This paper extends the idea of Lindstedt s frequencies expansion into the non-linear autonomous sys-tems with many degrees of freedom and describes a multiple-frequency perturbation approach of the phase-locked periodic solutions for the case of the internal resonance by way of the non-linear trasformation ofthe independent variable.

In the Black-Scholes framework, according to the risk-neutral valuation principal, we completely decompose a zero-coupon callable convertible bond with hard call period into four simple parts, and derive the analytic formula of each of these four parts.