The Annals of Applied Probability

Bubbles, convexity and the Black–Scholes equation

Erik Ekström and Johan Tysk

Full-text: Open access

Abstract

A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black–Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts.

Article information

Source
Ann. Appl. Probab. Volume 19, Number 4 (2009), 1369-1384.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700621

Digital Object Identifier
doi:10.1214/08-AAP579

Mathematical Reviews number (MathSciNet)
MR2538074

Zentralblatt MATH identifier
1219.91138

Subjects
Primary: 35K65: Degenerate parabolic equations 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91B28

Keywords
Parabolic equations stochastic representation preservation of convexity local martingales

Citation

Ekström, Erik; Tysk, Johan. Bubbles, convexity and the Black–Scholes equation. Ann. Appl. Probab. 19 (2009), no. 4, 1369--1384. doi:10.1214/08-AAP579. https://projecteuclid.org/euclid.aoap/1248700621


Export citation

References

  • [1] Andersen, L. B. G. and Piterbarg, V. V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11 29–50.
  • [2] Blei, S. and Engelbert, H.-J. (2008). On exponential local martingales associated with strong Markov continuous local martingales. Stoch. Proc. Appl. To appear.
  • [3] Brandt, A. (1969). Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle. Israel J. Math. 7 254–262.
  • [4] Cox, A. M. G. and Hobson, D. G. (2005). Local martingales, bubbles and option prices. Finance Stoch. 9 477–492.
  • [5] Delbaen, F. and Schachermayer, W. (1994). Arbitrage and free lunch with bounded risk for unbounded continuous processes. Math. Finance 4 343–348.
  • [6] Ekström, E. (2004). Properties of American option prices. Stochastic Process. Appl. 114 265–278.
  • [7] Ekström, E. and Tysk, J. (2008). The Black–Scholes equation in stochastic volatility models. Preprint.
  • [8] El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. E. (1998). Robustness of the Black and Scholes formula. Math. Finance 8 93–126.
  • [9] Friedman, A. (1976). Stochastic Differential Equations and Applications. Vols. 1 and 2. Probability and Mathematical Statistics 28. Academic Press, New York.
  • [10] Friedman, A. (1958). Boundary estimates for second order parabolic equations and their applications. J. Math. Mech. 7 771–791.
  • [11] Heston, S., Loewenstein, M. and Willard, G. (2007). Options and bubbles. Rev. Financ. Stud. 20 359–390.
  • [12] Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 193–205.
  • [13] Hobson, D. (2008). Comparison results for stochastic volatility models via coupling. Finance Stoch. To appear.
  • [14] Jarrow, R. A., Protter, P. and Shimbo, K. (2006). Asset price bubbles in complete markets. In Advances in Mathematical Finance 105–130. Birkhäuser, Boston.
  • [15] Jarrow, R. A., Protter, P. and Shimbo, K. (2007). Asset price bubbles in incomplete markets. Preprint.
  • [16] Janson, S. and Tysk, J. (2003). Volatility time and properties of option prices. Ann. Appl. Probab. 13 890–913.
  • [17] Janson, S. and Tysk, J. (2006). Feynman–Kac formulas for Black–Scholes-type operators. Bull. London Math. Soc. 38 269–282.
  • [18] Johnson, G. and Helms, L. L. (1963). Class D supermartingales. Bull. Amer. Math. Soc. 69 59–62.
  • [19] Knerr, B. F. (1980/81). Parabolic interior Schauder estimates by the maximum principle. Arch. Rational Mech. Anal. 75 51–58.
  • [20] Lewis, A. L. (2000). Option Valuation Under Stochastic Volatility: With Mathematica Code. Finance Press, Newport Beach, CA.
  • [21] Pal, S. and Protter, P. (2008). Strict local martingales, bubbles and no early exercise. Preprint.
  • [22] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [23] Sin, C. A. (1998). Complications with stochastic volatility models. Adv. in Appl. Probab. 30 256–268.