Implied volatility formula of European Power Option Pricing

We derive the implied volatility estimation formula in European power call options pricing, where the payoff functions are in the form of $V=(S^α_T-K)^{+}$ and $V=(S^α_T-K^α)^{+}$ ($α>0$)respectively. Using quadratic Taylor approximations, We develop the computing formula of implied volatility in European power call option and extend the traditional implied volatility formula of Charles J.Corrado, et al (1996) to general power option pricing. And the Monte-Carlo simulations are also given.