# A Cautionary Note on the Put-Call Parity under an Asset Pricing Model with a Lower Reflecting Barrier

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## Summary

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.

## JEL-Classification

E52 E58 F31 G13 G15## Keyword

Euro/Swiss franc floor hedging put-call parity reflected geometric Brownian motion risk-neutral parity## References

- Bakshi, Gurdip, Charles Cao, and Zhiwu Chen (1997), “Empirical Performance of Alternative Option Pricing Models”,
*Journal of Finance*, 52(5), pp. 2003–2049.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Bergman, Yaacov Z. (1996), “Equilibrium Asset Price Ranges”,
*International Review of Financial Analysis*, 5(3), pp. 161–169.CrossRefGoogle Scholar - Black, Fischer, and Myron Scholes (1973), “The Pricing of Options and Corporate Liabilities”,
*Journal of Political Economy*, 81(3), pp. 637–654.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Broome, Simon (2001), “The Lifetime of a Unilateral Target Zone: Some Extended Results”,
*Journal of International Money and Finance*, 20(3), pp. 419–438.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Castagna, Antonio, and Fabio Mercurio (2005), “Consistent Pricing of FX Options”, Internal Report, Banca IMI.Google Scholar
- Castagna, Antonio and Fabio Mercurio (2007), “The Vanna-Volga Method for Implied Volatilities”,
*Risk Magazine*, pp. 39–44.Google Scholar - 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.Google Scholar - Chen, Shuangshuang (2012), “The Implication of the Exchange Rate Floor in Current Times: The Swiss Experience”,
*University of California Working Paper*, pp. 1–34.Google Scholar - Cottarelli, Carlo and Peter Doyle (1999), “Disinflation in Transition, 1993–97”,
*International Monetary Fund Occasional Paper*, 179, pp. 1–45.Google Scholar - Cox, Alexander M. G., and David G. Hobson (2005), “Local Martingales, Bubbles and Option Prices”,
*Finance and Stochastics*, 9(4), pp. 477–492.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Czech National Bank (2013), “7th Situation Report on Economic and Monetary Developments”, in
*Press Release*(November 7, 2013).Google Scholar - Delbaen, Freddy, and Walter Schachermayer (1994), “A General Version of the Fundamental Theorem of Asset Pricing”,
*Mathematische Annalen*. 300(1), pp. 463–520.CrossRefGoogle Scholar - Delbaen, Freddy, and Walter Schachermayer (1998), “The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes”,
*Mathematische Annalen*, 312(2), pp. 215–250.CrossRefGoogle Scholar - Dhrymes, Phoebus J. (1998),
*Time Series, Unit Roots, and Cointegration*, San Diego: Academic Press.Google Scholar - Dillén, Hans, and Peter Sellin (2003), “Financial Bubbles and Monetary Policy”,
*Sveriges Riksbank Economic Review*, 3, pp. 119–144.Google Scholar - Doob, Joseph L. (1971), “What is a Martingale?”,
*The American Mathematical Monthly*, 78(5), pp. 451–463.CrossRefGoogle Scholar - Dumas, Bernard, and Lars E. O. Svensson (1994), “How Long do Unilateral Target Zones Last?”,
*Journal of International Economics*, 36(3), pp. 467–481.CrossRefGoogle Scholar - Dunis, Christian, and Pierre Lequeux (2001), “The Information Content of Risk Reversals”,
*Derivatives Use, Trading, and Regulation*, 7(2), pp. 98–117.Google Scholar - Ekström, Erik, and Johan Tysk (2009), “Bubbles, Convexity and the Black-Scholes Equation”,
*Annals of Applied Probability*, 19(4), pp. 1369–1384.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Garman, Mark B., and Steven W. Kohlhagen (1983), “Foreign Currency Option Values”,
*Journal of International Money and Finance*, 2(3), pp. 231–237.CrossRefGoogle Scholar - Geman, Hélyette (2015),
*Agricultural Finance: From Crops to Land, Water and Infrastructure*, Chichester: John Wiley & Sons.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Glasserman, Paul (2004),
*Monte Carlo Methods in Financial Engineering*, Heidelberg: Springer.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar
- Harrison, J. Michael (1985),
*Brownian Motion and Stochastic Flow Systems*. reprint 1990 edn., New York: John Wiley & Sons.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar
- Heston, Steven L., Mark Loewenstein, and Gregory A. Willard (2007), “Options and Bubbles”,
*Review of Financial Studies*, 20(2), pp. 359–390.CrossRefGoogle Scholar - Humpage, Owen F., and Javiera Ragnartz (2006), “Swedish Intervention and the Krona Float, 1993–2002”,
*Sveriges Riksbank Working Paper Series*. 192, pp. 1–40.Google Scholar - Imai, Junichi, and Phelim P. Boyle (2001), “Dynamic Fund Protection”,
*North American Actuarial Journal*, 5(3), pp. 31–49; Disscussion pp. 49–51.CrossRefGoogle Scholar - Ingersoll Jr., Jonathan E. (1987),
*Theory of Financial Decision Making*. Totowa: Rowman & Littlefield.Google Scholar - Ingersoll Jr., Jonathan E. (1997), “Valuing Foreign Exchange Rate Derivatives with a Bounded Exchange Process”,
*Review of Derivatives Research*, 1(2), pp. 159–181.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Jarrow, Robert A., and Philip Protter (2011), “Foreign Currency Bubbles”,
*Review of Derivatives Research*, 14(1), pp. 67–83.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Jarrow, Robert A., Philip Protter, and Kazuhiro Shimbo (2010), “Asset Price Bubbles in Incomplete Markets”,
*Mathematical Finance*, 20(2), pp. 145–185.CrossRefGoogle Scholar - Jermann, Urban J. (2015), “Financial Markets’ Views about the Euro-Swiss Franc Floor”, Working Paper, available at SSRN 2490086.Google Scholar
- 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.Google Scholar - 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.CrossRefGoogle Scholar - Longstaff, Francis A. (1995), “Option Pricing and the Martingale Restriction”,
*Review of Financial Studies*, 8(4), pp. 1091–1124.CrossRefGoogle Scholar - Maccioni, Alessandro Fiori (2011), “Endogenous Bubbles in Derivatives Markets: The Risk Neutral Valuation Paradox”, Working Paper, University of Sassari.Google Scholar
- 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.CrossRefGoogle Scholar - Merton, Robert C. (1973), “Theory of Rational Option Pricing”,
*Bell Journal of Economics and Management Science*, 4(1), pp. 141–183.CrossRefGoogle Scholar - Musiela, Marek, and Marek Rutkowski (2009),
*Martingale Methods in Financial Modelling*, Heidelberg: Springer.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - Reiswich, Dimitri, and Uwe Wystup (2010), “A Guide to FX Options Quoting Conventions”,
*Journal of Derivatives*, 18(2), pp. 58–68.CrossRefGoogle Scholar - Ruf, Johannes (2013), “Negative Call Prices”,
*Annals of Finance*, 9(4), pp. 787–794.CrossRefGoogle Scholar - Satchell, Stephen (2007),
*Forecasting Expected Returns in the Financial Markets*. London: Academic Press.Google Scholar - Shiller, Robert J. (2000), “Measuring Bubble Expectations and Investor Confidence”,
*Journal of Psychology and Financial Markets*, 1(1), pp. 49–60.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Skorokhod, Anatoliy V. (1961), “Stochastic Equations for Diffusion Processes in a Bounded Region”,
*Theory of Probability and its Applications*, 6(3), pp. 264–274.CrossRefGoogle Scholar - Stoll, Hans R (1969), “The Relationship Between Put and Call Option Prices”,
*Journal of Finance*, 24(5), pp. 801–824.CrossRefGoogle Scholar - Swiss National Bank (2011), “Nationalbank legt Mindestkurs von 1.20 Franken pro Euro fest”, in
*Press Release (September 6, 2011)*.Google Scholar - Veestraeten, Dirk (2008), “Valuing Stock Options when Prices are Subject to a Lower Boundary”,
*Journal of Futures Markets*, 28(3), pp. 231–247.CrossRefGoogle Scholar - Veestraeten, Dirk (2013), “Currency Option Pricing in a Credible Exchange Rate Target Zone”,
*Applied Financial Economics*, 23(11), pp. 951–962.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Whaley, Robert E. (1993), “Derivatives on Market Volatility: Hedging Tools Long Overdue”,
*Journal of Derivatives*, 1(1), pp. 71–84.CrossRefGoogle Scholar - Wystup, Uwe (2010a), “Foreign Exchange Symmetries”, in
*Encyclopedia of Quantitative Finance*, Rama Cont, ed., vol. 2, pp. 752–759, Chichester: John Wiley & Sons.Google Scholar - Wystup, Uwe (2010b), “Vanna-Volga Pricing”, in
*Encyclopedia of Quantitative Finance*, Rama Cont, ed., vol. 4, pp. 1867–1874, Chichester: John Wiley & Sons.Google Scholar