We study the continuous-time problem of hedging a generalized call option of the European and of the American type, in the presence of transaction costs. We show that if the price process of the relevant stock fluctuates with positive probability, then the only hedge that is possible for the American option is the trivial one. If the price of the stock, in addition to fluctuating with positive probability, is also stable with positive probability, then the same is true for the European option. We also show that in some sense, stable price with positive probability is a necessary condition for having only a trivial hedge for the European option. Our basic idea is to work with an appropriate discrete-time version of the problem which is transaction costs free. The mathematical tools that we use are elementary. A related result appears in Soner, Shreve and Cvitanic.
"On the possibility of hedging options in the presence of transaction costs." Ann. Appl. Probab. 7 (2) 410 - 443, May 1997. https://doi.org/10.1214/aoap/1034625338