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Local martingales, bubbles and option prices

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  • Alexander Cox
  • David Hobson

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. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    • Handle: RePEc:spr:finsto:v:9:y:2005:i:4:p:477-492
    DOI: 10.1007/s00780-005-0162-y
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