A perturbative formula to price barrier options with time dependent parameters in the Black and Scholes world

In this paper, using a perturbative method, a series expansion of the price of a (put up-and-out) barrier option with time-dependent parameters in the Black and Scholes world is obtained. The first three terms of this series are written explicitly as formulae involving some elementary and nonelementary transcendental functions. The formula obtained has been tested on some examples taken from the financial literature and a sufficient condition for the convergence of the perturbation series expansion is given. The numerical experience shows that, in the cases of practical interest considered, the use of the first two or three terms of the series expansion mentioned above guarantees three or four correct significant digits in the prices computed. Similar formulae can be obtained for other types of barrier options such as step options, step barrier options and so on. The website http://www.econ.univpm.it/recchioni/finance/w3 contains some auxiliary material that helps in understanding this paper and the computer programs needed to evaluate the formula obtained.