Currency Barrier Option Pricing with Mean Reversion

Currency option traders usually use the Black-Scholes model in which the exchange rate follows a lognormal process. However, it is found that exchange rates may follow a mean-reverting process instead, for example, certain currencies are constrained to move inside target zones or under a managed-floating regime. Different dynamical processes of exchange rates raise uncertainty on the choice of a pricing model for currency options. Such model risk would worsen the market condition when there is an adverse shock on the underlying currency. Financial instability could thus result, if pricing models are not chosen and used properly in the foreign exchange market. Barrier options have emerged as significant products for hedging and investment in the foreign exchange market since the late 1980s, largely in the over-the-counter markets and for structuring financial products (e.g., currency-linked notes). The existence of a barrier option depends upon whether the underlying exchange rate has crossed a predetermined barrier prior to the exercise time. The estimated daily turnover of currency barrier option trading is about US$12 billion. This paper develops a barrier-option pricing model in which the exchange rate follows a mean-reverting lognormal process. The corresponding closed-form solutions for the barrier options with time-dependent barriers are derived. The mean-reverting lognormal process keeps the exchange rate in a range around the mean level. The numerical results show that the parameters of the mean-reverting lognormal process make the valuation of currency barrier options and their hedge parameters different from those obtained from the conventional Black-Scholes model.