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Local martingales, bubbles and option prices

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Abstract.

In this article we are interested in option pricing in markets with bubbles. A bubble is defined to be a price process which, when discounted, is a local martingale under the risk-neutral measure but not a martingale. We give examples of bubbles both where volatility increases with the price level, and where the bubble is the result of a feedback mechanism. In a market with a bubble many standard results from the folklore become false. Put-call parity fails, the price of an American call exceeds that of a European call and call prices are no longer increasing in maturity (for a fixed strike). We show how these results must be modified in the presence of a bubble. It turns out that the option value depends critically on the definition of admissible strategy, and that the standard mathematical definition may not be consistent with the definitions used for trading.

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Author notes

  1. David G. Hobson

    Present address: Department of Operations Research and Financial Engineering, Princeton University, NJ 08544, Princeton, USA

Authors and Affiliations

  1. Department of Mathematics, University of York, Heslington, YO10 5DD, York, UK

    Alexander M. G. Cox

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, UK

    David G. Hobson

Authors
  1. Alexander M. G. Cox
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  2. David G. Hobson
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Corresponding author

Correspondence to Alexander M. G. Cox.

Additional information

Mathematics Subject Classification:

91B70, 60G44, 60G40

JEL Classification:

G13, D84

The second author is supported by an Advanced Fellowship from the EPSRC. Thanks are due to Matt Davison, Jonathan Evans and Walter Schachermayer and an anonymous referee for helpful comments and for suggesting references to the literature.

Manuscript received: August 2004; final version received: March 2005

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Cox, A.M.G., Hobson, D.G. Local martingales, bubbles and option prices. Finance Stochast. 9, 477–492 (2005). https://doi.org/10.1007/s00780-005-0162-y

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  • DOI: https://doi.org/10.1007/s00780-005-0162-y

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