January  2017, 2: 14 doi: 10.1186/s41546-017-0026-3

Financial asset price bubbles under model uncertainty

1 Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-Maximilians Universität, Theresienstraße 39, 80333 Munich, Germany;

2 Department of Mathematics, University of Oslo, Box 1053, Blindern, 0316 Oslo, Norway

Received  January 10, 2017 Revised  December 03, 2017 Published  June 2017

We study the concept of financial bubbles in a market model endowed with a set $\mathcal{P}$ of probability measures, typically mutually singular to each other. In this setting, we investigate a dynamic version of robust superreplication, which we use to introduce the notions of bubble and robust fundamental value in a way consistent with the existing literature in the classical case $\mathcal{P}$={$\mathbb{P}$}. Finally, we provide concrete examples illustrating our results.
Citation: Francesca Biagini, Jacopo Mancin. Financial asset price bubbles under model uncertainty. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 14-. doi: 10.1186/s41546-017-0026-3
References:
[1]

Ash, R:Real analysis and probability. Academic Press, New York (1972) Google Scholar

[2]

Beissner, P:Coherent Price Systems and Uncertainty-Neutral Valuation. Working Paper 464, Center for Mathematical Economics, Bielefeld University (2012) Google Scholar

[3]

Biagini, F, Föllmer, H, Nedelcu, S:Shifting martingale measures and the slow birth of a bubble as a submartingale. Finance Stochast. 18(2), 297-326 (2014) Google Scholar

[4]

Biagini, F, Nedelcu, S:The formation of financial bubbles in defaultable markets. SIAM J. Financ. Math. 6(1), 530-558 (2015) Google Scholar

[5]

Biagini, F, Zhang, Y:Reduced-form framework and superhedging for payment streams under model uncertainty. arXiv:1707.04475 (2017) Google Scholar

[6]

Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance. 27, 963-987 (2017). doi:10.1111/mafi.12110 Google Scholar

[7]

Bouchard, B, Nutz, M:Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25(2), 823-859 (2015) Google Scholar

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Burzoni, M, Riedel, F, Soner, HM:Viability and arbitrage under knightian uncertainty (2017).arXiv:1707.03335 Google Scholar

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Cohen, S:Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces. Electron. J. Probab. 17(62), 1-15 (2012) Google Scholar

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Cox, AMG, Hobson, DG:Local martingales, bubbles and option prices. Finance Stochast. 9(4), 477-492(2005) Google Scholar

[11]

Cox, AMG, Hou, Z, Obłój, J:Robust pricing and hedging under trading restrictions and the emergence of local martingale models. Finance Stoch. 20, 669 (2016). https://doi.org/10.1007/s00780-016-0293-3 Google Scholar

[12]

Delbaen, F:The structure of m-stable sets and in particular of the set of risk neutral measures, volume 1874 of Lecture Notes in Math, pp. 215-258. Springer, Berlin Heidelberg (2006) Google Scholar

[13]

Dellacherie, C, Meyer, P:Probabilities and potential B. North-Holland Publishing Co., Amsterdam (1982) Google Scholar

[14]

Elworthy, KD, Li, X-M, Yor, M:The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes. Probab. Theory Relat. Fields. 115(3), 325-355 (1999) Google Scholar

[15]

Föllmer, H, Protter, P:Local martingales and filtration shrinkage. ESAIM Probab. Stat. 15, S25-S38(2011) Google Scholar

[16]

Föllmer, H, Schied, A:Stochastic Finance. An Introduction in Discrete Time, 3rd edition. De Gruyter, Berlin (2011) Google Scholar

[17]

Herdegen, M, Schweizer, M:Strong bubbles and strict local martingales. Int. J. Theor. Appl. Finance. 19, 1650022 (2016) Google Scholar

[18]

Hugonnier, J:Rational asset pricing bubbles and portfolio constraints. J. Economic Theory. 147(6), 2260-2302 (2012) Google Scholar

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Jarrow, RA, Larsson, M:The meaning of market efficiency. Math. Financ. 22(1), 1-30 (2012) Google Scholar

[20]

Jarrow, RA, Protter, P, Roch, A:A liquidity based model for asset price bubble. Quant. Finan. 12(9), 1339-1349 (2012) Google Scholar

[21]

Jarrow, RA, Protter, P, Shimbo, K:Asset price bubbles in complete markets, pp. 97-121. Advances in Mathematical Finance, Birkhäuser Boston (2007) Google Scholar

[22]

Jarrow, RA, Protter, P, Shimbo, K:Asset price bubbles in incomplete markets. Math. Financ. 20(2), 145-185 (2010) Google Scholar

[23]

Kramkov, DO:Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields. 105(4), 459-479 (1996) Google Scholar

[24]

Loewenstein, M, Willard, GA:Rational equilibrium and asset-pricing bubbles in continuous trading models. J. Econ. Theory. 91(1), 17-58 (2000) Google Scholar

[25]

Luo, P, Wang, F:Stochastic differential equations driven by G-brownian motion and ordinary differential equations. Stoch. Process. Appl. 124(11), 3869-3885 (2014) Google Scholar

[26]

Merton, RC:Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141-183 (1973) Google Scholar

[27]

Nutz, M:Robust superhedging with jumps and diffusion. Stoch. Process. Appl. 125(12), 4543-4555(2015) Google Scholar

[28]

Nutz, M, Soner, HM:Superhedging and dynamic risk measures under volatility uncertainty. SIAM J Control. Optim. 50(4), 2065-2089 (2012) Google Scholar

[29]

Nutz, M, Van Handel, R:Constructing sublinear expectations on path space. Stoch. Process. Appl. 123(8), 3100-3121 (2013) Google Scholar

[30]

Pal, S, Protter, P:Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl. 120(8), 1424-1443 (2010) Google Scholar

[31]

Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stoch. Anal. Appl. 2, 541-567 (2007) Google Scholar

[32]

Protter, P:A mathematical theory of financial bubbles. Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Mathematics, vol 2081, pp. 1-108. Springer, Cham (2013) Google Scholar

[33]

Revuz, D, Yor, M:Continuous Martingales and Brownian Motion, third edition. Springer, Berlin Heidelberg (1999) Google Scholar

[34]

Soner, HM, Touzi, N, Zhang, J:Martingale representation theorem for the G-expectation. Stoch. Process.Appl. 121(2), 265-287 (2011a) Google Scholar

[35]

Soner, HM, Touzi, N, Zhang, J:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16, 1844-1879 (2011b) Google Scholar

[36]

Soner, HM, Touzi, N, Zhang, J:Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308-347 (2013) Google Scholar

[37]

Song, Y:Some properties of G-evaluation and its applications to G-martingale decomposition. Sci. China Math. 54(2), 287-300 (2011) Google Scholar

[38]

Stricker, C:Quasimartingales, martingales locales, semimartingales et filtration naturelle. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 39(1), 55-63 (1977) Google Scholar

[39]

Tutsch, D:Konsistente und konsequente dynamische Risikomaße und das Problem der Aktualisierung(2006). PhD thesis, Humboldt-Universität zu Berlin Google Scholar

show all references

References:
[1]

Ash, R:Real analysis and probability. Academic Press, New York (1972) Google Scholar

[2]

Beissner, P:Coherent Price Systems and Uncertainty-Neutral Valuation. Working Paper 464, Center for Mathematical Economics, Bielefeld University (2012) Google Scholar

[3]

Biagini, F, Föllmer, H, Nedelcu, S:Shifting martingale measures and the slow birth of a bubble as a submartingale. Finance Stochast. 18(2), 297-326 (2014) Google Scholar

[4]

Biagini, F, Nedelcu, S:The formation of financial bubbles in defaultable markets. SIAM J. Financ. Math. 6(1), 530-558 (2015) Google Scholar

[5]

Biagini, F, Zhang, Y:Reduced-form framework and superhedging for payment streams under model uncertainty. arXiv:1707.04475 (2017) Google Scholar

[6]

Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance. 27, 963-987 (2017). doi:10.1111/mafi.12110 Google Scholar

[7]

Bouchard, B, Nutz, M:Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25(2), 823-859 (2015) Google Scholar

[8]

Burzoni, M, Riedel, F, Soner, HM:Viability and arbitrage under knightian uncertainty (2017).arXiv:1707.03335 Google Scholar

[9]

Cohen, S:Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces. Electron. J. Probab. 17(62), 1-15 (2012) Google Scholar

[10]

Cox, AMG, Hobson, DG:Local martingales, bubbles and option prices. Finance Stochast. 9(4), 477-492(2005) Google Scholar

[11]

Cox, AMG, Hou, Z, Obłój, J:Robust pricing and hedging under trading restrictions and the emergence of local martingale models. Finance Stoch. 20, 669 (2016). https://doi.org/10.1007/s00780-016-0293-3 Google Scholar

[12]

Delbaen, F:The structure of m-stable sets and in particular of the set of risk neutral measures, volume 1874 of Lecture Notes in Math, pp. 215-258. Springer, Berlin Heidelberg (2006) Google Scholar

[13]

Dellacherie, C, Meyer, P:Probabilities and potential B. North-Holland Publishing Co., Amsterdam (1982) Google Scholar

[14]

Elworthy, KD, Li, X-M, Yor, M:The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes. Probab. Theory Relat. Fields. 115(3), 325-355 (1999) Google Scholar

[15]

Föllmer, H, Protter, P:Local martingales and filtration shrinkage. ESAIM Probab. Stat. 15, S25-S38(2011) Google Scholar

[16]

Föllmer, H, Schied, A:Stochastic Finance. An Introduction in Discrete Time, 3rd edition. De Gruyter, Berlin (2011) Google Scholar

[17]

Herdegen, M, Schweizer, M:Strong bubbles and strict local martingales. Int. J. Theor. Appl. Finance. 19, 1650022 (2016) Google Scholar

[18]

Hugonnier, J:Rational asset pricing bubbles and portfolio constraints. J. Economic Theory. 147(6), 2260-2302 (2012) Google Scholar

[19]

Jarrow, RA, Larsson, M:The meaning of market efficiency. Math. Financ. 22(1), 1-30 (2012) Google Scholar

[20]

Jarrow, RA, Protter, P, Roch, A:A liquidity based model for asset price bubble. Quant. Finan. 12(9), 1339-1349 (2012) Google Scholar

[21]

Jarrow, RA, Protter, P, Shimbo, K:Asset price bubbles in complete markets, pp. 97-121. Advances in Mathematical Finance, Birkhäuser Boston (2007) Google Scholar

[22]

Jarrow, RA, Protter, P, Shimbo, K:Asset price bubbles in incomplete markets. Math. Financ. 20(2), 145-185 (2010) Google Scholar

[23]

Kramkov, DO:Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields. 105(4), 459-479 (1996) Google Scholar

[24]

Loewenstein, M, Willard, GA:Rational equilibrium and asset-pricing bubbles in continuous trading models. J. Econ. Theory. 91(1), 17-58 (2000) Google Scholar

[25]

Luo, P, Wang, F:Stochastic differential equations driven by G-brownian motion and ordinary differential equations. Stoch. Process. Appl. 124(11), 3869-3885 (2014) Google Scholar

[26]

Merton, RC:Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141-183 (1973) Google Scholar

[27]

Nutz, M:Robust superhedging with jumps and diffusion. Stoch. Process. Appl. 125(12), 4543-4555(2015) Google Scholar

[28]

Nutz, M, Soner, HM:Superhedging and dynamic risk measures under volatility uncertainty. SIAM J Control. Optim. 50(4), 2065-2089 (2012) Google Scholar

[29]

Nutz, M, Van Handel, R:Constructing sublinear expectations on path space. Stoch. Process. Appl. 123(8), 3100-3121 (2013) Google Scholar

[30]

Pal, S, Protter, P:Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl. 120(8), 1424-1443 (2010) Google Scholar

[31]

Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stoch. Anal. Appl. 2, 541-567 (2007) Google Scholar

[32]

Protter, P:A mathematical theory of financial bubbles. Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Mathematics, vol 2081, pp. 1-108. Springer, Cham (2013) Google Scholar

[33]

Revuz, D, Yor, M:Continuous Martingales and Brownian Motion, third edition. Springer, Berlin Heidelberg (1999) Google Scholar

[34]

Soner, HM, Touzi, N, Zhang, J:Martingale representation theorem for the G-expectation. Stoch. Process.Appl. 121(2), 265-287 (2011a) Google Scholar

[35]

Soner, HM, Touzi, N, Zhang, J:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16, 1844-1879 (2011b) Google Scholar

[36]

Soner, HM, Touzi, N, Zhang, J:Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308-347 (2013) Google Scholar

[37]

Song, Y:Some properties of G-evaluation and its applications to G-martingale decomposition. Sci. China Math. 54(2), 287-300 (2011) Google Scholar

[38]

Stricker, C:Quasimartingales, martingales locales, semimartingales et filtration naturelle. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 39(1), 55-63 (1977) Google Scholar

[39]

Tutsch, D:Konsistente und konsequente dynamische Risikomaße und das Problem der Aktualisierung(2006). PhD thesis, Humboldt-Universität zu Berlin Google Scholar

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