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On the Hedging of Options On Exploding Exchange Rates

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  • Peter Carr
  • Travis Fisher
  • Johannes Ruf

Abstract

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).

Suggested Citation

  • Peter Carr & Travis Fisher & Johannes Ruf, 2012. "On the Hedging of Options On Exploding Exchange Rates," Papers 1202.6188, arXiv.org, revised Nov 2013.
  • Handle: RePEc:arx:papers:1202.6188
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    File URL: http://arxiv.org/pdf/1202.6188
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    References listed on IDEAS

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    1. Orlin J. Grabbe, "undated". "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.
    2. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    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. 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.
    5. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    6. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    7. 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.
    8. 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.
    9. Orlin J. Grabbe, "undated". "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.
    10. Robert Jarrow & Philip Protter, 2011. "Foreign currency bubbles," Review of Derivatives Research, Springer, vol. 14(1), pages 67-83, April.
    11. 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.
    12. Philipp J. Schönbucher, 2000. "A Libor Market Model with Default Risk," Bonn Econ Discussion Papers bgse15_2001, University of Bonn, Germany.
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    Cited by:

    1. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    2. Erhan Bayraktar & Xiang Yu, 2013. "On the Market Viability under Proportional Transaction Costs," Papers 1312.3917, arXiv.org, revised Jan 2017.
    3. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Working Papers hal-00814702, HAL.
    4. Johannes Ruf, 2012. "Negative Call Prices," Papers 1204.1903, arXiv.org, revised Jan 2013.
    5. Irina Penner & Anthony RĂ©veillac, 2015. "Risk measures for processes and BSDEs," Finance and Stochastics, Springer, vol. 19(1), pages 23-66, January.
    6. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Papers 1304.4853, arXiv.org.

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