\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Asset price bubbles in markets with transaction costs

  • *Corresponding author: Thomas Reitsam

    *Corresponding author: Thomas Reitsam

The financial support of the Verein zur Versicherungswissenschaft München e.V. is gratefully acknowledged

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • We study asset price bubbles in market models with proportional transaction costs $ \lambda\in (0, 1) $ and finite time horizon $ T $ in the setting of [49]. By following [29], we define the fundamental value $ F $ of a risky asset $ S $ as the price of a super-replicating portfolio for a position terminating in one unit of the asset and zero cash. We then obtain a dual representation for the fundamental value by using the super-replication theorem of [50]. We say that an asset price has a bubble if its fundamental value differs from the ask-price $ (1+\lambda)S $. We investigate the impact of transaction costs on asset price bubbles and show that our model intrinsically includes the birth of a bubble.

    Mathematics Subject Classification: Primary: 91G99, 91B70; Secondary: 60G44.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Example 4: Simulation of $ 253 $ days of $ S $, defined in 60 with $ \mu = 0.3 $, $ \sigma_0 = 0.4 $ and the starting time $ \gamma $ of the bubble being uniformly distributed on $ (0, 1) $. Before $ \gamma $ the fundamental value coincides with $ (1+\lambda)S $. When $ \gamma $ occurs, the fundamental value drops to $ 0 $

  • [1] D. Abreu and M. K. Brunnermeier, Bubbles and crashes, Econometrica, 71 (2003), 173-204.  doi: 10.1111/1468-0262.00393.
    [2] F. Allen and G. Gorton, Churning bubbles, The Review of Economic Studies, 60 (1993), 813-836. 
    [3] F. AllenS. Morris and A. Postlewaite, Finite bubbles with short sale constraints and asymmetric information, Journal of Economic Theory, 61 (1993), 206-229. 
    [4] J.-P. Ansel and C. Stricker, Couverture des actifs contingents et prix maximum, Ann. Inst. H. Poincaré Probab. Statist., 30 (1994), 303-315. 
    [5] J. Anthony, M. Bijlsma, A. Elbourne, M. Lever and G. Zwart, Financial transaction tax: Review and assessment, CPB Netherlands Bureau For Economic Policy Analysis. CPB Discussion Paper, 3 (2012).
    [6] E. Bayraktar and X. Yu, On the market viability under proportional transaction costs, Mathematical Finance, 28 (2018), 800-838.  doi: 10.1111/mafi.12155.
    [7] F. BiaginiH. Föllmer and S. Nedelcu, Shifting martingale measures and the birth of a bubble as a submartingale, Finance and Stochastics, 18 (2014), 297-326.  doi: 10.1007/s00780-013-0221-8.
    [8] F. Biagini and J. Mancin, Financial asset price bubbles under model uncertainty, Probability, Uncertainty and Quantitative Risk, 2 (2017), Paper No. 14, 29 pp. doi: 10.1186/s41546-017-0026-3.
    [9] F. BiaginiA. Mazzon and T. Meyer-Brandis, Liquidity induced asset bubbles via flows of ELMMs, SIAM Journal on Financial Mathematics, 9 (2018), 800-834.  doi: 10.1137/16M1107097.
    [10] F. Biagini and S. Nedelcu, The formation of financial bubbles in defaultable markets, SIAM Journal on Financial Mathematics, 6 (2015), 530-558.  doi: 10.1137/140960608.
    [11] L. Campi and W. Schachermayer, A super-replication theorem in Kabanov's model of transaction costs, Finance and Stochastics, 10 (2006), 579-596.  doi: 10.1007/s00780-006-0022-4.
    [12] A. M. G. Cox and D. G. Hobson, Local martingales, bubbles and option prices, Finance and Stochastics, 9 (2005), 477-492.  doi: 10.1007/s00780-005-0162-y.
    [13] J. Cvitanić and I. Karatzas, Hedging and portfolio optimization under transaction costs: A martingale approach, Mathematical Finance, 6 (1996), 133-165.  doi: 10.1111/j.1467-9965.1996.tb00075.x.
    [14] J. B. De LongA. ShleiferL. H. Summers and R. J. Waldmann, Noise trader risk in financial markets, Journal of Political Economy, 98 (1990), 703-738. 
    [15] J. B. De LongA. ShleiferL. H. Summers and R. J. Waldmann, Positive feedback investment strategies and destabilizing rational speculation, The Journal of Finance, 45 (1990), 379-395. 
    [16] F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Mathematische Annalen, 300 (1994), 463-520.  doi: 10.1007/BF01450498.
    [17] C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Publishing Co. 29, Amsterdam, 1978.
    [18] C. Dellacherie and P.-A. Meyer, Probabilities and Potential B, North-Holland Publishing Co. 72, Amsterdam, 1982.
    [19] N. El Karoui and M.-C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM Journal on Control and Optimization, 33 (1995), 29-66.  doi: 10.1137/S0363012992232579.
    [20] H. Föllmer and P. Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics, 15 (2011), 25-38.  doi: 10.1051/ps/2010023.
    [21] H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110218053.
    [22] E. F. Gerding, Laws against bubbles: An experimental-asset-market approach to analyzing financial regulation, Wisconsin Law Review, 977 (2007), 1339-1349. 
    [23] E. Gerding, Law, Bubbles, and Financial Regulation, Routledge, 2013. doi: 10.4324/9781315885049.
    [24] P. Guasoni and M. Rásonyi, Fragility of arbitrage and bubbles in local martingale diffusion models, Finance and Stochastics, 19 (2015), 215-231.  doi: 10.1007/s00780-015-0256-0.
    [25] P. GuasoniM. Rásonyi and W. Schachermayer, Consistent price systems and face-lifting pricing under transaction costs, The Annals of Applied Probability, 18 (2008), 491-520.  doi: 10.1214/07-AAP461.
    [26] P. GuasoniM. Rásonyi and W. Schachermayer, The fundamental theorem of asset pricing for continuous processes under small transaction costs, Annals of Finance, 6 (2010), 157-191. 
    [27] J. M. Harrison and D. M. Kreps, Speculative investor behavior in a stock market with heterogeneous expectations, The Quarterly Journal of Economics, 92 (1978), 323-336. 
    [28] M. Herdegen, No-arbitrage in a numéraire-independent modeling framework, Mathematical Finance, 27 (2017), 568-603.  doi: 10.1111/mafi.12088.
    [29] M. Herdegen and M. Schweizer, Strong bubbles and strict local martingales, International Journal of Theoretical and Applied Finance, 19 (2016), 1650022, 44 pp. doi: 10.1142/S0219024916500229.
    [30] S. L. HestonM. Loewenstein and G. A. Willard, Options and bubbles, The Review of Financial Studies, 20 (2006), 359-390. 
    [31] R. Jarrow, Asset price bubbles, Annual Review of Financial Economics, 7 (2015), 201-218. 
    [32] R. JarrowY. Kchia and P. Protter, How to detect an asset bubble, SIAM Journal on Financial Mathematics, 2 (2011), 839-865.  doi: 10.1137/10079673X.
    [33] R. A. Jarrow and P. Protter, Forward and futures prices with bubbles, International Journal of Theoretical and Applied Finance, 12 (2009), 901-924.  doi: 10.1142/S0219024909005518.
    [34] R. Jarrow and P. Protter, Foreign currency bubbles, Review of Derivatives Research, 14 (2009), 67-83. 
    [35] R. A. JarrowP. Protter and A. F. Roch, A liquidity-based model for asset price bubbles, Quantitative Finance, 12 (2012), 1339-1349.  doi: 10.1080/14697688.2011.620976.
    [36] R. A. JarrowP. Protter and K. Shimbo, Asset price bubbles in a complete market, Advances in Mathematical Finance, 18 (2006), 105-130. 
    [37] R. A. JarrowP. Protter and K. Shimbo, Asset price bubbles in incomplete markets, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 20 (2010), 145-185.  doi: 10.1111/j.1467-9965.2010.00394.x.
    [38] Y. Kabanov and M. Safarian,, Markets with Transaction Costs: Mathematical Theory, Springer Finance. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-68121-2.
    [39] Y. M. Kabanov, Hedging and liquidation under transaction costs in currency markets, Finance and Stochastics, 3 (1999), 237-248. 
    [40] Y. M. Kabanov and G. Last, Hedging under transaction costs in currency markets: A continuous-time model, Mathematical Finance, 12 (2002), 63-70.  doi: 10.1111/1467-9965.00004.
    [41] Y. M. Kabanov and C. Stricker, Hedging of Contingent Claims Under Transaction Costs, Advances in finance and stochastics, 125–136, Soringer, 2002.
    [42] T. Kaizoji and D. Sornette, Bubbles and crashes, Encyclopedia of Quantitative Finance, (2010).
    [43] D. O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 105 (1996), 459-479.  doi: 10.1007/BF01191909.
    [44] A. J. Levitin and S. M. Wachter, Explaining the housing bubble, Geo. LJ, 100 (2011).
    [45] M. Loewenstein and G. A. Willard, Rational equilibrium asset-pricing bubbles in continuous trading models, Journal of Economic Theory, 91 (2000), 17-58.  doi: 10.1006/jeth.1999.2589.
    [46] P. Protter, Stochastic Differential Equations, Springer, Berlin, Heidelberg, 2005.
    [47] P. Protter, A mathematical theory of financial bubbles, Paris-Princeton Lectures on Mathematical Finance, 2081 (2013), 1-108.  doi: 10.1007/978-3-319-00413-6_1.
    [48] T. Reitsam, Asset Price Bubbles in Market Models with Proportional Transaction Costs, Ph.D thesis, Ludwig-Maximilians-Universität München, 2021.
    [49] W. Schachermayer, Admissible trading strategies under transaction costs, Séminaire de Probabilités XLVI, 2123 (2014), 317–331. doi: 10.1007/978-3-319-11970-0_11.
    [50] W. Schachermayer, The super-replication theorem under proportional transaction costs revisited, Mathematics and Financial Economics, 8 (2014), 383-398.  doi: 10.1007/s11579-014-0129-x.
    [51] W. Schachermayer, Asymptotic Theory of Transaction Costs, European Mathematical Society, 2017. doi: 10.4171/173.
    [52] M. Schatz and D. Sornette, Inefficient bubbles and efficient drawdowns in financial markets, Int. J. Theor. Appl. Finance, 23 (2020), 2050047, 56 pp. doi: 10.1142/S0219024920500478.
    [53] J. A. Scheinkman and W. Xiong, Overconfidence and speculative bubbles, The University of Chicago Press, 111 (2003), 1183-1220. 
    [54] R. J. ShillerIrrational Exuberance: Revised and Expanded Third Edition, Princeton university press, 2015. 
    [55] A. Shleifer, Inefficient markets: An introduction to behavioural finance, Journal of Institutional and Theoretical Economics (JITE), 158 (2002), 369-374.  doi: 10.1628/0932456022975402.
    [56] A. Shleifer and L. H. Summers, The noise trader approach to finance, Journal of Economic Perspectives, 4 (1990), 19-33. 
    [57] A. Shleifer and R. W. Vishny, The limits of arbitrage, The Journal of Finance, 52 (1997), 35-55. 
    [58] D. Sornette, Critical market crashes, Physics Reports, 378 (2003), 1-98.  doi: 10.1016/S0370-1573(02)00634-8.
    [59] R. Šperka and M. Spišák, Transaction costs influence on the stability of financial market: Agent-based simulation, Journal of Business Economics and Management, 14 (2013), 1-12. 
    [60] E. Strasser, Necessary and sufficient conditions for the supermartingale property of a stochastic integral with respect to a local martingale, Séminaire de Probabilités XXXVII, 1832 (2003), 385–393. doi: 10.1007/978-3-540-40004-2_16.
    [61] W. Xiong, Bubbles, crises, and heterogeneous beliefs, National Bureau of Economic Research, (2013).
  • 加载中
Open Access Under a Creative Commons license

Figures(1)

SHARE

Article Metrics

HTML views(2015) PDF downloads(194) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return