The paper analyzes a family of dynamic trading strategies which do not rely on any stochastic process assumptions (aside from continuity and positivity) and in particular do not require predicting future volatilities. Derivative payoffs can still be replicated, except that this occurs at the stopping time at which the “realized cumulative squared volatility” hits a predetermined level. The application of these results to portfolio insurance is emphasized, and hedging strategies studied by Black and Jones and by Brennan and Schwartz are generalized. Classical results on European-style options arise as special cases. For example, the initial cost of replicating a call or a put under the new method is given by a generalized Black-Scholes formula, which yields the ordinary Black-Scholes formula when the volatility is derterministic.