Skip to main content
SpringerLink
Log in

Relative asset price bubbles

  • Research Article
  • Published:
Annals of Finance Aims and scope Submit manuscript

Abstract

In models of financial bubbles, the price of a stock is typically unbounded, and this plays a fundamental role in the analysis of finite horizon local martingale bubbles. It would seem that price bubbles do not apply to a priori bounded risky asset prices, such as bond prices. To avoid this limitation, to characterize, and to identify bond price mispricings consistent with an absence of arbitrage, we develop the concept of a relative asset price bubble. This notion uses a risky asset’s price as the numéraire instead of the money market account’s value. This change of numéraire generates some interesting mathematical complexities because many important numéraires, including risky bonds, can vanish with positive probability over the model’s horizon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Paul Krugman, “Bernanke, Blower of Bubbles?,” May 9, 2013, The New York Times.

  2. We will prove this statement a the beginning of Sect. 6.

  3. The Bessel(3) process is most easily constructed as the Euclidean norm of a standard 3 dimensional Brownian motion, starting (for example) at the point \((1,0,0)\in \mathbb {R}^{3}\).

  4. In a credit risk model satisfying NFLVR and using the numéraire \(R_{t}\), it is well known that under a local martingale measure \(Q_{R}\) both \(p(t,T)=E_{R}\left( \frac{R_{t}}{R_{T}}\right) \) and \(d(t,T)\le p(t,T)\). The second inequality follows because the payoff to the risky zero coupon bond is less than or equal to that of the default free bond with probability one. Since \(R_{0}=1\) and \(R_{t}\) is non-decreasing, this implies \(R_{T}\ge R_{t}\ge 1\). Thus, we have that \(p(t,T)=E_{R}\left( \frac{R_{t}}{R_{T}}\right) \le 1\) as claimed.

  5. This condition effectively states that the spot rate will never become too large, this follows because \(p(t,T)=E_{Q_{R}}\left( 1/R_{T}\right) R_{t}.\)

References

  • Biagini, F., Föllmer, H., Nedelcu, S.: Shifting martingale measures and the birth of a bubble as a submartingale. Finance Stoch 18, 297–326 (2014)

    Article  Google Scholar 

  • Bilina Falafala, R.: Mathematical Modeling of Insider Trading, PhD thesis in the Statistics Department, Columbia University (2014)

  • Carr, P., Fisher, T., Ruf, J.: On the hedging of options on exploding exchange rates. Finance Stoch 18(1), 115–144 (2014)

    Article  Google Scholar 

  • Cox, A.M.G., Hobson, D.G.: Local martingales, bubbles and option prices. Finance Stoch 9(4), 477–492 (2005)

    Article  Google Scholar 

  • Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage: Springer: Berlin (2008)

  • Delbaen, F., Schachermayer, W.: Arbitrage and free lunch with bounded risk for unbounded continuous processes. Math Finance 4(4), 343–348 (1994)

    Article  Google Scholar 

  • Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stoch Int J Prob Stoch Process 53(3 and 4), 213–226 (1995a)

    Article  Google Scholar 

  • Delbaen, F., Schachermayer, W.: Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab Theory Relat Fields 102, 357–366 (1995b)

    Article  Google Scholar 

  • Delbaen, F., Schachermayer, W.: A simple counterexample to several problems in the theory of asset pricing. Math Finance 8(1), 1–11 (1998)

    Article  Google Scholar 

  • Delbaen, F., Shirakawa, H.: No arbitrage condition for positive diffusion price processes. Asia-Pac Financ Markets 9, 159–168 (2002)

    Article  Google Scholar 

  • Fontana, C.: Weak and strong no-arbitrage conditions for continuous financial markets. Int J Theor Appl Finance 18, 1550005 (2015). doi:10.1142/S021902491550005

    Article  Google Scholar 

  • Geman, H., El Karoui, N., Rochet, J.: Changes of Numéraire. Changes of probability measure and option pricing. J Appl Probab 32, 443–458 (1995)

    Article  Google Scholar 

  • Geman, H.: From measure changes to time changes in asset pricing. J Bank Finance 29, 2701–2711 (2005)

    Article  Google Scholar 

  • Hong, H., Sraer, D.: Quiet bubbles. J Financ Econ 110, 596–606 (2013)

    Article  Google Scholar 

  • Hulley, H.: The economic plausibility of strict local martingales in financial modelling. In: Chiarella, C., Novikov, A. (eds.) Contemporary Quantitative Finance: Berlin, Heidelberg: Springer (2010). doi:10.1007/978-3-642-03479-4_4

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

    Article  Google Scholar 

  • Janson, S., M’Baye, S., Protter, P.: Absolutely continuous compensators. Int J Theor Appl Finance 14, 335–351 (2011)

    Article  Google Scholar 

  • Jarrow, R., Protter, P., Shimbo, K.: Asset price bubbles in a complete market. In: Fu, M.C., Jarrow, R.A., Yen, J.-Y.J., Elliott, R.J. (eds.) Advances in Mathematical Finance (In honor of Dilip B. Madan on the occasion of his 60th birthday). Birkhäuser, Boston, pp. 105–130 (2006)

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

  • Jarrow, R.: The role of ABS, CDS and CDOs in the credit crisis and the economy. In: Blinder, A., Lo, A., Solow, R. (eds.) Rethinking the Financial Crisis, pp. 210–234. New York: Russell Sage Foundation (2012)

  • Jarrow, R., Protter, P., Shimbo, K.: Asset price bubbles in an incomplete market. Math Finance 20(2), 145–185 (2010)

    Article  Google Scholar 

  • Jarrow, R., Kchia, Y., Protter, P.: How to Detect an Asset Bubble. SIAM J Financ Math 2, 839–865 (2011)

    Article  Google Scholar 

  • Jarrow, R., Larsson, M.: The meaning of market efficiency. Math Finance 22(1), 1–30 (2012)

    Article  Google Scholar 

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

  • Kardaras, C.: Valuation and Parity Formulas for Exchange Options (preprint). arXiv:1206.3220v1 [q-fin.PR] (2012)

  • Li, X., Lipkin, M., Sowers, R.: Dynamics of bankrupt stocks. SIAM J Financ Math 5(1), 232–257 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Protter, P.: Stochastic Integration and Differential Equations. Version 2.1, Second edn. Springer, Heidelberg (2005)

    Book  Google Scholar 

  • Protter, P.: A Mathematical Theory of Bubbles. Paris–Princeton Lecture Notes in Mathematical Finance, Springer Lecture Notes in Mathematics, vol. 2081, pp. 1–108 (2013)

  • Ruf, J.: Hedging under arbitrage. Math Finance 23(2), 297–317 (2013)

    Article  Google Scholar 

  • Yan, J.-A.: A new look at the fundamental theorem of asset pricing. J Korean Math Soc 35, 659–673 (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Goldman Sachs, New York, NY, USA

    Roseline Bilina Falafala

  2. Samuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, NY, 14853, USA

    Robert A. Jarrow

  3. Kamakura Corporation, Honolulu, HI, 96815, USA

    Robert A. Jarrow

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

    Philip Protter

Authors
  1. Roseline Bilina Falafala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert A. Jarrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philip Protter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philip Protter.

Additional information

Philip Protter supported in part by NSF Grant DMS-1308483.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bilina Falafala, R., Jarrow, R.A. & Protter, P. Relative asset price bubbles. Ann Finance 12, 135–160 (2016). https://doi.org/10.1007/s10436-016-0274-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10436-016-0274-8

Keywords

JEL Classification

Search

Navigation