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A Cautionary Note on the Put-Call Parity under an Asset Pricing Model with a Lower Reflecting Barrier
Swiss Journal of Economics and Statistics volume 151, pages227–260(2015)
The put-call parity is free from distributional assumptions. It is tempting to assume that this parity also holds when an asset pricing model includes reflecting barriers. This paper shows that in the case of geometric Brownian motion with reflection such barriers cause the standard put-call parity to differ from the riskneutral parity. This paper then analyzes the error that arises when the diffusion is bounded and the standard put-call parity is applied in a risk-neutral framework as a shortcut to impute put prices from call prices, and vice versa. The risk-neutral parity that is derived for a reflected geometric Brownian motion is then used to analyze the impact that the Swiss National Bank’s minimum exchange rate regime vis-à-vis the euro has had on foreign exchange hedging costs. The analysis shows that in the analyzed period domestic investors may have incurred substantial costs as a result of hedging exposure to the euro currency and may have been overexposed to foreign exchange risk.
Bakshi, Gurdip, Charles Cao, and Zhiwu Chen (1997), “Empirical Performance of Alternative Option Pricing Models”, Journal of Finance, 52(5), pp. 2003–2049.
Balyeat, R. Brian (2002), “Economic Significance of Risk Premiums in the S&P 500 Option Market”, Journal of Futures Markets, 22(12), pp. 1147–1178.
Bergman, Yaacov Z. (1996), “Equilibrium Asset Price Ranges”, International Review of Financial Analysis, 5(3), pp. 161–169.
Black, Fischer, and Myron Scholes (1973), “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81(3), pp. 637–654.
Bossens, Frédéric, Grégory Rayée, Nikos S. Skantzos, and Griselda Deelstra (2010), “Vanna-Volga Methods Applied to FX Derivatives: From Theory to Market Practice”, International Journal of Theoretical and Applied Finance, 13(8), pp. 1293–1324.
Broome, Simon (2001), “The Lifetime of a Unilateral Target Zone: Some Extended Results”, Journal of International Money and Finance, 20(3), pp. 419–438.
Campa, José M., and P.H. Kevin Chang (1996), “Arbitrage-Based Tests of Target-Zone Credibility: Evidence from ERM Cross-Rate Options”, American Economic Review, 86(4), pp. 726–740.
Campa, José M., P.H. Kevin Chang, and Robert L. Reider (1998), “Implied Exchange Rate Distributions: Evidence from OTC Option Markets”, Journal of International Money and Finance, 17(1), pp. 117–160.
Carr, Peter, Travis Fisher, and Johannes Ruf (2014), “On the Hedging of Options on Exploding Exchange Rates”, Finance and Stochastics, 18(1), pp. 115–144.
Castagna, Antonio, and Fabio Mercurio (2005), “Consistent Pricing of FX Options”, Internal Report, Banca IMI.
Castagna, Antonio and Fabio Mercurio (2007), “The Vanna-Volga Method for Implied Volatilities”, Risk Magazine, pp. 39–44.
Chaboud, Allen P., and Owen F. Humpage (2005), “An Assessment of the Impact of Japanese Foreign Exchange Intervention: 1991–2004”, International Finance Discussion Papers, 824, pp. 1–41.
Chen, Shuangshuang (2012), “The Implication of the Exchange Rate Floor in Current Times: The Swiss Experience”, University of California Working Paper, pp. 1–34.
Cottarelli, Carlo and Peter Doyle (1999), “Disinflation in Transition, 1993–97”, International Monetary Fund Occasional Paper, 179, pp. 1–45.
Cox, Alexander M. G., and David G. Hobson (2005), “Local Martingales, Bubbles and Option Prices”, Finance and Stochastics, 9(4), pp. 477–492.
Cox, John C., and Stephen A. Ross (1976), “The Valuation of Options for Alternative Stochastic Processes”, Journal of Financial Economics, 3(1), pp. 145–166.
Czech National Bank (2013), “7th Situation Report on Economic and Monetary Developments”, in Press Release (November 7, 2013).
Delbaen, Freddy, and Walter Schachermayer (1994), “A General Version of the Fundamental Theorem of Asset Pricing”, Mathematische Annalen. 300(1), pp. 463–520.
Delbaen, Freddy, and Walter Schachermayer (1998), “The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes”, Mathematische Annalen, 312(2), pp. 215–250.
Dhrymes, Phoebus J. (1998), Time Series, Unit Roots, and Cointegration, San Diego: Academic Press.
Dillén, Hans, and Peter Sellin (2003), “Financial Bubbles and Monetary Policy”, Sveriges Riksbank Economic Review, 3, pp. 119–144.
Doob, Joseph L. (1971), “What is a Martingale?”, The American Mathematical Monthly, 78(5), pp. 451–463.
Dumas, Bernard, and Lars E. O. Svensson (1994), “How Long do Unilateral Target Zones Last?”, Journal of International Economics, 36(3), pp. 467–481.
Dunis, Christian, and Pierre Lequeux (2001), “The Information Content of Risk Reversals”, Derivatives Use, Trading, and Regulation, 7(2), pp. 98–117.
Ekström, Erik, and Johan Tysk (2009), “Bubbles, Convexity and the Black-Scholes Equation”, Annals of Applied Probability, 19(4), pp. 1369–1384.
Elworthy, K. David, Xue-Mei Li, and Marc Yor (1999), “The Importance of Strictly Local Martingales; Applications to Radial Ornstein-Uhlenbeck Processes”, Probability Theory and Related Fields, 115(3), pp. 325–355.
Garman, Mark B., and Steven W. Kohlhagen (1983), “Foreign Currency Option Values”, Journal of International Money and Finance, 2(3), pp. 231–237.
Geman, Hélyette (2015), Agricultural Finance: From Crops to Land, Water and Infrastructure, Chichester: John Wiley & Sons.
Gerber, Hans U. and Gérard Pafumi (2000), “Pricing Dynamic Investment Fund Protection”, North American Actuarial Journal, 4(2), pp. 28–37; Discussion pp. 37–41.
Glasserman, Paul (2004), Monte Carlo Methods in Financial Engineering, Heidelberg: Springer.
Grabbe, J. Orlin (1983), “The Pricing of Call and Put Options on Foreign Exchange”, Journal of International Money and Finance, 2(3), pp. 239–253.
Hanke, Michael, Rolf Poulsen, and Alex Weissensteiner (2015), “Where Would the EUR/CHF Exchange Rate be Without the SNB’s Minimum Exchange Rate Policy?”, Working Paper, forthcoming in the Journal of Futures Markets.
Harrison, J. Michael (1985), Brownian Motion and Stochastic Flow Systems. reprint 1990 edn., New York: John Wiley & Sons.
Hertrich, Markus, and Dirk Veestraeten (2013), “Valuing Stock Options when Prices are Subject to a Lower Boundary: A Correction”, Journal of Futures Markets, 33(9), pp. 889–890.
Hertrich, Markus, and Heinz Zimmermann (2015), “On the Credibility of the Euro/Swiss Franc Floor: A Financial Market Perspective”, Working Paper, available at SSRN 2290997.
Heston, Steven L., Mark Loewenstein, and Gregory A. Willard (2007), “Options and Bubbles”, Review of Financial Studies, 20(2), pp. 359–390.
Humpage, Owen F., and Javiera Ragnartz (2006), “Swedish Intervention and the Krona Float, 1993–2002”, Sveriges Riksbank Working Paper Series. 192, pp. 1–40.
Imai, Junichi, and Phelim P. Boyle (2001), “Dynamic Fund Protection”, North American Actuarial Journal, 5(3), pp. 31–49; Disscussion pp. 49–51.
Ingersoll Jr., Jonathan E. (1987), Theory of Financial Decision Making. Totowa: Rowman & Littlefield.
Ingersoll Jr., Jonathan E. (1997), “Valuing Foreign Exchange Rate Derivatives with a Bounded Exchange Process”, Review of Derivatives Research, 1(2), pp. 159–181.
Jarrow, Robert A., and Philip Protter (2007), “An Introduction to Financial Asset Pricing”, in Handbooks in Operations Research and Management Science. vol. 15, chap. 1, pp. 13–69, Amsterdam: North-Holland.
Jarrow, Robert A., and Philip Protter (2011), “Foreign Currency Bubbles”, Review of Derivatives Research, 14(1), pp. 67–83.
Jarrow, Robert A., Philip Protter, and Kazuhiro Shimbo (2007), “Asset Price Bubbles in Complete Markets”, in Advances in Mathematical Finance. Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, eds., pp. 97–121, Basel: Birkhäuser Verlag.
Jarrow, Robert A., Philip Protter, and Kazuhiro Shimbo (2010), “Asset Price Bubbles in Incomplete Markets”, Mathematical Finance, 20(2), pp. 145–185.
Jermann, Urban J. (2015), “Financial Markets’ Views about the Euro-Swiss Franc Floor”, Working Paper, available at SSRN 2490086.
Ko, Bangwon, Elias S.W. Shiu, and Li Wei (2010), “Pricing Maturity Guarantee with Dynamic Withdrawal Benefit”, Insurance: Mathematics and Economics. 47(2), pp. 216–223.
Lamont, Owen A., and Richard H. Thaler (2003), “Can the Market Add and Subtract? Mispricing in Tech Stock Carve-outs”, Journal of Political Economy. 111(2), pp. 227–268.
Longstaff, Francis A. (1995), “Option Pricing and the Martingale Restriction”, Review of Financial Studies, 8(4), pp. 1091–1124.
Maccioni, Alessandro Fiori (2011), “Endogenous Bubbles in Derivatives Markets: The Risk Neutral Valuation Paradox”, Working Paper, University of Sassari.
Madan, Dilip B., and Marc Yor (2006), “Ito’s Integrated Formula for Strict Local Martingales”, in In Memoriam Paul-André Meyer, pp. 157–170, Heidelberg: Springer.
Merton, Robert C. (1973), “Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science, 4(1), pp. 141–183.
Musiela, Marek, and Marek Rutkowski (2009), Martingale Methods in Financial Modelling, Heidelberg: Springer.
Ofek, Eli, Matthew Richardson, and Robert F. Whitelaw (2004), “Limited Arbitrage and Short Sales Restrictions: Evidence from the Options Markets”, Journal of Financial Economics, 74(2), pp. 305–342.
Pennacchi, George (2008), Theory of Asset Pricing, Boston: Pearson Education.
Protter, Philip (2013), “A Mathematical Theory of Financial Bubbles”, in Paris-Princeton Lectures on Mathematical Finance 2013, Vicky Henderson and Ronnie Sircar, eds., vol. 2081 of Lecture Notes in Mathematics, pp. 1–108. Heidelberg: Springer.
Reiswich, Dimitri, and Uwe Wystup (2010), “A Guide to FX Options Quoting Conventions”, Journal of Derivatives, 18(2), pp. 58–68.
Ruf, Johannes (2013), “Negative Call Prices”, Annals of Finance, 9(4), pp. 787–794.
Satchell, Stephen (2007), Forecasting Expected Returns in the Financial Markets. London: Academic Press.
Shiller, Robert J. (2000), “Measuring Bubble Expectations and Investor Confidence”, Journal of Psychology and Financial Markets, 1(1), pp. 49–60.
Shonkwiler, J. Scott, and Gangadharrao S. Maddala (1985), “Modeling Expectations of Bounded Prices: An Application to the Market for Corn”, Review of Economics and Statistics, 67(4), pp. 697–702.
Skorokhod, Anatoliy V. (1961), “Stochastic Equations for Diffusion Processes in a Bounded Region”, Theory of Probability and its Applications, 6(3), pp. 264–274.
Stoll, Hans R (1969), “The Relationship Between Put and Call Option Prices”, Journal of Finance, 24(5), pp. 801–824.
Swiss National Bank (2011), “Nationalbank legt Mindestkurs von 1.20 Franken pro Euro fest”, in Press Release (September 6, 2011).
Veestraeten, Dirk (2008), “Valuing Stock Options when Prices are Subject to a Lower Boundary”, Journal of Futures Markets, 28(3), pp. 231–247.
Veestraeten, Dirk (2013), “Currency Option Pricing in a Credible Exchange Rate Target Zone”, Applied Financial Economics, 23(11), pp. 951–962.
Wang, Zhiguang, and Robert T. Daigler (2011), “The Performance of VIX Option Pricing Models: Empirical Evidence Beyond Simulation”, Journal of Futures Markets, 31(3), pp. 251–281.
Whaley, Robert E. (1993), “Derivatives on Market Volatility: Hedging Tools Long Overdue”, Journal of Derivatives, 1(1), pp. 71–84.
Wystup, Uwe (2010a), “Foreign Exchange Symmetries”, in Encyclopedia of Quantitative Finance, Rama Cont, ed., vol. 2, pp. 752–759, Chichester: John Wiley & Sons.
Wystup, Uwe (2010b), “Vanna-Volga Pricing”, in Encyclopedia of Quantitative Finance, Rama Cont, ed., vol. 4, pp. 1867–1874, Chichester: John Wiley & Sons.
I would like to thank Christian Kleiber, Klaus Neusser, Philip Protter, Dirk Veestraeten, Heinz Zimmermann and three anonymous referees for helpful comments and suggestions that improved the initial version. All remaining errors and omissions remain my responsibility. This work is dedicated to Alma Linnéa.
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Hertrich, M. A Cautionary Note on the Put-Call Parity under an Asset Pricing Model with a Lower Reflecting Barrier. Swiss J Economics Statistics 151, 227–260 (2015). https://doi.org/10.1007/BF03399417
- Euro/Swiss franc floor
- put-call parity
- reflected geometric Brownian motion
- risk-neutral parity