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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.

**Authors:**

**Keywords: **asymptotic expansion, algorithm, pricing,
Hardy field, exp-log function, Black–Scholes formula