Computations of complex options

In view of the good acceptance by practitioners of the Black-Scholes model it is important to investigate the possible extensions of the theory and its value as an approximation for computations in non-standard situations. Besides, this field is typically a source of adaptive identification of parameters, namely the “volatility”. The authors´ purpose is to study the valuation of complex options, to establish the necessary extensions, with respect to the BSM and to investigate the applicability of the BSM as an approximation. The authors study the volatility behaviour in important practical cases, like the valuation of warrants in equity or levered firms. Sharp and applicable approximations are given. Also, the authors solve completely the european option case in a large class of non-constant volatility and give explicit formulas, which have never appeared in the literature before, and constitute a useful extension of the BSM. They are applied to the case of levered firms, with warrants and debts