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: Cyril Grunspan, Joris van der Hoeven

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

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