Pricing and hedging barrier options

European options are a significant financial product. Barrier options, in turn, are European options with a barrier constraint. The investor may pay less buying the barrier option obtaining the same result as that of the European option whenever the barrier is not breached. Otherwise, the option's payoff cancels. In this paper, we obtain closed-form expressions of the exact no-arbitrage prices, delta hedges, and gammas of a call option with a moving barrier that tracks the prices of the risk-free asset. Besides the interest in its own right, this class of options constitutes the core element to obtain, via an original and simple technique, the closed-form expressions for the estimates of the prices of call options with barriers of arbitrary shape. Equally important is the fact that a bound for the worst associated error is provided, so the investor can evaluate beforehand if the accuracy provided is according to his/her needs or not. Discrete monitored barrier provisions are also allowed in the estimates. Simulations are performed illustrating the accuracy of the estimates. A quality of the aforementioned procedures is that the time consumed in computations is very small. In turn, we observe that the approximate prices, delta hedges, and gammas of the barrier option associated to the risk-free asset, obtained via a PDE approach in conjunction with a good finite difference method, converge to the closed-form expressions of the prices, hedges, and gammas of the option. This attests the correctness of the analytical results.