June  2022, 1(2): 161-188. doi: 10.3934/fmf.2021006

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 Published  June 2022 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, 2022, 1 (2) : 161-188. 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

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