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On the Hedging of Options On Exploding Exchange Rates

  • Peter Carr
  • Travis Fisher
  • Johannes Ruf
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    We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of num\'eraire (change of currency) technique when the underlying is only a local martingale modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as num\'eraire. Within this context, we interpret the lack of the martingale property of an exchange-rate as a reflection of the possibility that the num\'eraire currency may devalue completely against the asset currency (hyperinflation).

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    File URL: http://arxiv.org/pdf/1202.6188
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    Paper provided by arXiv.org in its series Papers with number 1202.6188.

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    Date of creation: Feb 2012
    Date of revision: Nov 2013
    Handle: RePEc:arx:papers:1202.6188
    Contact details of provider: Web page: http://arxiv.org/

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    1. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, January.
    2. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    3. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    4. Freddy Delbaen & Walter Schachermayer, 1998. "A Simple Counterexample to Several Problems in the Theory of Asset Pricing," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 1-11.
    5. Orlin Grabbe, J., 1983. "The pricing of call and put options on foreign exchange," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 239-253, December.
    6. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    7. Orlin J. Grabbe, . "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 6-83, Wharton School Rodney L. White Center for Financial Research.
    8. Philipp J. Schönbucher, 2000. "A Libor Market Model with Default Risk," Bonn Econ Discussion Papers bgse15_2001, University of Bonn, Germany.
    9. Orlin J. Grabbe, . "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 06-83, Wharton School Rodney L. White Center for Financial Research.
    10. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    11. Robert Jarrow & Philip Protter, 2011. "Foreign currency bubbles," Review of Derivatives Research, Springer, vol. 14(1), pages 67-83, April.
    12. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
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