It is known that an adaptation of Newton's method allows for the computation of functional inverses of formal power series. We show that it is possible to successfully use a similar algorithm in a fairly general analytical framework. This is well suited for functions that are highly tangent to identity and that can be expanded with respect to asymptotic scales of ‘‘exp-log functions''. We next apply our algorithm to various well-known functions coming from the world of quantitative finance. In particular, we deduce asymptotic expansions for the inverses of the Gaussian and the Black–Scholes functions.