Effective asymptotic analysis for finance


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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