doi: 10.3934/fmf.2021006
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Asset price bubbles: Invariance theorems

1. 

Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, N.Y. 14853, USA

2. 

Kamakura Corporation, Honolulu, Hawaii 96815

3. 

Statistics Department, Columbia University, New York, NY 10027, USA

4. 

CMM-DIM, UMI-CNRS 2807, Universidad de Chile, Chile

* Corresponding author: Robert Jarrow

Received  January 2021 Revised  June 2021 Early access August 2021

Fund Project: The second author is supported in part by NSF Grant DMS-2106433. The third author is supported by BASAL project AFB170001, ANID

This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.

Citation: Robert Jarrow, Philip Protter, Jaime San Martin. Asset price bubbles: Invariance theorems. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021006
References:
[1]

L. B. G. Andersen and V. Piterbarg, Moment explosions in stochastic volatility models, Finance Stoch., 11 (2007), 29-50.  doi: 10.1007/s00780-006-0011-7.  Google Scholar

[2]

A. M. G. Cox and D. G. Hobson, Local martingales, bubbles and option prices, Finance Stoch., 9 (2005), 477-492.  doi: 10.1007/s00780-005-0162-y.  Google Scholar

[3]

D. Criens, Deterministic criteria for the absence and existence of arbitrage in multi-dimensional diffusion markets, Int. J. Theor. Appl. Finance, 21 (2018), 41pp. doi: 10.1142/S0219024918500024.  Google Scholar

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A. Dandapani and P. Protter, Strict local martingales and the Khasminskii test for explosions, Stochastic Process. Appl., (2019). doi: 10.1016/j.spa.2019.03.009.  Google Scholar

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F. Delbaen and H. Shirakawa, No arbitrage condition for positive diffusion price processes, Asia-Pacific Financial Markets, 9 (2002), 159-168.  doi: 10.1023/A:1024173029378.  Google Scholar

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C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

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M. Herdegen and M. Schweizer, Strong bubbles and strict local martingales, Int. J. Theor. Appl. Finance, 19 (2016), 44pp. doi: 10.1142/S0219024916500229.  Google Scholar

[8]

S. L. HestonM. Loewenstein and G. A. Willard, Options and bubbles, Rev. Financial Stud., 20 (2007), 359-390.  doi: 10.1093/rfs/hhl005.  Google Scholar

[9]

J. Jacod and P. Protter, Risk-neutral compatibility with option prices, Finance Stoch., 14 (2010), 285-315.  doi: 10.1007/s00780-009-0109-9.  Google Scholar

[10]

R. JarrowY. Kchia and P. Protter, How to detect an asset bubble, SIAM J. Financial Math., 2 (2011), 839-865.  doi: 10.1137/10079673X.  Google Scholar

[11]

R. A. Jarrow, Continuous-Time Asset Pricing Theory. A Martingale Based Approach, Springer Finance Textbooks, Springer, Cham, 2018. doi: 10.1007/978-3-319-77821-1.  Google Scholar

[12]

R. A. Jarrow and M. Larsson, The meaning of market efficiency, Math. Finance, 22 (2012), 1-30.  doi: 10.1111/j.1467-9965.2011.00497.x.  Google Scholar

[13]

R. A. Jarrow, P. Protter and K. Shimbo, Asset price bubbles in complete markets, in Advances in Mathematical Finance, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007, 97–121. doi: 10.1007/978-0-8176-4545-8_7.  Google Scholar

[14]

R. A. JarrowP. Protter and K. Shimbo, Asset price bubbles in incomplete markets, Math. Finance, 20 (2010), 145-185.  doi: 10.1111/j.1467-9965.2010.00394.x.  Google Scholar

[15]

N. Kashkari, Monetary policy and bubbles, (2017)., Available from: https://www.minneapolisfed.org/news-and-events/messages/monetary-policy-and-bubbles. Google Scholar

[16]

P.-L. Lions and M. Musiela, Correlations and bounds for stochastic volatility models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1-16.  doi: 10.1016/j.anihpc.2005.05.007.  Google Scholar

[17]

M. Loewenstein and G. A. Willard, Consumption and bubbles, J. Econom. Theory, 148 (2013), 563-600.  doi: 10.1016/j.jet.2012.07.001.  Google Scholar

[18]

M. Loewenstein and G. A. Willard, Local martingales, arbitrage, and viability. Free snacks and cheap thrills, Econom. Theory, 16 (2000), 135-161.  doi: 10.1007/s001990050330.  Google Scholar

[19]

M. Loewenstein and G. A. Willard, Rational equilibrium asset-pricing bubbles in continuous trading models, J. Econom. Theory, 91 (2000), 17-58.  doi: 10.1006/jeth.1999.2589.  Google Scholar

[20]

A. Mijatović and M. Urusov, On the martingale property of certain local martingales, Probab. Theory Related Fields, 152 (2012), 1-30.  doi: 10.1007/s00440-010-0314-7.  Google Scholar

[21]

Y. ObayashiP. Protter and S. Wang, The lifetime of a financial bubble, Math. Financ. Econ., 11 (2017), 45-62.  doi: 10.1007/s11579-016-0170-z.  Google Scholar

[22]

P. Protter, A mathematical theory of financial bubbles, in Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math., 2081, Springer, Cham, 2013, 1–108. doi: 10.1007/978-3-319-00413-6_1.  Google Scholar

[23]

P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Stochastic Modelling and Applied Probability, 21, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.  Google Scholar

[24]

A. Riveros, Propiedad de Martingala, Volatilidad Estocástica y Burbujas Financieras, Masters thesis, Universidad de Chile, 2020. Available from: http://repositorio.uchile.cl/handle/2250/175773. Google Scholar

[25]

J. Ruf, The martingale property in the context of stochastic differential equations, Electron. Commun. Probab., 20 (2015), 10pp. doi: 10.1214/ECP.v20-3449.  Google Scholar

[26]

C. A. Sin, Complications with stochastic volatility models, Adv. in Appl. Probab., 30 (1998), 256-268.  doi: 10.1239/aap/1035228003.  Google Scholar

show all references

References:
[1]

L. B. G. Andersen and V. Piterbarg, Moment explosions in stochastic volatility models, Finance Stoch., 11 (2007), 29-50.  doi: 10.1007/s00780-006-0011-7.  Google Scholar

[2]

A. M. G. Cox and D. G. Hobson, Local martingales, bubbles and option prices, Finance Stoch., 9 (2005), 477-492.  doi: 10.1007/s00780-005-0162-y.  Google Scholar

[3]

D. Criens, Deterministic criteria for the absence and existence of arbitrage in multi-dimensional diffusion markets, Int. J. Theor. Appl. Finance, 21 (2018), 41pp. doi: 10.1142/S0219024918500024.  Google Scholar

[4]

A. Dandapani and P. Protter, Strict local martingales and the Khasminskii test for explosions, Stochastic Process. Appl., (2019). doi: 10.1016/j.spa.2019.03.009.  Google Scholar

[5]

F. Delbaen and H. Shirakawa, No arbitrage condition for positive diffusion price processes, Asia-Pacific Financial Markets, 9 (2002), 159-168.  doi: 10.1023/A:1024173029378.  Google Scholar

[6]

C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[7]

M. Herdegen and M. Schweizer, Strong bubbles and strict local martingales, Int. J. Theor. Appl. Finance, 19 (2016), 44pp. doi: 10.1142/S0219024916500229.  Google Scholar

[8]

S. L. HestonM. Loewenstein and G. A. Willard, Options and bubbles, Rev. Financial Stud., 20 (2007), 359-390.  doi: 10.1093/rfs/hhl005.  Google Scholar

[9]

J. Jacod and P. Protter, Risk-neutral compatibility with option prices, Finance Stoch., 14 (2010), 285-315.  doi: 10.1007/s00780-009-0109-9.  Google Scholar

[10]

R. JarrowY. Kchia and P. Protter, How to detect an asset bubble, SIAM J. Financial Math., 2 (2011), 839-865.  doi: 10.1137/10079673X.  Google Scholar

[11]

R. A. Jarrow, Continuous-Time Asset Pricing Theory. A Martingale Based Approach, Springer Finance Textbooks, Springer, Cham, 2018. doi: 10.1007/978-3-319-77821-1.  Google Scholar

[12]

R. A. Jarrow and M. Larsson, The meaning of market efficiency, Math. Finance, 22 (2012), 1-30.  doi: 10.1111/j.1467-9965.2011.00497.x.  Google Scholar

[13]

R. A. Jarrow, P. Protter and K. Shimbo, Asset price bubbles in complete markets, in Advances in Mathematical Finance, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007, 97–121. doi: 10.1007/978-0-8176-4545-8_7.  Google Scholar

[14]

R. A. JarrowP. Protter and K. Shimbo, Asset price bubbles in incomplete markets, Math. Finance, 20 (2010), 145-185.  doi: 10.1111/j.1467-9965.2010.00394.x.  Google Scholar

[15]

N. Kashkari, Monetary policy and bubbles, (2017)., Available from: https://www.minneapolisfed.org/news-and-events/messages/monetary-policy-and-bubbles. Google Scholar

[16]

P.-L. Lions and M. Musiela, Correlations and bounds for stochastic volatility models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1-16.  doi: 10.1016/j.anihpc.2005.05.007.  Google Scholar

[17]

M. Loewenstein and G. A. Willard, Consumption and bubbles, J. Econom. Theory, 148 (2013), 563-600.  doi: 10.1016/j.jet.2012.07.001.  Google Scholar

[18]

M. Loewenstein and G. A. Willard, Local martingales, arbitrage, and viability. Free snacks and cheap thrills, Econom. Theory, 16 (2000), 135-161.  doi: 10.1007/s001990050330.  Google Scholar

[19]

M. Loewenstein and G. A. Willard, Rational equilibrium asset-pricing bubbles in continuous trading models, J. Econom. Theory, 91 (2000), 17-58.  doi: 10.1006/jeth.1999.2589.  Google Scholar

[20]

A. Mijatović and M. Urusov, On the martingale property of certain local martingales, Probab. Theory Related Fields, 152 (2012), 1-30.  doi: 10.1007/s00440-010-0314-7.  Google Scholar

[21]

Y. ObayashiP. Protter and S. Wang, The lifetime of a financial bubble, Math. Financ. Econ., 11 (2017), 45-62.  doi: 10.1007/s11579-016-0170-z.  Google Scholar

[22]

P. Protter, A mathematical theory of financial bubbles, in Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math., 2081, Springer, Cham, 2013, 1–108. doi: 10.1007/978-3-319-00413-6_1.  Google Scholar

[23]

P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Stochastic Modelling and Applied Probability, 21, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.  Google Scholar

[24]

A. Riveros, Propiedad de Martingala, Volatilidad Estocástica y Burbujas Financieras, Masters thesis, Universidad de Chile, 2020. Available from: http://repositorio.uchile.cl/handle/2250/175773. Google Scholar

[25]

J. Ruf, The martingale property in the context of stochastic differential equations, Electron. Commun. Probab., 20 (2015), 10pp. doi: 10.1214/ECP.v20-3449.  Google Scholar

[26]

C. A. Sin, Complications with stochastic volatility models, Adv. in Appl. Probab., 30 (1998), 256-268.  doi: 10.1239/aap/1035228003.  Google Scholar

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