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)

[2]

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

[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)

[4]

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

[5]

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

[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

[7]

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

[8]

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

[9]

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

[10]

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

[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

[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)

[13]

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

[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)

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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)

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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)

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

show all references

References:
[1]

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

[2]

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

[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)

[4]

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

[5]

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

[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

[7]

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

[8]

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

[9]

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

[10]

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

[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

[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)

[13]

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

[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)

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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)

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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)

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

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