Currency option pricing in a credible exchange rate target zone

Abstract

This article examines currency option pricing within a credible target zone arrangement where interventions at the boundaries push the exchange rate back into its fluctuation band. Valuation of such options is complicated by the requirement that the reflection mechanism should prevent the arbitrage opportunities that would arise if the exchange rate were to spend finite time on the boundaries. To prevent the latter, we superimpose instantaneously reflecting boundaries upon the familiar geometric Brownian motion (GBM) framework. We derive closed-form expressions for European call and put option prices and show that prices for the GBM model of Garman and Kohlhagen (1983) arise as the limit case for infinitely wide bands. We also illustrate that taking account of boundaries is of considerable economic value as erroneously using the unbounded-domain model of Garman and Kohlhagen (1983) easily overprices options by more than 100%.

Keywords:

• currency option pricing
• exchange rate target zones
• geometric Brownian motion
• reflecting boundaries
• Brownian motion
• risk-neutral valuation

JEL Classification:

• F31
• G13.

Notes

We are indebted to Hans Dewachter, Henk Jager, Franc Klaassen, Piet Sercu and Koen Vermeylen for stimulating discussions. The usual disclaimer applies.

1 For an interpretation of the reflection functions in terms of local time, we can refer to Ikeda and Watanabe (1981) and Harrison (1985).

2 Alternative reflection mechanisms such as slow reflection (Revuz and Yor, 1994), delayed reflection (Skorokhod, 1961) and reflection at so-called sticky barriers (Karlin and Taylor, 1981) would then clearly not be acceptable for derivative pricing since the speed of return from the boundaries in these mechanisms is finite and thus positive time is spent on them.

3 We will come back to this property when giving a formal justification for the boundary behaviour of the option price.

4 Complete derivations of all results in this article can be obtained from the author upon simple demand.

5 The prices for the unrestricted-domain model of Garman and Kohlhagen (1983) in Table 1 will be used below to discuss the impact of applying the latter model also to target zone exchange rates.

6 As mentioned earlier, underpricing is also possible. However, the required tight range of parameter values of exchange rates and exercise prices that both have to be (very) near to the lower target zone limit makes this possibility of probably limited practical relevance.

7 Veestraeten (2008) reports the call option price when a lower barrier restricts the fluctuation range of the stock price.

8 The risk-neutral valuation Equation 7 in the case of a sole lower boundary can be written as , whereas the price under a sole upper boundary is given by