  It is well-known that using delta hedging to hedge financial options is not
feasible in practice. Traders often rely on discrete-time hedging strategies
based on fixed trading times or fixed trading prices (i.e., trades only occur
if the underlying asset's price reaches some predetermined values). Motivated
by this insight and with the aim of obtaining explicit solutions, we consider
the seller of a perpetual American put option who can hedge her portfolio once
until the underlying stock price leaves a certain range of values $(a,b)$. We
determine optimal trading boundaries as functions of the initial stock holding,
and an optimal hedging strategy for a bond/stock portfolio. Optimality here
refers to the variance of the hedging error at the (random) time when the stock
leaves the interval $(a,b)$. Our study leads to analytical expressions for both
the optimal boundaries and the optimal stock holding, which can be evaluated
numerically with no effort.
