Local martingales, bubbles and option prices

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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Correspondence to Alexander M. G. Cox.

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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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Keywords:

  • Bubbles
  • feedback
  • local martingales
  • derivative pricing
  • put-call parity