Acknowledgements
ZK would like to thank Eyal Neuman for discussions on Brownian motion with reflecting barriers, which prompted him to think about this topic. We would also like to thank Travis Fisher for discussions.
Notes
No potential conflict of interest was reported by the authors.
1 For a literature survey, see, e.g. (Duarte,Andrade and Duarte, Citation2013). For a partial list (with some related literature, including on option pricing), see, e.g. (Andersen,Bollerslev,Diebold and Labys, Citation2001), (Anthony and MacDonald, Citation1998), (Ayuso and Restoy, Citation1996), (Ball and Roma, Citation1994), (Bauer,De Grauwe and Reitz, Citation2009), (Beetsma and Van Der Ploeg, Citation1994), (Bekaert and Gray, Citation1998), (Bertola and Caballero, Citation1992), (Bertola and Svensson, Citation1993), (Black and Scholes, Citation1973), (Bo,Wang and Yang, Citation2011), (Bo,Li,Ren,Wang and Yang, Citation2011), (Campa and Chang, Citation1996), (Carr,Ellis and Gupta, Citation1998), (Carr and Jarrow, Citation1990), (Carr and Linetsky, Citation2000), (Cavaliere, Citation1998), (Chinn, Citation1991), (Cornell, Citation2003), (Christensen,Lando and Miltersen, Citation1998), (De Jong, Citation1994), (De Jong,Drost and Werker, Citation2001), (Delgado,Dumas, Citation1992), (Dominquez and Kenen, Citation1992), (Driffill and Sola, Citation2006), (Duarte,Andrade and Duarte, Citation2010), Dumas et al. (Citation1995a, Citationb), (Edin and Vredin, Citation1993), (Edison,Miller and Williamson, Citation1987), (Flood and Garber, Citation1991), (Flood,Rose and Mathieson, Citation1991), (Garman and Kohlhagen, Citation1983), (Grabbe, Citation1983), (Harrison, Citation1985), (Harrison and Pliska, Citation1981), (Honogan, Citation1998), (Hull and White, Citation1987), (Kempa and Nelles, Citation1999), (Klaster and Knot, Citation2002), (Klein and Lewis, Citation1993), (Koedijk,Stork and de Vries, Citation1998), Krugman (Citation1991, Citation1992), (Lai,Fang and Chang, Citation2008), (Larsen and Sørensen, Citation2007), (Lin, Citation2008), Lindberg and Söderlind (Citation1994a, Citationb), (Lindberg,Söderlind and Svensson, Citation1993), (Linetsky, Citation2005), (Lundbergh and Teräsvirta, Citation2006), (Magnier, Citation1992), McKinnon (Citation1982, Citation1984), Meese and Rose (Citation1990, Citation1991), Merton (Citation1973, Citation1976), (Miller and Weller, Citation1991), (Mizrach, Citation1995), (Obstfeld and Rogoff, Citation1995), (Rangvid and Sørensen, Citation2001), (Rose and Svensson, Citation1995), (Saphores, Citation2005), (Serrat, Citation2000), (Slominski, Citation1994), (Smith,Spencer, Citation1992), (Sutherland, Citation1994), Svensson (Citation1991a, Citationb, Citation1992a, Citationb, Citation1993, Citation1994), (Taylor and Iannizzotto, Citation2001), Torres (Citation2000a, Citationb), (Tronzano,Psaradakis and Sola, Citation2003), (Veestraeten, Citation2008), (Vlaar and Palm, Citation1993), (Ward and Glynn, Citation2003), (Werner, Citation1995), Williamson (Citation1985, Citation1986, Citation1987a, Citationb, Citation1989, Citation2002), (Williamson and Miller, Citation1987), (Yu, Citation2007), (Zhang, Citation1994), (Zhu, Citation1996), and references therein.
2 E.g. the USD/HKD spread trades between 7.75 and 7.85, a band fixed by the Hong Kong Monetary Authority.
5 FX has an analog in equities. Consider a stock with a continuous dividend rate 
. Then the risk-free interest rate is analogous to the domestic short-rate, the dividend rate is analogous to the foreign short-rate, and the stock is analogous to the foreign currency (so the stock price is analogous to the worth of one unit of the foreign currency in terms of the domestic currency).
7 In the log-normal Black–Scholes model the discounted process is given by (Equation11(11) ).
8 We consider time-homogeneous dynamics so the problem is analytically tractable (see below).
11 Thus, Dirichlet or Robin boundary conditions would be inconsistent with being a martingale. See appendix 2 for the transition density and martingales for Robin boundary conditions.
12 Monotonicity is assumed so we can price claims (see below).
13 Otherwise, barring any contrived time dependence, generically will break the band.
15 We assume that U(x) is bounded on , so the spectrum is bounded from below.
19 This is where the monotonicity of f(x) is important.
28 Usually, the numeraire is chosen to be a cash bond, but it can be any tradable instrument.
29 Otherwise, the market would be incomplete and we would not be able to hedge claims.
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