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A Mathematical Theory of Financial Bubbles

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Paris-Princeton Lectures on Mathematical Finance 2013

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2081))

Abstract

Over the last 10 years or so a mathematical theory of bubbles has emerged, in the spirit of a martingale theory based on an absence of arbitrage, as opposed to an equilibrium theory. This paper attempts to explain the major developments of the theory as it currently stands, including equities, options, forwards and futures, and foreign exchange. It also presents the recent development of a theory of bubble detection. Critiques of the theory are presented, and a defense is offered. Alternative theories, especially for bubble detection, are sketched.

Recurrent speculative insanity and the associated financial deprivation and larger devastation are, I am persuaded, inherent in the system. Perhaps it is better that this be recognized and accepted.

–John Kenneth Galbraith, A Short History of Financial Euphoria, Forward to the 1993 Edition, p. viii.

Supported in part by NSF grant DMS-0906995.

The author wishes to thank the hospitality of the Courant Institute of Mathematical Sciences, of NYU, as well as INRIA at Sophia Antipolis, for their hospitality during the writing of this paper.

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Notes

  1. 1.

    More precisely, J.P. Morgan’s role was to organize and pressure a group of important bankers to themselves add liquidity to the system and help to stem the panic. Ron Chernow describes the scene dramatically, as a crucible in which every minute counted [28, pp. 124–125] as the 70 year old J.P. Morgan’s prestige and personality prevailed to save the day.

  2. 2.

    Even in 1907, in his December 30 speech in Boston, President Taft pointed out that an impediment to resolving the crisis was the government’s inability to increase rapidly and temporarily the money supply; one can infer from his remarks that he was already thinking along the lines of creating a Federal Reserve system [149].

  3. 3.

    Classic economic theory tells us that it makes no difference in the short run whether or not a company pays out dividends or reinvests its returns in the company in order to grow, in terms of wealth produced for the stockholders. However eventually investors are going to want a cash flow, as even Apple has recently discovered [160], and dividends will be issued.

  4. 4.

    A strict local martingale is a local martingale which is not a martingale. More precisely, a process M with M 0 = 1, is a local martingale if there exists a sequence of stopping times \((\tau _{n})_{n\geq 1}\) increasing to ∞ a.s. such that for each n one has that the process \((M_{t\wedge \tau _{n}})_{t\geq 0}\) is a martingale.

  5. 5.

    The “usual hypotheses” are defined in [128]. For convenience, what they are is that on the underlying space \((\Omega ,\mathcal{F},P)\) with filtration \(\mathbb{F} = (\mathcal{F}_{t})_{t\geq 0}\), the filtration \(\mathbb{F}\) is right continuous in the sense that \(\mathcal{F}_{t} = \cap _{u>t}\mathcal{F}_{u}\), and also \(\mathcal{F}_{0}\) contains all the P null sets of \(\mathcal{F}\). For all other unexplained stochastic calculus terms and notation, please see [128].

  6. 6.

    No company, government, or economic system can last forever. Of the original 12 companies from the 1896 Dow Jones Industrial Average, only General Electric and Laclede Gas still exist under the same name with remarkable continuity. National Lead is now NL Industries, and Laclede Gas (a utility in St. Louis) was removed from the DJIA in 1899. See [51] for more details.

  7. 7.

    Càdlàg paths refers to paths that are right continuous and have left limits, a.s.

  8. 8.

    In the general case one must also consider sigma martingales, but if the price process is assumed to be nonnegative, we use the fortuitous fact that sigma martingales bounded from below are local martingales (see for example [76] or [128]).

  9. 9.

    One way to think of risk aversion is to consider the following game one time, and one time only: I toss a fair coin, and you pay me $2 if it comes up heads, and I pay you $5 if it comes up tails. Most people would gladly play such a game. But if the stakes were raised to $20,000 and $50,000, most people short of the 2012 US Presidential candidate and über rich Mitt Romney would not play the game, unwilling to risk losing $20,000 in one toss of a coin. (In a 2012 presidential race debate Romney offered to bet $10,000 about something an opponent said; he did it casually, as if this were a frequent type of bet for him.) Exceptions it is easy to imagine are Wall Street and Connecticut Hedge Fund traders, who deal with large sums of other people’s money; they might well take advantage of such an opportunity for a quick profit (or loss) since the game is a good bet, irrespective of the high stakes. The hedge fund traders are still risk averse of course, but in ways quite different from the small “retail” investor.

  10. 10.

    One can ask if it is not possible to have bubbles which are negative? In our models, for stocks, the answer is no. However for risky assets other than stocks, such as foreign exchange, it is possible to have negative bubbles. For example when the dollar is in a bubble relative to the euro, then the euro would be in a negative bubble relative to the dollar.

  11. 11.

    We thank Etienne Tanre of INRIA for making these simulations of paths of the inverse Bessel process.

  12. 12.

    We thank a referee for suggesting we include this remark.

  13. 13.

    What we mean by this is that if NFLVR holds, then one can neither find not exploit an arbitrage opportunity in the short run by strategies of buying and selling the asset, or by using financial derivatives

  14. 14.

    Although the definition of the fundamental price as given depends on the construction of the extended economy, one could have alternatively used expression (50) as the initial definition. This alternative approach relaxes the extrinsic uncertainty restriction explicit in our extended economy.

  15. 15.

    To be precise, we note that the strike price is quoted in units of the numéraire for all of these derivative securities.

  16. 16.

    In an analogous theorem in Jarrow et al. [89], they used the implicit assumption that T = τ which would imply that \(E_{{Q}^{t{\ast}}}\left [\left .\beta _{T}^{3}\right \vert \mathcal{F}_{t}\right ] = 0\).

  17. 17.

    We wish to thank Roy DeMeo of Morgan Stanley for stimulating discussion on bubbles and foreign exchange.

  18. 18.

    To consider foreign currency derivatives, one would want to include trading in default free zero-coupon bonds in both dollars and euros. Then, the no arbitrage condition would be extended to include the discounted dollar values of the dollar zero-coupon bonds and the dollar value of the euro zero-coupon bonds (see Amin and Jarrow [2]).

  19. 19.

    The author spent over 20 years at Purdue University in Indiana, and there he developed an appreciation for the importance of pork belly futures, for example.

  20. 20.

    When considering non-financial commodities, this expression implicitly assumes that the risky asset is storable.

  21. 21.

    \({\mathbb{L}}^{0}(\Omega ,\mathcal{F}_{t},P)\) is the collection of finite valued \(\mathcal{F}_{t}\) measurable functions on \(\Omega .\)

  22. 22.

    Sigma martingales are defined and discussed for example in [76, 128]. When a sigma martingale is continuous, or bounded below, it is a local martingale. Otherwise, in general, local martingales are a proper subset of sigma martingales.

  23. 23.

    How one measures capital at risk (involving Value at Risk and the theory of risk measures) is another thorny issue that we do not even attempt to address in this article.

  24. 24.

    See for example [65] for the details of how to go about this.

  25. 25.

    “CEV” stands for constant elasticity of volatility.

  26. 26.

    We thank Arun Verma of Bloomberg for quickly providing us with high quality tick data.

  27. 27.

    Hereafter referred to as Zmirou’s estimator.

  28. 28.

    We thank Arun Verma of Bloomberg, again, for providing us with data.

  29. 29.

    That is, there are no continuous functions except for trivialities such as using the discrete topology and thereby making all functions continuous.

  30. 30.

    It is shown in [104] that the two methods are equivalent.

References

  1. Y. Aït-Sahalia, J. Yu, High frequency market microstructure noise estimates and liquidity measures. Ann. Appl. Stat. 3, 422–457 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Amin, R. Jarrow, Pricing Foreign currency options under stochastic interest rates. J. Int. Money Financ. 10(3), 310–329 (1991)

    Article  Google Scholar 

  3. L.B.G. Andersen, V. Piterbarg, Moment explosions in stochastic volatility models. Financ. Stoch. 11, 29–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.V. Andersen, D. Sornette, Fearless versus fearful speculative financial bubbles. Phys. A 337, 565–585 (2004)

    Article  MathSciNet  Google Scholar 

  5. N. Aronszajin, La théorie générale des noyaux reproduisants et ses applications, Première Partie. Proc. Camb. Philos. Soc. 39, 133–153 (1943)

    Article  Google Scholar 

  6. N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Balasko, D. Cass, K. Shell, Market participation and sunspot equilibria. Rev. Econ. Stud. 62, 491–512 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Baldwin, LinkedIn shares were a bubble: Academic model (2 June 2011). Available (for example) at the URL http://www.easybourse.com/bourse/international/news/918606/linkedin-shares-were-a-bubble-academic-model.html,

  9. E. Bayraktar, C. Kardaras, H. Xing, Strict local martingale deflators and valuing American call-type options. Financ. Stoch. 16, 275–291 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Bastiaensen, P. Cauwels, Z.-Q. Jiang, D. Sornette, R. Woodard, W.-X. Zhou, Bubble diagnosis and prediction of the 2005–2007 and 2008–2009 Chinese stock market bubbles. J. Econ. Behav. Organ. 74, 149–162 (2010)

    Article  Google Scholar 

  11. A. Bentata, M. Yor, Ten notes on three lectures: From Black–Scholes and Dupire formulae to last passage times of local martingales, in Notes from a Course at the Bachelier Seminar, 2008

    Google Scholar 

  12. B. Bernanke, Senate Confirmation Hearing, December 2009. This is quoted in many places; one example is Dealbook, edited by Andrew Ross Sorkin (January 6, 2010)

    Google Scholar 

  13. J. Berman, Apple not bringing overseas cash back home, Blames U.S. tax policy. The Huffington Post (March 19, 2012), online at the URL: http://www.huffingtonpost.com/2012/03/19/apple-us-tax-law_n_1362934.html

  14. F. Biagini, H. Föllmer, S. Nedelcu, Shifting martingale measures and the slow birth of a bubble, Working paper, 2012

    Google Scholar 

  15. P. Biane, M. Yor, Quelques Précisions sur le Méandre Brownien. Bull. de la Soc. Math. 2 Sér. 112, 101–109 (1988)

    Google Scholar 

  16. M. Blais, P. Protter, An analysis of the supply curve for liquidity risk through book data. Int. J. Theor. Appl. Financ. 13(6), 821–838 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Bris, W.N. Goetzmann, N. Zhu, Efficiency and the bear: Short sales and markets around the world. J. Financ. 62(3), 1029–1079 (2007)

    Article  Google Scholar 

  18. R. Caballero, E. Fahri, P.-O. Gourinchas, Financial crash, commodity prices and global imbalances. Brookings Pap. Econ. Act. Fall, 1–55 (2008)

    Google Scholar 

  19. C. Camerer, Bubbles and fads in asset prices. J. Econ. Sur. 3(1), 3–41 (1989)

    Article  Google Scholar 

  20. R. Carmona (ed.), Indifference Pricing: Theory and Applications (Princeton University Press, Princeton, 2008)

    MATH  Google Scholar 

  21. P. Carr, T. Fisher, J. Ruf, On the hedging of options On exploding exchange rates, preprint (2012)

    Google Scholar 

  22. D. Cass, K. Shell, Do sunspots matter? J. Polit. Econ. 91(2), 193–227 (1983)

    Article  Google Scholar 

  23. U. Çetin, R. Jarrow, P. Protter, Liquidity risk and arbitrage pricing theory. Financ. Stoch. 8, 311–341 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. U. Çetin, M. Soner, N. Touzi, Option hedging for small investors under liquidity costs. Financ. Stoch. 14, 317–341 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Charoenrook, H. Daouk, A study of market-wide short-selling restrictions (February 2005). Available at SSRN: http://ssrn.com/abstract=687562orhttp://dx.doi.org/10.2139/ssrn.687562

    Google Scholar 

  26. J. Chen, H. Hong, J. Stein, Breadth of ownership and stock returns. J. Financ. Econ. 66, 171–205 (2002)

    Article  Google Scholar 

  27. P. Cheridito, D. Filipovic, M. Yor, Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Probab. 15, 1713–1732 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Chernow, The House of Morgan (Atlantic Monthly Press, New York, 1990)

    Google Scholar 

  29. K.L Chung, R.J.Williams, Introduction to Stochastic Integration, 2nd edn. (Birkhäuser, Boston, 1990)

    Google Scholar 

  30. A. Cox, D. Hobson, Local martingales, bubbles and option prices. Financ. Stoc. 9, 477–492 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Cox, J. Ingersoll, S. Ross, The relationship between forward prices and futures prices. J. Financ. Econ. 9(4), 321–346 (1981)

    Article  Google Scholar 

  32. J. Creswell, Analysts are Wary of LinkedIn’s stock Surge. New York Times Digest 4 (Monday, May 23, 2011)

    Google Scholar 

  33. S.M. Davidoff, How to deflate a gold bubble (that might not even exist). Dealbook; The New York Times (August 30, 2011)

    Google Scholar 

  34. F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. F. Delbaen, W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312(2), 215–250 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Delbaen, W. Schachermayer, A simple counter-example to several problems in the theory of asset pricing. Math. Financ. 8, 1–12 (1998)

    Article  MATH  Google Scholar 

  37. F. Delbaen, W. Schachermayer, in The Mathematics of Arbitrage. Springer Finance (Springer, Heidelberg, 2005)

    Google Scholar 

  38. F. Delbaen, H. Shirakawa, No arbitrage condition for positive diffusion price processes. Asia Pacific Financ. Mark. 9, 159–168 (2002)

    Article  MATH  Google Scholar 

  39. C. Dellacherie, Capacités et Processus Stochastiques (Springer, Berlin, 1972)

    MATH  Google Scholar 

  40. H. Dengler, R.A. Jarrow, Option pricing using a binomial model with random time steps (a formal model of gamma hedging). Rev. Deriv. Res. 1, 107–138 (1997)

    Article  MATH  Google Scholar 

  41. K. Diether, C. Malloy, A. Scherbina, Differences of opinion and the cross section of stock returns. J. Financ. 52, 2113–2141 (2002)

    Article  Google Scholar 

  42. D. Duffie, Dynamic Asset Pricing Theory, 3rd edn. (Princeton University Press, Princeton, 2001)

    MATH  Google Scholar 

  43. B. Dupire, Pricing and hedging with smiles, in Mathematics of Derivative Securities (Cambridge University Press, Cambridge, 1997), pp. 103–112

    MATH  Google Scholar 

  44. E. Ekström, J. Tysk, Convexity and the Black–Scholes equation. Ann. Appl. Probab. 19, 1369–1384 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Ekström, P. Lötstedt, L. Von Sydow, J. Tysk, Numerical option pricing in the presence of bubbles. Quant. Financ. 11(8), 1125–1128 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. K.D. Elworthy, X.M. Li, M. Yor, The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Probab. Theory Relat. Fields 115, 325–355 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. H.J. Engelbert, W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, Parts I, II, and III. Math. Nachr. 143, 167–184; 144, 241–281; 151, 149–197 (1989/1991)

    Google Scholar 

  48. G. Evans, A test for speculative bubbles in the sterling-dollar exchange rate: 1981–1984. Am. Econ. Rev. 76(4), 621–636 (1986)

    MathSciNet  Google Scholar 

  49. J.D. Farmer, The economy needs agent-based modelling. Nature 460, 680–681 (2009)

    Article  Google Scholar 

  50. R. Fernholz, I. Karatzas, Relative arbitrage in volatility stabilized markets. Ann. Financ. 1, 149–177 (2005)

    Article  MATH  Google Scholar 

  51. Finfacts web site: http://www.finfacts.com/Private/curency/djones.htm. Accessed 2012

  52. D. Florens-Zmirou, On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30, 790–804 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  53. H. Föllmer, M. Schweizer, Hedging of contingent claims under incomplete information, in Applied Stochastic Analysis, ed. by M.H.A. Davis, R.J. Elliott (Gordon and Breach, New York, 1991), pp. 389–414

    Google Scholar 

  54. K. Froot, M. Obstfeld, Intrinsic bubbles: The case of stock prices. Am. Econ. Rev. 81(5), 1189–1214 (1991)

    Google Scholar 

  55. J.K. Galbraith, A Short History of Financial Euphoria (Penguin Books, New York, 1993)

    Google Scholar 

  56. V. Genon-Catalot, J. Jacod, On the estimation of the diffusion coefficient for multi dimensional diffusion processes. Ann. de l’I.H.P B 29, 119–151 (1993)

    Google Scholar 

  57. C. Gilles, Charges as equilibrium prices and asset bubbles. J. Math. Econ. 18, 155–167 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  58. C. Gilles, S.F. LeRoy, Bubbles and charges. Int. Econ. Rev. 33(2), 323–339 (1992)

    Article  Google Scholar 

  59. J. Goldstein, Interview with William Dudley, the President of the New York Federal Reserve. Planet Money (April 9, 2010)

    Google Scholar 

  60. P. Grandits, T. Rheinlander, On the minimal entropy martingale measure. Ann. Probab. 30, 1003–1038 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Guasoni, M. Rasonyi, Fragility of arbitrage and bubbles in diffusion models, preprint (2011). Available at SSRN: http://ssrn.com/abstract=1856223orhttp://dx.doi.org/10.2139/ssrn.1856223

    Google Scholar 

  62. M. Guidolina, A. Timmermann, Asset allocation under multivariate regime switching. J. Econ. Dyn. Control 31, 3503–3544 (2007)

    Article  MATH  Google Scholar 

  63. S. Heston, M. Loewenstein, G.A. Willard, Options and bubbles. Rev. Financ. Stud. 20(2), 359–390 (2007)

    Article  Google Scholar 

  64. M. Hoffmann, L p Estimation of the diffusion coefficient. Bernoulli 5, 447–481 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  65. T. Hollebeek, T.S. Ho, H. Rabitz, Constructing multidimensional molecular potential energy surfaces from AB initio data. Annu. Rev. Phys. Chem. 50, 537–570 (1999)

    Article  Google Scholar 

  66. C.H. Hommes, A. Huesler, D. Sornette, Super-exponential bubbles in lab experiments: evidence for anchoring over-optimistic expectations on price, preprint (2012). Available at http://arxiv.org/abs/1205.0635

  67. H. Hong, J. Scheinkman, W. Xiong, Advisors and asset prices: A model of the origins of bubbles. J. Financ. Econ. 89, 268–287 (2008)

    Article  Google Scholar 

  68. H. Hulley, The economic plausibility of strict local martingales in financial modeling, in Contemporary Quantitative Finance, ed. by C. Chiarella, A. Novikov (Springer, Berlin, 2010)

    Google Scholar 

  69. H. Hulley, E. Platen, A visual criterion for identifying Itô diffusions as martingales or strict local martingales, in Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol. 63, Part 1 (2011), pp. 147–157

    Google Scholar 

  70. A. Hüsler, D. Sornette, C.H. Hommes, Super-exponential bubbles in lab experiments: evidence for anchoring over-optimistic expectations on price, Swiss Finance Institute Research Paper No. 12–20 (2012). Available at SSRN: http://ssrn.com/abstract=2060978orhttp://dx.doi.org/10.2139/ssrn.2060978

    Google Scholar 

  71. J. Jacod, Rates of convergence to the local time of a diffusion. Ann. de l’Institut Henri Poincaré Sect. B 34, 505–544 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  72. J. Jacod, Non-parametric kernel estimation of the coefficient of a diffusion. Scand. J. Stat. 27, 83–96 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  73. J. Jacod, P. Protter, Risk neutral compatibility with option prices. Financ. Stoch. 14, 285–315 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  74. J. Jacod, P. Protter, Discretization of Processes (Springer, Heidelberg, 2012)

    Book  MATH  Google Scholar 

  75. J. Jacod, A. Shiryaev, Local martingales and the fundamental asset pricing theorems in the discrete-time case. Financ. Stoch. 2, 259–273 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  76. J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Heidelberg, 2003)

    Book  MATH  Google Scholar 

  77. J. Jacod, Y. Li, P. Mykland, M. Podolskij, M. Vetter, Microstructure noise in the continuous case: The pre-averaging approach. Stoch. Process. Their Appl. 119, 2249–2276 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  78. R. Jarrow, Y. Kchia, P. Protter, How to detect an asset bubble. SIAM J. Financ. Math. 2, 839–865 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  79. R. Jarrow, Y. Kchia, P. Protter, Is there a bubble in LinkedIn’s stock price? J. Portf. Manag. 38, 125–130 (2011)

    Article  Google Scholar 

  80. R. Jarrow, Y. Kchia, P. Protter, Is gold in a bubble? Bloomberg’s Risk Newsletter 8–9 (October 26, 2011)

    Google Scholar 

  81. R. Jarrow, D. Madan, Arbitrage, martingales, and private monetary value. J. Risk 3(1), 73–90 (2000)

    Article  Google Scholar 

  82. R. Jarrow, G. Oldfield, Forward contracts and futures contracts. J. Financ. Econ. 9(4), 373–382 (1981)

    Article  Google Scholar 

  83. R. Jarrow, P. Protter, An introduction to financial asset pricing theory, in Handbook in Operation Research and Management Science: Financial Engineering, vol. 15, ed. by J. Birge, V. Linetsky. (North Holland, New York, 2007), pp. 13–69

    Google Scholar 

  84. R. Jarrow, P. Protter, Forward and futures prices with bubbles. Int. J. Theor. Appl. Financ. 12(7), 901–924 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  85. R. Jarrow, P. Protter, Foreign currency bubbles. Rev. Deriv. Res. 14(1), 67–83 (2011)

    Article  MATH  Google Scholar 

  86. R. Jarrow, P. Protter, Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory. Financ. Res. Lett. 9, 58–62 (2012)

    Article  Google Scholar 

  87. R. Jarrow, Y. Yildirim, Pricing treasury inflation protected securities and related derivatives using an HJM model. J. Financ. Quant. Anal. 38(2), 337–358 (2003)

    Article  Google Scholar 

  88. R. Jarrow, P. Protter, K. Shimbo, Asset price bubbles in a complete market, in Adv. Math. Financ. [in Honor of Dilip B. Madan], ed. by M.C. Fu, D. Madan (2006), pp. 105–130

    Google Scholar 

  89. R. Jarrow, P. Protter, K. Shimbo, Asset price bubbles in incomplete markets. Math. Financ. 20, 145–185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  90. R. Jarrow, P. Protter, A. Roch, A liquidity-based model for asset price bubbles. Quant. Financ. (2011). doi:10.1080/14697688.2011.620976

    MATH  Google Scholar 

  91. R. Jarrow, P. Protter, S. Pulido, The effect of trading futures on short sales constraints, preprint (2012)

    Google Scholar 

  92. A. Johansen, D. Sornette, Modeling the stock market prior to large crashes. Eur. Phys. J. B 9, 167–174 (1999)

    Article  Google Scholar 

  93. G. Johnson, L.L. Helms, Class D supermartingales. Bull. Am. Math. Soc. 14, 59–61 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  94. B. Jourdain, Loss of martingality in asset price models with lognormal stochastic volatility, Preprint CERMICS 2004-267 (2004). http://cermics.enpc.fr/reports/CERMICS-2004/CERMICS-2004-267.pdf

  95. Y. Kabanov, In discrete time a local martingale is a martingale under an equivalent probability measure. Financ. Stoch. 12, 293–297 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  96. I. Karatzas, C. Kardaras, The numéraire portfolio in semimartingale financial models. Financ. Stoch. 11, 447–493 (2007)

    Article  MATH  Google Scholar 

  97. I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)

    MATH  Google Scholar 

  98. I. Karatzas, S. Shreve, Methods of Mathematical Finance (Springer, New York, 2010)

    MATH  Google Scholar 

  99. C. Kardaras, Valuation and parity formulas for exchange options, preprint (2012). Available at arXiv: arXiv:1206.3220v1 [q-fin.PR]

    Google Scholar 

  100. C. Kardaras, D. Kreher, A. Nikeghbali, Strict local martingales and bubbles, preprint (2012). Available at arXiv:1108.4177v1 [math.PR]

    Google Scholar 

  101. S. Kotani, On a condition that one dimensional diffusion processes are martingales, in In Memoriam Paul-André Meyer. Séminaire de Probabilitiés XXXIX. Lecture Notes in Mathematics, vol. 1874 (2006), pp. 149–156

    Google Scholar 

  102. D. Kramkov, Discussion on how to detect an asset bubble, by R. Jarrow, Y. Kchia, and P. Protter; Remarks given after the presentation of P. Protter at the meeting, Contemporary Issues and New Directions in Quantitative Finance, Oxford, England, 10 July 2010

    Google Scholar 

  103. T.G. Kurtz, P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19s, 1035–1070 (1991)

    Google Scholar 

  104. T.G. Kurtz, P. Protter, Characterizing the weak convergence of stochastic integrals, in Stochastic Analysis, ed. by M. Barlow, N. Bingham (Cambridge University Press, Cambridge, 1991), pp. 255–259

    Chapter  Google Scholar 

  105. O.A. Lamont, R.H. Thaler, Can the market add and subtract? Mispricing in tech stock Carve-outs. J. Polit. Econ. 111, 227–268 (2003)

    Google Scholar 

  106. X. Li, M. Lipkin, R. Sowers, Dynamics of Bankrupt stocks, preprint (2012). Available at SSRN: http://ssrn.com/abstract=2043631

  107. P.L. Lions, M. Musiela, Correlations and bounds for stochastic volatility models. Ann. Inst. Henri Poincaré, (C) Nonlinear Anal. 24(1), 1–16 (2007)

    Google Scholar 

  108. M. Loewenstein, G.A. Willard, Rational equilibrium asset-pricing bubbles in continuous trading models. J. Econ. Theory 91, 17–58 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  109. H.P. McKean, Jr., Stochastic Integrals (AMS Chelsea Publishing, 2005) [Originally published in 1969 by Academic Press, New York]

    Google Scholar 

  110. D. Madan, M. Yor, Itôs integrated formula for strict local martingales, in In Memoriam Paul-André Meyer, Séminaire de Probabilités XXXIX, ed. by M. Emery, M. Yor. Lecture Notes in Mathematics, vol. 1874 (Springer, Berlin, 2006), pp. 157–170

    Google Scholar 

  111. J. Markham, A Financial History of Modern U.S. Corporate Scandals: From Enron to Reform (M.E. Sharpe, Armonk, 2005)

    Google Scholar 

  112. R. Meese, Testing for bubbles in exchange markets: A case of sparkling rates? J. Polit. Econ. 94(2), 345–373 (1986)

    Article  MathSciNet  Google Scholar 

  113. J. Mémin, L. Slominski, Condition UT et Stabilité en Loi des Solutions dEquations Différentielles Stochastiques, in Sém. de Proba. XXV. Lecture Notes in Mathematics, vol. 1485 (1991), pp. 162–177

    Google Scholar 

  114. R. Merton, Theory of rational option pricing. Bell J. Econ. 4(1), 141–183 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  115. P.A. Meyer, in Martingales and Stochastic Integrals. Lecture Notes in Mathematics, vol. 284 (Springer, Berlin, 1972/1973)

    Google Scholar 

  116. A. Mijatovic, M. Urusov, On the martingale property of certain local martingales. Probab. Theory Relat. Fields 152, 1–30 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  117. A. Mijatovic, M. Urusov, Convergence of integral functionals of one-dimensional diffusions. Probab. Theory Relat. Fields 152, 1–30 (2012)

    Article  MATH  Google Scholar 

  118. E. Miller, Risk, uncertainty and divergence of opinion. J. Financ. 32, 1151–1168 (1977)

    Article  Google Scholar 

  119. P. Monat, C. Stricker, Föllmer-Schweizer decomposition and mean-variance hedging for general claims. Ann. Probab. 23, 605–628 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  120. A. Nikeghbali, An essay on the general theory of stochastic processes. Probab. Sur. 3, 345–412 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  121. E. Ofek, M. Richardson, R.F. Whitelaw, Limited arbitrage and short sales restrictions: Evidence from the options markets. J. Financ. Econ. 74(2), 305–342 (2004)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  123. E. Parzen, Statistical inference on time series by RKHS methods, in Proceedings 12th Biennial Seminar, ed. by R. Pyke. (Canadian Mathematical Congress, Montreal, 1970), pp. 1–37

    Google Scholar 

  124. C. Profeta, B. Roynette, M. Yor, Option Prices as Probabilities: A New Look at Generalized Black-Scholes Formulae (Springer, Heidelberg, 2010)

    Book  MATH  Google Scholar 

  125. P.C.B. Phillips, J. Yu, Dating the timeline of financial bubbles during the subprime crisis. Quant. Econ. 2, 455–491 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  126. P.C.B. Phillips, Y. Wu, J. Yu, Explosive behavior in the 1990s Nasdaq: When did exuberance escalate asset values? Int. Econ. Rev. 52, 201–226 (2011)

    Article  Google Scholar 

  127. P. Protter, A partial introduction to financial asset pricing theory. Stoch. Process. Their Appl. 91, 169–203 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  128. P. Protter, Stochastic Integration and Differential Equations, version 2.1, 2nd edn. (Springer, Heidelberg, 2005)

    Google Scholar 

  129. P. Protter, The financial meltdown. Gazette de la Soc. des Math. de Fr. 119, 76–82 (2009)

    Google Scholar 

  130. P. Protter, K. Shimbo, No arbitrage and general semimartingales, in Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz. IMS Lecture Notes–Monograph Series, vol. 4 (2008), pp. 267–283

    Google Scholar 

  131. S. Pulido, The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions. Ann. Appl. Probab., preprint (2011). Available at http://arxiv.org/abs/1012.3102 (to appear)

  132. R. Rebib, D. Sornette, R. Woodard, W. Yan, Detection of crashes and rebounds in major equity markets. Int. J. Portf. Anal. Manag. 1(1), 59–79 (2012)

    Google Scholar 

  133. A. Roch, Liquidity risk, volatility, and financial bubbles, Ph.D. Thesis, Applied Mathematics, Cornell University, 2009. Available at the URL dspace.library.cornell.edu/bitstream/1813/.../Roch,\%20Alexandre.pdf

  134. A. Roch, Liquidity risk, price impacts and the replication problem. Financ. Stoch. 15(3), 399–419 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  135. A. Roch, M. Soner, Resilient price impact of trading and the cost of illiquidity (2011). Available at SSRN: http://ssrn.com/abstract=1923840orhttp://dx.doi.org/10.2139/ssrn.1923840.

    Google Scholar 

  136. L.C.G. Rogers, S. Singh, The costs of illiquidity and its effect on hedging. Math. Financ. 20(4), 597–615 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  137. J. Ruf, Hedging under arbitrage. Math. Financ. 23, 297–317 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  138. J. Scheinkman, W. Xiong, Overconfidence and speculative bubbles. J. Polit. Econ. 111(6), 1183–1219 (2003)

    Article  Google Scholar 

  139. J. Scheinkman, W. Xiong, Heterogeneous beliefs, speculation and trading in financial markets, in Paris-Princeton Lecture Notes on Mathematical Finance 2003. Lecture Notes in Mathematics, vol. 1847 (2004), pp. 217–250

    Article  MATH  Google Scholar 

  140. P. Schönbucher, A market model for stochastic implied volatility. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Engr. Sci. 357, 2071–2092 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  141. M. Schweizer, J. Wissel, Term structures of implied volatilities: Absence of arbitrage and existence results. Math. Financ. 18, 77–114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  142. M. Schweizer, J. Wissel, Arbitrage-free market models for option prices: The multi-strike case. Financ. Stoch. 12, 469–505 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  143. A.N. Shiryaev, Probability (Springer, Heidelberg, 1984)

    Book  MATH  Google Scholar 

  144. S. Shreve, Stochastic Calculus for Finance II: Continuous Time Models (Springer, New York, 2004)

    MATH  Google Scholar 

  145. C. Sin, Complications with stochastic volatility models. Adv. Appl. Probab. 30, 256–268 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  146. D. Sondermann, in Introduction to Stochastic Calculus for Finance: A New Didactic Approach. Lecture Notes in Economic and Mathematical Systems (Springer, Berlin, 2006)

    Google Scholar 

  147. D. Sornette, R. Woodard, W. Yan, W-X. Zhou, Clarifications to questions and criticisms on the Johansen-Ledoit-Sornette bubble model, preprint (2011). Available at arXiv:1107.3171v1

    Google Scholar 

  148. M. Swayne, How to detect a gold bubble, in eHow Money (November 28, 2010), at the URL http://www.ehow.com/how_7415180_detect-gold-bubble.html

  149. W.H. Taft, Present Day Problems: A Collection of Addresses Delivered on Various Occasions. (Books for Libraries Press, Freeport, 1967); Originally published in 1908

    Google Scholar 

  150. M. Taylor, Purchasing power parity. Rev. Int. Econ. 11(3), 436–452 (2003)

    Article  Google Scholar 

  151. A. Taylor, M. Taylor, The purchasing power parity debate. J. Econ. Perspect. 18(4), fall, 135–158 (2004)

    Google Scholar 

  152. C. Thomas-Agnan, Computing a family of reproducing kernels for statistical applications. Numer. Algorithms 13, 21–32 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  153. J. Tirole, On the possibility of speculation under rational expectations. Econometrica 50(5), 1163–1182 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  154. J. Tirole, Asset bubbles and overlapping generations. Econometrica 53(5), 1071–1100 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  155. V. Todorov, Estimation of continuous-time stochastic volatility models with jumps using high-frequency data. J. Econom. 148, 131–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  156. G. Wahba, Spline models for observational data, in CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59 (SIAM, Philadelphia, 1990)

    Google Scholar 

  157. G. Wahba, An introduction to model building With reproducing kernel Hilbert spaces (2000). Available for free download at http://www.stat.wisc.edu/~wahba/ftp1/interf/index.html

  158. P. Weil, On the possibility of price decreasing bubbles. Econometrica 58(6), 1467–1474 (1990)

    Article  MATH  Google Scholar 

  159. Wikipedia page http://en.wikipedia.org/wiki/Dot-com_bubble. Last updated 29 May 2013

  160. N. Wingfield, Flush with cash, apple plans buyback and dividend. New York Times (March 19, 2012)

    Google Scholar 

  161. WRDS, Lastminute.com, bid prices, from 14 March 2000 to 30 July 2004; WRDS, eToys, bid prices, from 20 May 1999 to 26 February 2001; WRDS, Infospace, bid prices, from 15 December 1998 to 19 September 2002; WRDS, Geocities, bid prices, from 11 August 1998 to 28 May 1999

    Google Scholar 

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Acknowledgements

The author wishes to thank Vicky Henderson and Ronnie Sircar for their kind invitation to do this project. He also wishes to thank Celso Brunetti, Roy DeMeo, John Hall, Jean Jacod, Robert Jarrow, Alexander Lipton, Wesley Phoa, and for helpful comments and criticisms, and his co-authors on the subject of bubbles, Younes Kchia, Soumik Pal, Sergio Pulido, Alexandre Roch, Kazuhiro Shimbo, and especially Robert Jarrow. He has also benefited from discussions with Sophia (Xiaofei) Liu, Johannes Ruf, Etienne Tanre, and especially Denis Talay. An anonymous referee gave us many useful suggestions and we thank this referee for his or her careful reading of this work. The author is grateful to the NSF for its financial support. He is also grateful for a sabbatical leave from Columbia University, and the author thanks the hospitality of the Courant Institute of NYU, and INRIA, Sophia–Antipolis, where he spent parts of his sabbatical leave.

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    Philip Protter

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Protter, P. (2013). A Mathematical Theory of Financial Bubbles. In: Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Mathematics, vol 2081. Springer, Cham. https://doi.org/10.1007/978-3-319-00413-6_1

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