-0286-7

Download citation

Keywords

  • Transaction costs
  • Arbitrage of the second kind
  • Consistent price system
  • Model uncertainty

Mathematics Subject Classification (2010)

  • 60G42
  • 91B25
  • 93E20
  • 49L20

JEL Classification

  • G13