Introduction
The results reported in this paper were motivated by the problem of pricing and hedging a particular exotic option, a call which knocks out in the money. More specifically, we assume a geometric Brownian motion model dS(t) = (rd − rf )S(t) dt + S(t)dW(t), S(0) > 0, (1) for the exchange rate. The domestic interest rate rd 2 R, the foreign interest rate rf 2 R, the volatility  > 0 and the planning horizon T > 0 are assumed to be constant. The process (W(t); 0  t  T) is a Brownian motion under a probability measure P which is risk-neutral, i.e., is chosen so that the foreign currency has mean rate of return r , rd − rf . In an equity model, where S denotes the stock price, we can think of rf as a continuously paid dividend rate.
Consider an up-and-out call whose payoff is (S(T) − K)+I{max0tT S(t) B, then the dangerous region of large negative delta and gamma can be moved above B, and the option will knock out before the exchange rate reaches this region. Of course, the computed price of the option increases with increasing B0, and there is no clear procedure for choosing an appropriate value for B0.
The risk of loss and the hedging problem of barrier options have been recognized by trading practitioners as well as academics. There are various ways to limit this risk and this hedging difficulty as for example
(i) Include rebates, see Section 4.2.2.
(ii) Modify the knock-out regulation as follows. The final payoff loses its value at a rate